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Jun 17, 2010, 9:55:51 AM6/17/10

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Actually, now that you mention it, won't we have to change all the axioms to include "line or arc or circle"? This reminds me of inversions through circle, inversive geometry: http://www.cut-the-knot.org/Curriculum/Geometry/InversionDemo.shtml Except folding paper circles is not inversive, being of this physical world. It would be interesting to extend, though.

Original axioms:

1. same

2. same

3. Given two lines OR TWO ARCS l1 and l2, there is a fold that places l1 onto l2.

4. Given a point p1 and a line OR THE CIRCLE l1, there is a unique fold perpendicular to l1 that passes through point p1.

I don't know what to do with the rest - please help:

5. Given two points p1 and p2 and a line l1, there is a fold that places p1 onto l1 and passes through p2.

6. Given two points p1 and p2 and two lines l1 and l2, there is a fold that places p1 onto l1 and p2 onto l2.

7. Given one point p and two lines l1 and l2, there is a fold that places p onto l1 and is perpendicular to l2.

Cheers,

Maria Droujkova

http://www.naturalmath.com

Make math your own, to make your own math.

Original axioms:

- Given two points
*p*_{1}and*p*_{2}, there is a unique fold that passes through both of them. - Given two points
*p*_{1}and*p*_{2}, there is a unique fold that places*p*_{1}onto*p*_{2}. - Given two lines
*l*_{1}and*l*_{2}, there is a fold that places*l*_{1}onto*l*_{2}. - Given a point
*p*_{1}and a line*l*_{1}, there is a unique fold perpendicular to*l*_{1}that passes through point*p*_{1}. - Given two points
*p*_{1}and*p*_{2}and a line*l*_{1}, there is a fold that places*p*_{1}onto*l*_{1}and passes through*p*_{2}. - Given two points
*p*_{1}and*p*_{2}and two lines*l*_{1}and*l*_{2}, there is a fold that places*p*_{1}onto*l*_{1}and*p*_{2}onto*l*_{2}. - Given one point
*p*and two lines*l*_{1}and*l*_{2}, there is a fold that places*p*onto*l*_{1}and is perpendicular to*l*_{2}.

1. same

2. same

3. Given two lines OR TWO ARCS l1 and l2, there is a fold that places l1 onto l2.

4. Given a point p1 and a line OR THE CIRCLE l1, there is a unique fold perpendicular to l1 that passes through point p1.

I don't know what to do with the rest - please help:

5. Given two points p1 and p2 and a line l1, there is a fold that places p1 onto l1 and passes through p2.

6. Given two points p1 and p2 and two lines l1 and l2, there is a fold that places p1 onto l1 and p2 onto l2.

7. Given one point p and two lines l1 and l2, there is a fold that places p onto l1 and is perpendicular to l2.

Cheers,

Maria Droujkova

http://www.naturalmath.com

Make math your own, to make your own math.

On Thu, Jun 17, 2010 at 8:41 AM, Alexander Bogomolny <abo...@gmail.com> wrote:

Perhaps, one more axiom is needed. That you can fold the circle to a given point.

On Thu, Jun 17, 2010 at 6:41 AM, Maria Droujkova <drou...@gmail.com> wrote:

--I tried to make circle-folding axioms, similar to Huzita-Hatori axioms for origami. To the best of my understanding, the seven origami axioms work in their entirety for circles, and there's no need to make more. I wonder if some can be removed because of circle properties, but I could not find any to remove, either. This leads me to believe folding circles isn't all that different from folding squares. I'd like to pinpoint differences Brad mentions somehow, but I was unable to do it with axioms, so far.

Finding centers of circles (and thus radii) is quite useful, and easy: you just fold the circle twice. That is the first step in folding tetrahedrons. We were doing it the other day to make a 3d Sierpinski:

Brad, this is an easier construction that what you recommend here, because trisection (while allowed by folding axioms) is hard on humans: http://www.wholemovement.com/index.php?option=com_content&view=article&id=51&Itemid=43

- Find the center (fold the circle into quarters)
- Fold a segment by making circumference touch the center
- Fold another segment, with end point at an end of the first one, and circumference touching the center.
- Fold the third segment with end points formed by the first two, finishing an equilateral triangle
- ...
- Profit

The above construction only requires folding in halves, not in thirds - because it uses the center.

Cheers,

Maria Droujkova

http://www.naturalmath.com

Make math your own, to make your own math.

On Thu, Jun 17, 2010 at 7:27 AM, Bradford Hansen-Smith <wholem...@sbcglobal.net> wrote:Alex,

>Your circle, i.e., the one you paperfold, has a radius. It may not be any crease or line drawn on paper, >but a radius a circle has.Okay, it has a radius because we give it a radius, even when one is not there. The radius of a circle is a function of measurement starting with the center point which is what we do when we draw the image using a compass. Using Euclid 's definitions to define a circle we are calling a circle by how a compass works. The circle we draw is an image and not the circle at all, not any more than the picture of a tree is the reality of a tree. My experience finds the circle so much more than image or definition of construction.

>So folding a circle as you have appeared to suggest is not the way to even illustrate the Pythagorean >theorem.

Folding circles and drawing pictures of circle are different systems. Confusion comes about by trying to fit folding circles to the generalized illustrations used to picture concepts. In attending to my experience brings conflict to what I am told can and cannot be done as it is thus written, leads me to question what can not be done and deeper into my own observations.

To say there is a center point and at the same time acknowledge relative scale as the concentric nature of the circle creates conflict where circles move both into and out from themselves. No one smallest circle can be identified as the center any more than the one largest circle can be identified, except by the limitations of the tools we use. In this regard I see the circle as self-referencing, a self-centering movement by nature of what it is. The circle is itself center, so where exactly is the radius? Words represent useful concepts which always indicate a greater reality.

.

My experiences bring in question some of the constructed logic of mathematics. It is not a negation or putting down what has been developed by people with a great passion for mathematics, rather an effort to understand what is missing that makes mathematics a problem in how we teach it; which reflects how we understand it. I wonder what is missing from the construction, the abstracted short cuts, that makes it difficult for many students at the lower grade levels, which I include myself, to "get it".

Brad

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Jun 17, 2010, 10:57:19 AM6/17/10

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Actually, now that you mention it, won't we have to change all the axioms to include "line or arc or circle"?

You can't replace "lines' by "arcs" (or their parts) in #3 and #6

In #3, consider two arcs that are translates of each other in the direction of the radius at the midpoint

In #6, consider two close circles and one point near by, the other far away.

Jun 17, 2010, 1:52:54 PM6/17/10

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Were I to set about making a set of axioms for folding the circle it would be as follows:

1. Given two points,*p1* and *p2 *anywhere on the circumference, when touched together generate a unique fold perpendicular to and dividing the distance between p1 and p2, and the circle, in half.

This fold is a semi-spherical pattern of movement. Given movement goes in both directions, it is a spherical pattern of movement reflecting circle origin and is a minimum description of a dual tetrahedron pattern. This is unique to folding the circle.

2. All other axioms pertaining to the squares also follows for the circle, given the square is a truncated circle.

Brad

Bradford Hansen-Smith

Wholemovement

4606 N. Elston #3

Chicago Il 60630

www.wholemovement.com

**From:** Maria Droujkova <drou...@gmail.com>

**To:** mathf...@googlegroups.com

**Sent:** Thu, June 17, 2010 8:55:51 AM

**Subject:** [Math 2.0]
Circle-folding axioms

1. Given two points,

This fold is a semi-spherical pattern of movement. Given movement goes in both directions, it is a spherical pattern of movement reflecting circle origin and is a minimum description of a dual tetrahedron pattern. This is unique to folding the circle.

2. All other axioms pertaining to the squares also follows for the circle, given the square is a truncated circle.

Brad

Wholemovement

4606 N. Elston #3

Chicago Il 60630

www.wholemovement.com

Jun 23, 2010, 6:22:13 PM6/23/10

to mathf...@googlegroups.com, alicet...@gmail.com, Colin McAllister

I made a list of circle origami axioms, using comments by Alex and Brad on my first version. I have not started to work on proving this is a complete set, yet. My daughter Katherine and I made equivalent GeoGebra constructions (based on Euclid's axioms) for all circle axioms except 1-6-C, which is a third-degree equation, Wikipedia says. We are working on this and can use help.

The complete axioms page is here: http://mathfuture.wikispaces.com/Circle+origami+axioms

Can you please check it for making sense? I want to submit this as an article, including everybody who helps (Brad and Alex, so far) as co-authors. I will also send it to "square origami" people to get some feedback.

Circle origami adds these five axioms to the Huzita-Hatori list. I numbered them according to the Huzita-Hatori axioms they supplant.

### Axiom 2 - 3 - C

Given two points*p*1 and *p*2 on the circle, there is a
unique fold that places *p*1 onto *p*2 and places the
half-circles *l*1 and *l*2 defined by the fold onto one
another.

This is equivalent to finding a bisector of the angle between radii to*p1*
and *p2*.

### Axiom 4-C

Given a point *p*1, there is a unique fold that passes through
point *p*1 and places the half-circles *l*1 and *l*2
defined by the fold onto one another.

This is equivalent to finding a diameter through*p*1.

### Axiom 5-C

Given two points*p*1 and *p*2, there is a fold that
places *p*1 onto the circle and passes through *p*2. If
the distance between *p1* and *p2* is greater than the
distance between *p2* and the circle, there are two such folds,
if the distances are equal, one such fold. If the distance between *p1*
and *p2* is smaller than the distance between *p2* and
the circle, the fold is impossible.

The two-solution situation is shown above.

### Axiom 1-6-C

Given two points *p*1 and *p*2, of which at least one is
not on the circle, there are two and only two folds that place both
points on the circle.

**Need a drawing (Euclid's
construction) here.**

### Axiom 7-C

Given one point *p* and a line *l*1, there is a fold that
places *p* onto the circle and is perpendicular to *l*1.
There is one such fold if *p* lies on the diameter perpendicular
to *l1*, and two such folds otherwise.

This is equivalent to finding midpoints of segments between*p*
and the intersections of the circle with the line parallel to *l1*
and passing through *p*.

The complete axioms page is here: http://mathfuture.wikispaces.com/Circle+origami+axioms

Can you please check it for making sense? I want to submit this as an article, including everybody who helps (Brad and Alex, so far) as co-authors. I will also send it to "square origami" people to get some feedback.

Circle origami adds these five axioms to the Huzita-Hatori list. I numbered them according to the Huzita-Hatori axioms they supplant.

Given two points

This is equivalent to finding a bisector of the angle between radii to

This is equivalent to finding a diameter through

Given two points

The two-solution situation is shown above.

This is equivalent to finding midpoints of segments between

Jun 24, 2010, 1:02:54 AM6/24/10

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Hi Maria,

I am interested in your efforts to come up with axioms for folding circles. While you are trying to give mathematical expression to folding, you must realize that folding circles has little to do with Euclid's axioms based on the construction of the circle image of points and lines. The inverse function of points and lines do not necessarily hold up for the circle since there is no way of knowing where the lines are until the points touch and there is a crease. While there are generalizations to be made about the folds, the circle is very specific and can not be disregarded for any or no shape context You can not assume a radius or center when it is not shown, that is a constructionist perspective. With the circle the lines are a direct result of the touching of points, whether they are parallel or angled. As the axioms stand they are confuse in trying to understand folding the circle. Origami is construction using a set of a specific sequence of predetermined folds where as this is not the case with the circle.

I am out of town until next week and can not take the time to do more than give a quick response. It is not clear to me what you are describing.

I will give this some thought and get back to you upon return.

I am interested in your efforts to come up with axioms for folding circles. While you are trying to give mathematical expression to folding, you must realize that folding circles has little to do with Euclid's axioms based on the construction of the circle image of points and lines. The inverse function of points and lines do not necessarily hold up for the circle since there is no way of knowing where the lines are until the points touch and there is a crease. While there are generalizations to be made about the folds, the circle is very specific and can not be disregarded for any or no shape context You can not assume a radius or center when it is not shown, that is a constructionist perspective. With the circle the lines are a direct result of the touching of points, whether they are parallel or angled. As the axioms stand they are confuse in trying to understand folding the circle. Origami is construction using a set of a specific sequence of predetermined folds where as this is not the case with the circle.

I am out of town until next week and can not take the time to do more than give a quick response. It is not clear to me what you are describing.

I will give this some thought and get back to you upon return.

Jun 24, 2010, 12:17:38 AM6/24/10

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Sol Lederman pointed out that, although there was a lot of info on the wiki page about Calc Nation, Illuminations, and me, there wasn't really anything about what content would be covered during the webinar. So, I just went in and added a paragraph at the top that gives a general overview.

Just wanted you to be aware of why I made the change.

Also, just to make sure. The page says we'll start at 9:30pm ET, but I had written 9:00pm ET in my notes. Please confirm so I make sure to show up at the correct time.

Thanks,

Patrick

Just wanted you to be aware of why I made the change.

Also, just to make sure. The page says we'll start at 9:30pm ET, but I had written 9:00pm ET in my notes. Please confirm so I make sure to show up at the correct time.

Thanks,

Patrick

Jun 24, 2010, 6:57:59 AM6/24/10

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Patrick,

Thank you for the thoughtful edits. It really helps people to see what is going on! http://mathfuture.wikispaces.com/Calculation+Nation I will try to have short descriptions like you included at the top for everybody from now on.

There are two times listed because the virtual room opens at 9pm, and some people come earlier to test the software and hardware, or to meet and chat. We start the recording and the main conversation at 9:30pm, though. Usually hosts come some ten minutes before that time.

Talk to you soon!

Thank you for the thoughtful edits. It really helps people to see what is going on! http://mathfuture.wikispaces.com/Calculation+Nation I will try to have short descriptions like you included at the top for everybody from now on.

There are two times listed because the virtual room opens at 9pm, and some people come earlier to test the software and hardware, or to meet and chat. We start the recording and the main conversation at 9:30pm, though. Usually hosts come some ten minutes before that time.

Talk to you soon!

Jun 24, 2010, 7:28:10 AM6/24/10

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On Thu, Jun 24, 2010 at 1:02 AM, Bradford Hansen-Smith <wholem...@sbcglobal.net> wrote:

Hi Maria,

I am interested in your efforts to come up with axioms for folding circles. While you are trying to give mathematical expression to folding, you must realize that folding circles has little to do with Euclid's axioms based on the construction of the circle image of points and lines. The inverse function of points and lines do not necessarily hold up for the circle since there is no way of knowing where the lines are until the points touch and there is a crease. While there are generalizations to be made about the folds, the circle is very specific and can not be disregarded for any or no shape context You can not assume a radius or center when it is not shown, that is a constructionist perspective. With the circle the lines are a direct result of the touching of points, whether they are parallel or angled. As the axioms stand they are confuse in trying to understand folding the circle. Origami is construction using a set of a specific sequence of predetermined folds where as this is not the case with the circle.

I am out of town until next week and can not take the time to do more than give a quick response. It is not clear to me what you are describing.

I will give this some thought and get back to you upon return.

Brad

The fold themselves, as described, are not "based" on Euclid's axioms. However, being another full set of construction axioms, Euclid's set provides means of constructing any fold, say, in GeoGebra - or using compass and straightedge. I find the descriptions of correspondences valuable.

What we can do to underlie the independence: make little (5-10 second) videos of every fold, showing how it's done, instead of current illustrations. Then, have the current illustrations listed separately, as a list of correspondences.

Two more people commented on list. Here are the comments, which will go into the next version, as well:

~*~*~*~*~*

Colin McAlister:

There is one exception to 4-C; when p1 at the center of the circle.

For points not at the center, there is only one diameter, so you would say "This is equivalent to finding

5c

This sentence is false:

"Given two points

as
there are two cases where is is not true. (i) distance between *p1*
and *p2* is smaller than the distance between *p2* and
the circle. (ii) when there are two folds, as "a fold" could be casually
read as meaning only one fold.

So suggest that you rewrite 5c. It would be more accurate to say:

"Given two points*p*1 and *p*2,
**consider the possibility of** a fold (or folds) through p2 that
place(s) *p*1 onto the circle."

~*~*~*~*~*

The video Linda linked assumes the use of right angles of the paper to construct the center of a circle. The seven Huzita-Hatori origami axioms don't use the edge of the paper at all. Circle-folding axioms we are working on use the fact we know the edge is a circle. Will a set of origami axioms using right angles of the regular origami paper be different from Huzita-Hatori ones, or derivative?

Maria D

So suggest that you rewrite 5c. It would be more accurate to say:

"Given two points

In 5C, you have omitted to show two arrowed lines that show the
direction of the fold. I must presume that you use the dashed lines to
indicate folds, as you did in the previous diagram. It is not clear if
you are using them to indicate folds, or simply lines through pairs of
points. The line in the 1 o'clock direction is a solution fold, but the
line in the 5 o'clock is not. I think the second solution would be
folded along a line near the four o'clock direction.

In 7-C

"if p lies on the diameter perpendicular to l1" there are two folds
and not one, as you state. Name the line l instead of l1 as there is no
l2 in the diagram. "l" looks like "i" or a slash (/) if italicised. I
suggest handscript l and p or or bold l and p for lines and points.
Perhaps you could use "f" to indicate a fold, as there are other lines
on your diagrams, such as radii and lines through pairs of points.

I suggest generalising the media beyond circles. 5-C applies to
shapes other than circles. What family of shapes does the axiom define?
Does it restrict radius of curvature of arcs of the shape? Would a
starfish shape be excluded?

In 7-C, you could consider stadium shapes, a rectangle bounded on
each end by a semicircle. Could you consider any irregular shaped piece
of paper, with one perimeter and no holes?

A soccer ball can be folded along a circumference against itself,
by letting all of the air out. The surface of a sphere has an
interesting difference from a circle. It has no centre; every point is
equivalent. How does axiom 4-C differ for this situation?

~*~*~*~*~*

LFS:

~*~*~*~*~*

LFS:

Marija – I know this has nothing to do with your axioms, but it is fun about finding the center of a circle drawn on a piece of paper by folding the paper.

http://www.youtube.com/watch?v=7kMFjXtAWAY

Linda

I found it when I happened upon the inverse of Thales theorem.: http://geogebrawiki.wikispaces.com/Thales+Inverse~*~*~*~*~*

The video Linda linked assumes the use of right angles of the paper to construct the center of a circle. The seven Huzita-Hatori origami axioms don't use the edge of the paper at all. Circle-folding axioms we are working on use the fact we know the edge is a circle. Will a set of origami axioms using right angles of the regular origami paper be different from Huzita-Hatori ones, or derivative?

Maria D

Jun 24, 2010, 9:43:05 AM6/24/10

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Maria, hello:

It's all very interesting and I wish I could see how you actually worked with your daughter.

I shall be well satisfied if you acknoweldge my help/advice/whatever without claiming the authorship.

This said, here's my two cents concerning the axioms:

The existence of the fold follows from Hurita's 2. The essence of 2-3-C is that this fold cuts the circle into two. This raises a question: do we know in advance whether the piece we are folding is a circle in a conventional sense or it deserves to emerge as such through the axioms. In the former case, 2-3-C is not needed at all. The latter case makes a rather delicate point. Free usage of the words "circle" and "semicircle" is viciously circular (pun intended.)

This is problematic. As Colin observed, the center of the circle requires special attention. However, as Brad would want it, the center of the piece should not be mentioned explicitly until identified somehow by folding.

I suggest that the point should be picked on the "circle": For any point p on the boundary of the piece there is a fold that maps an arc of the boundary on its complement.

5-C

Needs correction as per Colin.

Hurita-6 is indeed of order 3. For example, it allows for angle trisection, see

Alex

Jun 25, 2010, 11:30:19 PM6/25/10

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Maria, hi:

I do think 3-2-C is unnecessary and 4-C could be reduced to handle only points on the border. This actually is a basis for proving that the folding piece is indeed a circle. I am not sure that even that is necessary. When you work with a square, the sides of the square are assumed to be lines with the same rights as any other crease. By analogy, you know in advance that what you fold is a circle.

Nonetheless, let 4-C-1 mean the restriction of 4-C to points on the circle. I am going to prove 4-C from 4-C-1.

Let P be a point inside a circle. Let AB, with A and B on the circle be any crease through P. From 4-C-1, there exists A' and B' such that AA' and BB' are two diameters of the circle. If A' = B then also B' = A and P is the center of the circle. Then AB is a diameter through P.

If A' != B, then AA' and BB' define the center O of the circle. Then OP is the required diameter.

All the best,

Alex

Jul 4, 2010, 10:01:02 AM7/4/10

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Hi Maria, Alex, and others interested in circle folding axioms.

__ __

Having gone through the seven folding axioms and considered the implications as they apply to circles; it all seems counter productive. The nature and benefits of folding circles would be lost by the dismembering of parts and isolating individual movements just for the sake of constructing a set of rules for making representations of flat folds using paper.

__ __

The origami axioms are based on construction methods and the separation
of parts using images to diagram isolated relationships. The axioms have little
to do with facilitating the experience, understanding, and practical nature of
folding circles. It makes little sense to me to make generalized rules for what is obviously far
more in experience. The compass and straight edge construction, or using GeoGebra can only suggest spatial movement in a diagrammatically way. It is not folding, nor does it show the circle as more than just another 2-D constructed shape.

Euler’s formula for polygons will not work for a circle, nor will his formula for polyhedra work for a circle disk (a compressed sphere.) There is a great difference between polyhedral parts and circle/sphere unity. They are not interchangeable. You can not add a constant to what is already Whole.

__ __

There are so many different functions revealed by one fold in the circle that are not addressed by any of the Huzita-Hatori axioms, or other interpretations of what constitutes folding paper.

__ __

Polygons can be taken apart and reassembled in many different ways, naturally leading to construction methods bases on fixed parts. The circle demonstrates a different system of logic. Circles have nothing to take apart; there are no rules; only principles and the order of structural pattern generation and transformation. The self-referential movement of the circle reveals endless relationships of parts that are inherent to unity only found in the circle form, and only reflected in part.

__ __

A few primary observations about folding the circle;

__ __

- First there are no lines, no points unless you consider the circle as a large point.
- Folding does not start with a straight line, only with two arbitrary, imaginary points anywhere on the circumference. There is no relationship without movement between the two locations.
- Without creasing, touch these two points and a circle with a smaller diameter is formed showing a cylinder with the circle surface parallel or a cone pattern with the surface out of parallel.
- Touching the two points and creasing will form a hemisphere pattern; reciprocal in both directions forming a full spherical pattern of movement.
- Touching these points and creasing always generates a chord, one straight line with two end points. (It is possible to fold lines segments without the chord which isolates the function from the consistency of process, eliminating information, thus limiting choices and decreasing reforming potential.)
- The chord is always a perpendicular bisector to the distance between the two points touching. This is consistent to right angle compression of sphere to circle.
- The diameter/axis and two points is a dual tetrahedron pattern.
- One diameter and one circle can be reformed into a closed tetrahedron showing the curve and straight edge can be congruent, with three edges and one surface.
- In folding the circle there is no point, line, or plane in isolation, each is context for all others, formed and unformed. The division and individualization of parts is without separation; reflecting circle unity.

__ __

There are so many interrelated functions in this first fold that just to make the statement that “there exist a single fold connecting two distinct points” is a meaningless generalization about drawing a line between two points. “Given two points P1 and P2 there exist a unique fold that maps P1 onto P2.” seems to be a complicated way of describing the nature of reflective symmetry. Mapping a position fixes what it is. Axioms are intellectually comfortable. The circle is not a planer construction and the movement of the circle is difficult to describe in linear flat terms.

__ __

It can be said that the information is the result of certain movements of relationship; and cause should not be confused with the effects. I say when we create unnecessary separation, and cut up a process of in-formation into fragmented pieces, much is lost and confusion will result. What ever rules we decide upon are always compromised by the greater context beyond the limitations we set for ourselves. On those grounds I would caution you about trying to craft axioms about the movement of selected parts in the circle. While there is precedents for doing this, I see little value beyond intellectual stimulation, which might be reason enough. My suggestion is to fold a few hundred circles, and then you have a start in understanding how maybe to develop axioms that might have some value in relationship to better understanding circle folding that possibly would be worth publishing.

Jul 4, 2010, 10:39:54 AM7/4/10

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Axioms are not a good start of any activity, in my mind, and starting to teach geometry from axioms is counter-productive in any form of geometry you pick.

The axiomatic method, though, I find valuable. I also find valuable making bridges between different ways of doing geometry. Huzita-Hatori is a bridge between Euclidian geometry and origami, for example.

"Science" and "scissors" come from the same root, so analytic approaches are very prevalent everywhere in STEM, and powerful set of tools for that have been developed in math, in particular.

Can synthetic tools, like metaphor, be developed for building mathematics? Does anyone know how to do that? I definitely do not. I don't know who builds mathematics without dismembering everything into parts and making construction kits for conjectures out of them.

The axiomatic method, though, I find valuable. I also find valuable making bridges between different ways of doing geometry. Huzita-Hatori is a bridge between Euclidian geometry and origami, for example.

"Science" and "scissors" come from the same root, so analytic approaches are very prevalent everywhere in STEM, and powerful set of tools for that have been developed in math, in particular.

Can synthetic tools, like metaphor, be developed for building mathematics? Does anyone know how to do that? I definitely do not. I don't know who builds mathematics without dismembering everything into parts and making construction kits for conjectures out of them.

Jul 4, 2010, 11:02:11 AM7/4/10

to mathf...@googlegroups.com

Maria this is a very good question and I am going to think about it. It is very interesting poit.

Cheers from Caracas...

P.D: I fallow you always, it is just that I am shy to participate...

Enviado desde mi dispositivo movil BlackBerry® de Digitel.

Jul 4, 2010, 4:11:17 PM7/4/10

to mathf...@googlegroups.com

On Sun, Jul 4, 2010 at 7:39 AM, Maria Droujkova <drou...@gmail.com> wrote:

> Axioms are not a good start of any activity, in my mind, and starting to

> teach geometry from axioms is counter-productive in any form of geometry you

> pick.

>

> Axioms are not a good start of any activity, in my mind, and starting to

> teach geometry from axioms is counter-productive in any form of geometry you

> pick.

>

The way I think about it, geometry involves geography, where by

"geography" I mean that navigable space wherein people fall

down, cut their fingers in, have to develop skills in.

Philosophers call it "the real world" (versus their dream ones

I suppose -- like Narnia).

The world of geography (ala National Geographic, the magazine)

includes microbes and stars, atoms and galaxies, and everything

in between (Planet Earth, ourselves aboard her).

Geometry by itself, as distinct from geography, is more about

timeless and sizeless patterns, meaning we don't need to

know "where / when" in order to understand the communication

(e.g. "the sum of a triangle's angles adds to 180 degrees" is

true "in pure principle" assuming a planar triangle and a shared

concept of angular degree measure (both are geographic

assumptions, but we tend to ignore that when idealizing)).

Early math education ala Montessori involves taking "jobs" off

the shelf and exploring them, alone, with peers ("peers"

includes teachers).

This design integrates manipulation and play with the idea

of working according to schedules. The jobs are somewhat

standardized and have been designed to give insights.

When designing a curriculum, I like to think in terms of what

"jobs" it might include -- clearly I've been influenced by the

Montessori way.

Will they need to use GPS devices?

Will they need to cook over an open flame?

Initial exposure to a job might involve undirected activity, just

"poking around" (we assume this is safe, that the students have

enough experience to not hurt themselves or one another).

This initial phase or opening is like playing with chess pieces,

counting and sorting them perhaps, without anyone starting

off with a lecture on "the rules of chess". That lecture may

come, but it's not how we need to start -- how about we name

them first, learn to recognize a "bishop" as distinct from a

"rook".

> The axiomatic method, though, I find valuable. I also find valuable making

> bridges between different ways of doing geometry. Huzita-Hatori is a bridge

> between Euclidian geometry and origami, for example.

>

The bridge I see twixt (geometry + geography) and origami is precisely

that origami is a set of skills, as well as abstractions.

Folding paper takes concentration and time.

If cutting is involved, say to get the initial squares, then there might be

sharp objects in the picture, more skills to master.

Where the rubber meets the road in geometry is with plane nets.

These are the creasable / foldable origami-like "graphs" (in the sense of

"connected networks of edges") that one might cut out and tape together.

Examples:

http://www.rwgrayprojects.com/synergetics/s09/figs/f1601.html

http://www.rwgrayprojects.com/synergetics/s09/figs/f5012.html

Yes, the software gurus have also taken these fold-up operations to

screen, so you can project them in class, or assign as Youtubes

for homework (to discuss the next day?).[1]

However, there's no substitute for actually sitting down, alone or

with peers, and using scissors to cut out and fold up the

following (diving into the non-Euclidean geometry we

introduce early):

Mite = AAB

Mite + Mite = Sytes

Sytes:

Lite (skewed trigonal dipyramid)

Bite (monorectangular tetrahedron)

Rite (disphenoid tetrahedron)

Kites:

Kit = 2 Lites

Kate = 2 Bites

Kat = 2 Rites or 2 Bites

Coupler = 2 Kits, 2 Kates or 2 Kats.

Volumes

A 1/24

B 1/24

Mite 1/8

Sytes 1/4

Kites 1/2

Coupler 1

Cube 3

...

and so on.

http://www.flickr.com/photos/17157315@N00/4738169322/in/photostream/lightbox/

> "Science" and "scissors" come from the same root, so analytic approaches are

> very prevalent everywhere in STEM, and powerful set of tools for that have

> been developed in math, in particular.

>

I remember this workshop I delivered to mostly 13-15 year olds, that

involved wooden dowels, rubber tubing, and yes, cutting with scissors.

This was a geometry class.

I was somewhat shocked and surprised how poorly some operated with

scissors. They'd obviously been given little to no opportunity to master

this implement, probably because of the "not in my back yard" syndrome:

with Art and Shop largely phased out in a lot of schools, there's no one

left who sees teaching the use of scissors as a part of the job, especially

if the students are as old as 13-15.

[ Some schools may not have a budget for scissors. In many schools,

supplies simply "walk away" as they're simply too valuable to kept

locked up in supply closets -- a better model might be to give each

student some useful craft tools as theirs to keep (and bring to practice),

provided there's a working government in this picture, able to disperse

resources by means of public institutions (not everyone gets that,

have to join the army to get government issue scissors) ]

> Can synthetic tools, like metaphor, be developed for building mathematics?

> Does anyone know how to do that? I definitely do not. I don't know who

> builds mathematics without dismembering everything into parts and making

> construction kits for conjectures out of them.

>

The geometry I'm talking about does come with construction kits,

or such kits may be adapted for exploration in this space. Zome

is a good example, or if you want to explore virtually, vZome.

What is vZome? I recently made a less frenetic Youtube than that

"lightning talk" (about "quacks"), featuring vZome by Scott Vorthmann.

The HP Pavilion I'm using is a bit short on RAM, especially given

this new digital monitor, which needs a big footprint (video RAM

is shared with CPU RAM in this model, unless you insert a

different card, which I did, high nVidia, but it blew out right away

-- ran extremely hot, wow). So the video doesn't show off vZome

in its full glory. Maybe someone else reading this has a better

setup and wants to give it a try.

Anyway, here's a blog post with a Youtube about vZome

by Scott "not a quack" Vorthmann, who said he liked it [2]:

http://mybizmo.blogspot.com/2010/06/working-with-vzome.html

Well, I've rambled enough. Cut! <-- movie director voice

More later I hope,

Kirby

>

> Cheers,

> Maria Droujkova

> http://www.naturalmath.com

>

> Make math your own, to make your own math.

>

[1] Sal Khan of Khan Academy was on a community access TV show

recently in Portland, with Keith Devlin et al, suggesting that many

schools have it backwards: have students get those lectures at

home or in their dorms e.g. through tcp/ip and/or on DVD, and

go to school to get help with the exercises, the homework, the

manual tasks that take teamwork (many do).

http://www.flickr.com/photos/17157315@N00/4761556180/sizes/l/

(some details on the TV show, doesn't give the date I notice, last

week I think it was)

[2] there's an inside joke among users of dynamically

typed languages that their objects are more like "ducks"

meaning if they "quack and walk like ducks" then they

qualify as such. One might say our criteria relate to

outward behavior and appearance, less so to "essence"

The statically typed languages, such as Java, require their

users to declare what type an object "really is" and this

credential has to match at every "door" or the compiler

will complain. Python employs "duck typing" (considered

more agile, and not strictly typeless, but with more of

a testing burden in the runtime environment, vs. at the

compilation stage (hence "test driven development")).

If you wanna get subtle and technical, the Java notion

of an "interface" is closer to agile, implies a kind of

contract which, again, the compiler is empowered to

enforce in that language. vZome is written in Java by

the way.

Jul 4, 2010, 4:34:48 PM7/4/10

to mathf...@googlegroups.com

Changing directions somewhat here...

So if Maria says starting to teach geometry from axioms is counter-productive, and based on many previous discussions about 4d geometry, tessellations, mites, origami and other geometric topics, I'm becoming more and more confused about the purpose of highschool geometry.ï¿½ I homeschool a rising 9th grader and I'm looking ahead to university requirements, etc. as I think about different opportunities and directions I'd like us to cover in the next 4 years.ï¿½

Most universities require highschool geometry on a student's transcript -- but what skills or knowledge are they expecting students to have based on that line-item?ï¿½ Is it a thought process or a knowledge base or something else?ï¿½ And what opportunites are best approaches for achieving those skills/knowledge to make them interesting and motivational to explore further?

Some traditional highschool textbooks approach it primarily from a axiom/property/proof direction while others look more at constructs with very little proofs.ï¿½ Most textbooks don't seem very motivational (and they certainly weren't when I was that age).

Any suggestions?

Linda

Maria Droujkova wrote:

So if Maria says starting to teach geometry from axioms is counter-productive, and based on many previous discussions about 4d geometry, tessellations, mites, origami and other geometric topics, I'm becoming more and more confused about the purpose of highschool geometry.ï¿½ I homeschool a rising 9th grader and I'm looking ahead to university requirements, etc. as I think about different opportunities and directions I'd like us to cover in the next 4 years.ï¿½

Most universities require highschool geometry on a student's transcript -- but what skills or knowledge are they expecting students to have based on that line-item?ï¿½ Is it a thought process or a knowledge base or something else?ï¿½ And what opportunites are best approaches for achieving those skills/knowledge to make them interesting and motivational to explore further?

Some traditional highschool textbooks approach it primarily from a axiom/property/proof direction while others look more at constructs with very little proofs.ï¿½ Most textbooks don't seem very motivational (and they certainly weren't when I was that age).

Any suggestions?

Linda

Maria Droujkova wrote:

Axioms are not a good start of any activity, in my mind, and starting to teach geometry from axioms is counter-productive in any form of geometry you pick.

The axiomatic method, though, I find valuable. I also find valuable making bridges between different ways of doing geometry. Huzita-Hatori is a bridge between Euclidian geometry and origami, for example.

"Science" and "scissors" come from the same root, so analytic approaches are very prevalent everywhere in STEM, and powerful set of tools for that have been developed in math, in particular.

Can synthetic tools, like metaphor, be developed for building mathematics? Does anyone know how to do that? I definitely do not. I don't know who builds mathematics without dismembering everything into parts and making construction kits for conjectures out of them.

Cheers,

Maria Droujkova

http://www.naturalmath.com

Make math your own, to make your own math.

On Sun, Jul 4, 2010 at 10:01 AM, Bradford Hansen-Smith <wholem...@sbcglobal.net> wrote:

Hi Maria, Alex, and others interested in circle folding axioms.

ï¿½

Having gone through the seven folding axioms and considered the implications as they apply to circles; it all seems counter productive. The nature and benefits of folding circles would be lost by the dismembering of parts and isolating individual movements just for the sake of constructing a set of rules for making representations of flat folds using paper. ï¿½

ï¿½

The origami axioms are based on construction methods and the separation of parts using images to diagram isolated relationships. The axioms have little to do with facilitating the experience, understanding, and practical nature of folding circles. It makes little sense to me to make generalized rules for what is obviously far more in experience. The compass and straight edge construction, or using GeoGebra can only suggest spatial movement in a diagrammatically way. It is not folding, nor does it show the circle as more than just another 2-D constructed shape.

Eulerï¿½s formula for polygons will not work for a circle, nor will his formula for polyhedra work for a circle disk (a compressed sphere.)ï¿½ There is a great difference between polyhedral parts and circle/sphere unity. They are not interchangeable. You can not add a constant to what is already Whole.

ï¿½

There are so many different functions revealed by one fold in the circle that are not addressed by any of the Huzita-Hatori axioms, or other interpretations of what constitutes folding paper.ï¿½ï¿½

ï¿½

Polygons can be taken apart and reassembled in many different ways, naturally leading to construction methods bases on fixed parts. The circle demonstrates a different system of logic. ï¿½Circles have nothing to take apart; there are no rules; only principles and the order of structural pattern generation and transformation. The self-referential movement of the circle reveals endless relationships of parts that are inherent to unity only found in the circle form, and only reflected in part.

ï¿½

A few primary observations about folding the circle;

ï¿½

- First there are no lines, no points unless you consider the circle as a large point.
- Folding does not start with a straight line, only with two arbitrary, imaginary points anywhere on the circumference. There is no relationship without movement between the two locations.
- Without creasing, touch these two points and a circle with a smaller diameter is formed showing a cylinder with the circle surface parallel or a cone pattern with the surface out of parallel.
- Touching the two points and creasing will form a hemisphere pattern; reciprocal in both directions forming a full spherical pattern of movement.
- Touching these points and creasing always generates a chord, one straight line with two end points. (It is possible to fold lines segments without the chord which isolates the function from the consistency of process, eliminating information, thus limiting choices and decreasing reforming potential.)
- The chord is always a perpendicular bisector to the distance between the two points touching. This is consistent to right angle compression of sphere to circle.
- The diameter/axis and two points is a dual tetrahedron pattern.
- One diameter and one circle can be reformed into a closed tetrahedron showing the curve and straight edge can be congruent, with three edges and one surface.
- In folding the circle there is no point, line, or plane in isolation, each is context for all others, formed and unformed. The division and individualization of parts is without separation; reflecting circle unity.

ï¿½

There are so many interrelated functions in this first fold that just to make the statement that ï¿½there exist a single fold connecting two distinct pointsï¿½ is a meaningless generalization about drawing a line between two points. ï¿½Given two points P1 and P2 there exist a unique fold that maps P1 onto P2.ï¿½ seems to be a complicated way of describing the nature of reflective symmetry. Mapping a position fixes what it is. Axioms are intellectually comfortable. The circle is not a planer construction and the movement of the circle is difficult to describe in linear flat terms. ï¿½

ï¿½

It can be said that the information is the result of certain movements of relationship; and cause should not be confused with the effects. I say when we create unnecessary separation, and cut up a process of in-formation into fragmented pieces, much is lost and confusion will result. What ever rules we decide upon are always compromised by the greater context beyond the limitations we set for ourselves. On those grounds I would caution you about trying to craft axioms about the movement of selected parts in the circle. While there is precedents for doing this, I see little value beyond intellectual stimulation, which might be reason enough. My suggestion is to fold a few hundred circles, and then you have a start in understanding how maybe to develop axioms that might have some value in relationship to better understanding circle folding that possibly would be worth publishing.

ï¿½

Brad

ï¿½

Sent:Thu, June 24, 2010 6:28:10 AM

Subject:Re: [Math 2.0] Circle-folding axioms

On Thu, Jun 24, 2010 at 1:02 AM, Bradford Hansen-Smith <wholem...@sbcglobal.net> wrote:

Hi Maria,

I am interested in your efforts to come up with axioms for folding circles. While you are trying to give mathematical expression to folding, you must realize that folding circles has little to do with Euclid's axioms based on the construction of the circle image of points and lines. The inverse function of points and lines do not necessarily hold up for the circle since there is no way of knowing where the lines are until the points touch and there is a crease. While there are generalizations to be made about the folds, the circle is very specific and can not be disregarded for any or no shape context You can not assume a radius or center when it is not shown, that is a constructionist perspective. With the circle the lines are a direct result of the touching of points, whether they are parallel or angled. As the axioms stand they are confuse in trying to understand folding the circle. Origami is construction using a set of a specific sequence of predetermined folds where as this is not the case with the circle.

I am out of town until next week and can not take the time to do more than give a quick response. It is not clear to me what you are describing.

I will give this some thought and get back to you upon return.

Brad

ï¿½

The fold themselves, as described, are not "based" on Euclid's axioms. However, being another full set of construction axioms, Euclid's set provides means of constructing any fold, say, in GeoGebra - or using compass and straightedge. I find the descriptions of correspondences valuable.

What we can do to underlie the independence: make little (5-10 second) videos of every fold, showing how it's done, instead of current illustrations. Then, have the current illustrations listed separately, as a list of correspondences.

Two more people commented on list. Here are the comments, which will go into the next version, as well:

~*~*~*~*~*

Colin McAlister:

There is one exception to 4-C; when p1 at the center of the circle.

For points not at the center, there is only one diameter, so you would say "This is equivalent to findingthediameter throughp1." instead ofï¿½ "This is equivalent to findingadiameter throughp1." You are only discussing one point in this axiom, so you could call it simply "p" instead of "p1".

5c

This sentence is false:as there are two cases where is is not true. (i) distance between

"Given two pointsp1 andp2, there isa foldthat placesp1 onto the circle and passes throughp2."

p1andp2is smaller than the distance betweenp2and the circle. (ii) when there are two folds, as "a fold" could be casually read as meaning only one fold.

So suggest that you rewrite 5c. It would be more accurate to say:

"Given two pointsp1 andp2,consider the possibility ofa fold (or folds) through p2 that place(s)p1 onto the circle."

In 5C, you have omitted to show two arrowed lines that show the direction of the fold. I must presume that you use the dashed lines to indicate folds, as you did in the previous diagram. It is not clear if you are using them to indicate folds, or simply lines through pairs of points. The line in the 1 o'clock direction is a solution fold, but the line in the 5 o'clock is not. I think the second solution would be folded along a lineï¿½near the four o'clock direction.ï¿½

In 7-C"if p lies on the diameter perpendicular to l1" there are two folds and not one, as you state. Name the line l instead of l1 as there is no l2 in the diagram. "l" looks like "i" or a slash (/) if italicised. I suggest handscript l and p or or bold l and p for lines and points. Perhaps you could use "f" to indicate a fold, as there are other lines on your diagrams, such as radii and lines through pairs of points.

I suggest generalising the media beyond circles. 5-C applies to shapes other than circles. What family of shapes does the axiom define? Does it restrict radius of curvature of arcs of the shape? Would a starfish shape be excluded?

In 7-C, you could consider stadium shapes, a rectangle bounded on each end by a semicircle. Could you consider any irregular shaped piece of paper, with oneï¿½perimeter and no holes?

A soccer ball can be folded along a circumference against itself, by letting all of the air out. The surface of a sphere has an interesting difference from a circle. It has no centre; every point is equivalent. How does axiom 4-C differ for this situation?

~*~*~*~*~*

LFS:

Marija ï¿½ I know this has nothing to do with your axioms, but it is fun about finding the center of a circle drawn on a piece of paper by folding the paper.

http://www.youtube.com/watch?v=7kMFjXtAWAY

Linda

I found it when I happened upon the inverse of Thales theorem.: http://geogebrawiki.wikispaces.com/Thales+Inverse

~*~*~*~*~*

The video Linda linked assumes the use of right angles of the paper to construct the center of a circle. The seven Huzita-Hatori origami axioms don't use the edge of the paper at all. Circle-folding axioms we are working on use the fact we know the edge is a circle. Will a set of origami axioms using right angles of the regular origami paper be different from Huzita-Hatori ones, or derivative?

Maria D

Jul 4, 2010, 4:58:59 PM7/4/10

to mathf...@googlegroups.com

Hello! I'm Ali, the new member of group.

I have an event on 25August, it is: "Laerning Math by Making" . I think this intereting discussion can be connected to that event. So I will use your insightful comments.

Best Wishes.

Ali.

On Mon, Jul 5, 2010 at 1:04 AM, Linda Whipker <lwhi...@earthlink.net> wrote:

Changing directions somewhat here...

So if Maria says starting to teach geometry from axioms is counter-productive, and based on many previous discussions about 4d geometry, tessellations, mites, origami and other geometric topics, I'm becoming more and more confused about the purpose of highschool geometry. I homeschool a rising 9th grader and I'm looking ahead to university requirements, etc. as I think about different opportunities and directions I'd like us to cover in the next 4 years.

Most universities require highschool geometry on a student's transcript -- but what skills or knowledge are they expecting students to have based on that line-item? Is it a thought process or a knowledge base or something else? And what opportunites are best approaches for achieving those skills/knowledge to make them interesting and motivational to explore further?

Some traditional highschool textbooks approach it primarily from a axiom/property/proof direction while others look more at constructs with very little proofs. Most textbooks don't seem very motivational (and they certainly weren't when I was that age).

Any suggestions?

Linda

Maria Droujkova wrote:

Axioms are not a good start of any activity, in my mind, and starting to teach geometry from axioms is counter-productive in any form of geometry you pick.

The axiomatic method, though, I find valuable. I also find valuable making bridges between different ways of doing geometry. Huzita-Hatori is a bridge between Euclidian geometry and origami, for example.

"Science" and "scissors" come from the same root, so analytic approaches are very prevalent everywhere in STEM, and powerful set of tools for that have been developed in math, in particular.

Can synthetic tools, like metaphor, be developed for building mathematics? Does anyone know how to do that? I definitely do not. I don't know who builds mathematics without dismembering everything into parts and making construction kits for conjectures out of them.

Cheers,

Maria Droujkova

http://www.naturalmath.com

Make math your own, to make your own math.

On Sun, Jul 4, 2010 at 10:01 AM, Bradford Hansen-Smith <wholem...@sbcglobal.net> wrote:

Hi Maria, Alex, and others interested in circle folding axioms.

Having gone through the seven folding axioms and considered the implications as they apply to circles; it all seems counter productive. The nature and benefits of folding circles would be lost by the dismembering of parts and isolating individual movements just for the sake of constructing a set of rules for making representations of flat folds using paper.

The origami axioms are based on construction methods and the separation of parts using images to diagram isolated relationships. The axioms have little to do with facilitating the experience, understanding, and practical nature of folding circles. It makes little sense to me to make generalized rules for what is obviously far more in experience. The compass and straight edge construction, or using GeoGebra can only suggest spatial movement in a diagrammatically way. It is not folding, nor does it show the circle as more than just another 2-D constructed shape.

Euler’s formula for polygons will not work for a circle, nor will his formula for polyhedra work for a circle disk (a compressed sphere.) There is a great difference between polyhedral parts and circle/sphere unity. They are not interchangeable. You can not add a constant to what is already Whole.

There are so many different functions revealed by one fold in the circle that are not addressed by any of the Huzita-Hatori axioms, or other interpretations of what constitutes folding paper.

Polygons can be taken apart and reassembled in many different ways, naturally leading to construction methods bases on fixed parts. The circle demonstrates a different system of logic. Circles have nothing to take apart; there are no rules; only principles and the order of structural pattern generation and transformation. The self-referential movement of the circle reveals endless relationships of parts that are inherent to unity only found in the circle form, and only reflected in part.

A few primary observations about folding the circle;

- First there are no lines, no points unless you consider the circle as a large point.

- Folding does not start with a straight line, only with two arbitrary, imaginary points anywhere on the circumference. There is no relationship without movement between the two locations.
- Without creasing, touch these two points and a circle with a smaller diameter is formed showing a cylinder with the circle surface parallel or a cone pattern with the surface out of parallel.
- Touching the two points and creasing will form a hemisphere pattern; reciprocal in both directions forming a full spherical pattern of movement.
- Touching these points and creasing always generates a chord, one straight line with two end points. (It is possible to fold lines segments without the chord which isolates the function from the consistency of process, eliminating information, thus limiting choices and decreasing reforming potential.)
- The chord is always a perpendicular bisector to the distance between the two points touching. This is consistent to right angle compression of sphere to circle.
- The diameter/axis and two points is a dual tetrahedron pattern.
- One diameter and one circle can be reformed into a closed tetrahedron showing the curve and straight edge can be congruent, with three edges and one surface.
- In folding the circle there is no point, line, or plane in isolation, each is context for all others, formed and unformed. The division and individualization of parts is without separation; reflecting circle unity.

There are so many interrelated functions in this first fold that just to make the statement that “there exist a single fold connecting two distinct points” is a meaningless generalization about drawing a line between two points. “Given two points P1 and P2 there exist a unique fold that maps P1 onto P2.” seems to be a complicated way of describing the nature of reflective symmetry. Mapping a position fixes what it is. Axioms are intellectually comfortable. The circle is not a planer construction and the movement of the circle is difficult to describe in linear flat terms.

It can be said that the information is the result of certain movements of relationship; and cause should not be confused with the effects. I say when we create unnecessary separation, and cut up a process of in-formation into fragmented pieces, much is lost and confusion will result. What ever rules we decide upon are always compromised by the greater context beyond the limitations we set for ourselves. On those grounds I would caution you about trying to craft axioms about the movement of selected parts in the circle. While there is precedents for doing this, I see little value beyond intellectual stimulation, which might be reason enough. My suggestion is to fold a few hundred circles, and then you have a start in understanding how maybe to develop axioms that might have some value in relationship to better understanding circle folding that possibly would be worth publishing.

Brad

Sent:Thu, June 24, 2010 6:28:10 AMSubject:Re: [Math 2.0] Circle-folding axioms

On Thu, Jun 24, 2010 at 1:02 AM, Bradford Hansen-Smith <wholem...@sbcglobal.net> wrote:

Hi Maria,

I am interested in your efforts to come up with axioms for folding circles. While you are trying to give mathematical expression to folding, you must realize that folding circles has little to do with Euclid's axioms based on the construction of the circle image of points and lines. The inverse function of points and lines do not necessarily hold up for the circle since there is no way of knowing where the lines are until the points touch and there is a crease. While there are generalizations to be made about the folds, the circle is very specific and can not be disregarded for any or no shape context You can not assume a radius or center when it is not shown, that is a constructionist perspective. With the circle the lines are a direct result of the touching of points, whether they are parallel or angled. As the axioms stand they are confuse in trying to understand folding the circle. Origami is construction using a set of a specific sequence of predetermined folds where as this is not the case with the circle.

I am out of town until next week and can not take the time to do more than give a quick response. It is not clear to me what you are describing.

I will give this some thought and get back to you upon return.

Brad

The fold themselves, as described, are not "based" on Euclid's axioms. However, being another full set of construction axioms, Euclid's set provides means of constructing any fold, say, in GeoGebra - or using compass and straightedge. I find the descriptions of correspondences valuable.

What we can do to underlie the independence: make little (5-10 second) videos of every fold, showing how it's done, instead of current illustrations. Then, have the current illustrations listed separately, as a list of correspondences.

Two more people commented on list. Here are the comments, which will go into the next version, as well:

~*~*~*~*~*

Colin McAlister:

There is one exception to 4-C; when p1 at the center of the circle.

For points not at the center, there is only one diameter, so you would say "This is equivalent to findingthediameter throughp1." instead of "This is equivalent to findingadiameter throughp1." You are only discussing one point in this axiom, so you could call it simply "p" instead of "p1".

5c

This sentence is false:as there are two cases where is is not true. (i) distance between

"Given two pointsp1 andp2, there isa foldthat placesp1 onto the circle and passes throughp2."p1andp2is smaller than the distance betweenp2and the circle. (ii) when there are two folds, as "a fold" could be casually read as meaning only one fold.

So suggest that you rewrite 5c. It would be more accurate to say:

"Given two pointsp1 andp2,consider the possibility ofa fold (or folds) through p2 that place(s)p1 onto the circle."

In 5C, you have omitted to show two arrowed lines that show the direction of the fold. I must presume that you use the dashed lines to indicate folds, as you did in the previous diagram. It is not clear if you are using them to indicate folds, or simply lines through pairs of points. The line in the 1 o'clock direction is a solution fold, but the line in the 5 o'clock is not. I think the second solution would be folded along a line near the four o'clock direction.

In 7-C"if p lies on the diameter perpendicular to l1" there are two folds and not one, as you state. Name the line l instead of l1 as there is no l2 in the diagram. "l" looks like "i" or a slash (/) if italicised. I suggest handscript l and p or or bold l and p for lines and points. Perhaps you could use "f" to indicate a fold, as there are other lines on your diagrams, such as radii and lines through pairs of points.

I suggest generalising the media beyond circles. 5-C applies to shapes other than circles. What family of shapes does the axiom define? Does it restrict radius of curvature of arcs of the shape? Would a starfish shape be excluded?

In 7-C, you could consider stadium shapes, a rectangle bounded on each end by a semicircle. Could you consider any irregular shaped piece of paper, with one perimeter and no holes?

A soccer ball can be folded along a circumference against itself, by letting all of the air out. The surface of a sphere has an interesting difference from a circle. It has no centre; every point is equivalent. How does axiom 4-C differ for this situation?

~*~*~*~*~*

LFS:

Marija – I know this has nothing to do with your axioms, but it is fun about finding the center of a circle drawn on a piece of paper by folding the paper.

http://www.youtube.com/watch?v=7kMFjXtAWAY

Linda

I found it when I happened upon the inverse of Thales theorem.: http://geogebrawiki.wikispaces.com/Thales+Inverse

~*~*~*~*~*

The video Linda linked assumes the use of right angles of the paper to construct the center of a circle. The seven Huzita-Hatori origami axioms don't use the edge of the paper at all. Circle-folding axioms we are working on use the fact we know the edge is a circle. Will a set of origami axioms using right angles of the regular origami paper be different from Huzita-Hatori ones, or derivative?

Maria D

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Jul 4, 2010, 5:09:30 PM7/4/10

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On Sun, Jul 4, 2010 at 4:34 PM, Linda Whipker <lwhi...@earthlink.net> wrote:

Changing directions somewhat here...

So if Maria says starting to teach geometry from axioms is counter-productive, and based on many previous discussions about 4d geometry, tessellations, mites, origami and other geometric topics, I'm becoming more and more confused about the purpose of highschool geometry. I homeschool a rising 9th grader and I'm looking ahead to university requirements, etc. as I think about different opportunities and directions I'd like us to cover in the next 4 years.

Most universities require highschool geometry on a student's transcript -- but what skills or knowledge are they expecting students to have based on that line-item? Is it a thought process or a knowledge base or something else? And what opportunities are best approaches for achieving those skills/knowledge to make them interesting and motivational to explore further?

Some traditional highschool textbooks approach it primarily from a axiom/property/proof direction while others look more at constructs with very little proofs. Most textbooks don't seem very motivational (and they certainly weren't when I was that age).

Any suggestions?

Linda

Linda,

You may want to look at Van Hiele levels as to where I am coming from when I go onto my "don't start with axiom" soapbox: http://en.wikipedia.org/wiki/Van_Hiele_model This model overall is hugely helpful in planning instruction.

Recently I invited four girls to a geometry project. I thought we'd make a video about connections among different construction systems. But we ended up making a video about an artist, an engineer, a Western philosopher/mathematician (quoting Shakespeare in a toga), an Eastern philosopher/mathematician folding origami, and a programmer building a rhombus, using their tools of the trade. Sasha and Katherine are still editing the video, but filming it was meaningful in itself.

How do you make a rhombus? What IS a rhombus, then? How do you know that what you made is a rhombus? How do you convince others (and cross-discipline) that what you made is a rhombus?

I don't know what universities want (I expect different ones want different things). I want my geometry students:

- To make things using different tool sets (software and physical - fold paper neatly, program in style, use scissors well, know how to operate a compass and a straightedge decently)

- To use logical reasoning in general and to recognize different "truth claim" systems; know similarities and differences between construction and reasoning

- Get to the level of theory-building within several different existing axiom systems

- Work a bit on building their own axiomatics

- Be able to recognize an interesting problem, solve interesting problems, compose interesting problems

- Communicate their work using appropriate media (diagrams, spreadsheets, interactives, stories) and participate in some communities using geometry (from 3d modeling to carpenters)

- As they solve interesting problems, don't claim their hypotenuse is longer than a leg, don't make self-referential loops in software, and in general, know simple facts of life related to their activities

I am probably forgetting something. Note that this sounds like a lot, but these are not separate items. You can do all that within one rich activity.

Jul 4, 2010, 6:17:25 PM7/4/10

to mathf...@googlegroups.com

Linda,

there is a huge difference how you start teaching (or rather learning geometry) and the level you achieve by the end of high school. College requirements being what they are this is where (under the circumstances) the study of geometry should lead. The fundamental element of that level of geometry is proof which is a hard thing to teach and to learn. Proof by itself is an important step in problem solving. The idea is then to accustom students to solving problems before the formal requirements of college preparation become overarching.

Proof and its presentation should not be confused. However, they are being confused implicitly in the expressions "2-column proof" and "paragraph proof." A proof is a sequence of ideas that evolve purposefully. "2-column proof" and "paragraph proof" are two ways of communicating those ideas. It is impossible to teach either unless there is a degree of problem solving mastery. Not that they (system, teachers, texts) do not try.

Alex B

--

Jul 4, 2010, 6:58:18 PM7/4/10

to mathf...@googlegroups.com

On Sun, Jul 4, 2010 at 1:34 PM, Linda Whipker <lwhi...@earthlink.net> wrote:

> Changing directions somewhat here...

>

> So if Maria says starting to teach geometry from axioms is

> counter-productive, and based on many previous discussions about 4d

> geometry, tessellations, mites, origami and other geometric topics, I'm

> becoming more and more confused about the purpose of highschool geometry. I> Changing directions somewhat here...

>

> So if Maria says starting to teach geometry from axioms is

> counter-productive, and based on many previous discussions about 4d

> geometry, tessellations, mites, origami and other geometric topics, I'm

> homeschool a rising 9th grader and I'm looking ahead to university

> requirements, etc. as I think about different opportunities and directions

> I'd like us to cover in the next 4 years.

>

Good questions.

Given this group is called MathFuture, I'm going on the assumption

that futuristic flavors of math get some air time, whereas the status

quo is more in the background, not so much the focus.

None of the mass-published high school textbooks that are widely

available in 2010 have anything about Mites, Sytes 'n Kites!

I used to work at McGraw-Hill by the way, and know how K-12

mathematics is seen as a stash of relatively changeless content, with

only shallow cosmetic changes needed to keep up with what's

fashionable (in which case, maybe just pad the book with new material,

make it even thicker, heavier and more expensive).

Writing from scratch is just not in the cards, too much inertia.

The NCTM lesson plan suggesting a tetrahedron as a "non-traditional"

unit of volume is deeply buried on the Illuminates web site (a needle

in a haystack) and gets pulled up mostly by esoteric postings by "out

there" math teachers such as myself.

You'll find the tie to Alexander Graham Bell buried here:

I've had opportunities to test this newfangled material, which I often

mix with a computer language.

Here's a remnant of a course I gave at Winterhaven PPS, Portland's

"geek hogwarts":

http://www.4dsolutions.net/ocn/winterhaven/section3.html

(note Geometry + Geography is my approach, even then...).

Portland, Oregon is a pioneering town and still has some of that

experimental spirit.

> Most universities require highschool geometry on a student's transcript --

> but what skills or knowledge are they expecting students to have based on

> that line-item? Is it a thought process or a knowledge base or something

> else? And what opportunites are best approaches for achieving those

> skills/knowledge to make them interesting and motivational to explore

> further?

>

Some of us look askance at universities for not showing more leadership.

Harvard has been especially disappointing lately, in the news for one

retro concept after another it seems.

MIT has been more forward thinking in my book, serving as a base for

several promising initiatives. OpenCourseWare, One Laptop per

Child...

Princeton, my alma mater, is being slow with its Woodrow Wilson School

of Public Affairs, sigh.

Stanford is also on my radar, although I think we're in the lead in

Oregon in some respects.

> Some traditional highschool textbooks approach it primarily from a

> axiom/property/proof direction while others look more at constructs with

> very little proofs. Most textbooks don't seem very motivational (and they

> certainly weren't when I was that age).

>

> Any suggestions?

>

I'm assuming the Euclidean axioms and theorems will continue to get a

lot of air time, and support this.

Ralph Abraham did a lot of work on getting the proofs on the web.

Old editions of Euclid's Elements are also on the web and highly worth perusing:

http://sunsite.ubc.ca/DigitalMathArchive/Euclid/byrne.html

However, one has little clue what axioms and definitions if one is not

conscious of alternatives.

Non-Euclidean geometries have been somewhat out of reach of high

school aged students, given they branch off from the fifth postulate.

In the experimental curriculum I'm concerned with, we follow Karl

Menger's suggestion that we might have a "geometry of lumps" wherein

we define basic concepts differently. (Karl was a dimension theorist.

His daughter Eve is a chemistry teacher in our science teaching

community.)

http://coffeeshopsnet.blogspot.com/2009/03/res-extensa.html

So I agree with Maria that exposure to multiple systems is apropos.

In our "geometry of lumps", points, lines, planes and "solids" are all

"lumps" (like "of clay") and differ only in their topological

characteristics (planes are flatter, lines are longer).

Nothing is "infinite" and we have no "continua" (it's a discrete

math). Donald Knuth gave a lecture at MIT with a likewise Finite /

Discrete universe model. I'm not pretending Karl Menger is the only

influence here -- we're talking about a lineage or philosophical

tradition.

http://newsgroups.derkeiler.com/Archive/Soc/soc.culture.romanian/2007-05/msg01934.html

http://www-cs-faculty.stanford.edu/~knuth/things.html

Philosophy goes ahead of mathematics a lot of the time. If you have a

9th grader, here's a comic book I recommend (you could read it first,

decide if it's suitable):

Kirby

Message has been deleted

Jul 5, 2010, 3:59:16 AM7/5/10

to mathf...@googlegroups.com

Hello Ali - it is great to see
you in this group! I look forward to your event and
its connection to this thread - which I must admit to not following
very closely (surprise, surprise not to all of you who know my single-mindedness
about using technology in standard classroom curricula). Ali and
his wife Shirin and I have collaborated in GeoGebra (another

surprise not). So I think this event should be
quite interesting. Again - well come!

Linda
(LFS)

__ __

**From:** mathf...@googlegroups.com [mailto:mathf...@googlegroups.com]
**On Behalf Of **S. Ali Ghasempouri

**Sent:** Sunday, July 04, 2010 10:59 PM

**To:** mathf...@googlegroups.com

**Subject:** Re: [Math 2.0] Circle-folding axioms

__ __

Hello! I'm Ali, the new member of group.

Jul 5, 2010, 4:05:40 AM7/5/10

to mathf...@googlegroups.com

Thank you Linda! You always have suprise for me with full of energy in your words.

I would like to thank Linda for her support. thank-you again. Ali.

--

Jul 5, 2010, 8:31:14 AM7/5/10

to mathf...@googlegroups.com

Maria, you said

"Can synthetic tools, like metaphor, be developed for building
mathematics? Does anyone know how to do that? I definitely do not. I
don't know who builds mathematics without dismembering everything into
parts and making construction kits for conjectures out of them."

Yes, thoughts for conjecture. I never did much like construction kits.

It has been put forth that infants first perceive everything as one continuous
whole without differentiation. As senses begin to click in the brain using
these perceptions to grow connections necessary to function as it does. If this
is so our brains are programmed to recognize differences and form
individualized meaning. We construct in math and all understanding, learning
from part-to-part, supposedly towards understanding something larger we call whole. The problem is
that unless we start with the Whole we will never get there because of the
infinite nature, scale, and complexity of parts on a magnitude far beyond human
comprehension. We all start out knowing nothing of differences. At some point
we must acknowledge both building knowledge through experience and beyond
experience by having faith in what already is. Faith is the bases for any
axiom, agreed upon as first principle, thus requires no proof and becomes
foundational for the theoretical development of constructing understanding.

My approach is to start comprehensively with the Whole, represented by the
sphere-compressed-to-circle-everything. The first movement is that of spherical
compression and therefore is principle to all that follows. Folding the circle
decompresses spherical information. To sum up the first axiom for folding the circle,
I would have to say, Wholemovement.

__ __

- A circle results from the right angle
compression of any two opposites points on a sphere, that is consistent to
contained spherical unity.

__ __

It would then follow to list
observations about compression as principle, not rules, to all that follows in folding
the circle. Without having to invent connections or rely on the constructed abstractions of
mathematical language I have then to observe what is revealed through folding the
circle and recognize connections to my experience.

__ __

This axiomatic approach stimulates
my imagination to go to the largest place I can and to make an assumption about
this first movement as a place to observe and begin to make meaningful connections
to what I discover in little places. I
do not see nature dismembering; that seems to come from animal curiosity with a
lack of understanding about what we see, having limited insight to the nature
of the larger context. The only place I have to start anything is where I am right
now. My map needs to be inclusive to my understanding since my experience will
always be local.

Hi I am Carola. Live in Venezuela, a country in south america with beatifull weather!!! Sunny and warm.

Maria this is a very good question and I am going to think about it. It is very interesting poit.

Cheers from Caracas...

P.D: I fallow you always, it is just that I am shy to participate...

Enviado desde mi dispositivo movil BlackBerry® de Digitel.

Axioms are not a good start of any activity, in my mind, and starting to teach geometry from axioms is counter-productive in any form of geometry you pick.

The axiomatic method, though, I find valuable. I also find valuable making bridges between different ways of doing geometry. Huzita-Hatori is a bridge between Euclidian geometry and origami, for example.

"Science" and "scissors" come from the same root, so analytic approaches are very prevalent everywhere in STEM, and powerful set of tools for that have been developed in math, in particular.

Can synthetic tools, like metaphor, be developed for building mathematics? Does anyone know how to do that? I definitely do not. I don't know who builds mathematics without dismembering everything into parts and making construction kits for conjectures out of them.

Cheers,

Maria Droujkova

http://www.naturalmath.com

Make math your own, to make your own math.

The axiomatic method, though, I find valuable. I also find valuable making bridges between different ways of doing geometry. Huzita-Hatori is a bridge between Euclidian geometry and origami, for example.

"Science" and "scissors" come from the same root, so analytic approaches are very prevalent everywhere in STEM, and powerful set of tools for that have been developed in math, in particular.

Can synthetic tools, like metaphor, be developed for building mathematics? Does anyone know how to do that? I definitely do not. I don't know who builds mathematics without dismembering everything into parts and making construction kits for conjectures out of them.

Cheers,

Maria Droujkova

http://www.naturalmath.com

Make math your own, to make your own math.

Subject:Re: [Math 2.0] Circle-folding axioms

Jul 5, 2010, 8:59:12 AM7/5/10

to mathf...@googlegroups.com

On Mon, Jul 5, 2010 at 8:31 AM, Bradford Hansen-Smith <wholem...@sbcglobal.net> wrote:

Maria, you said"Can synthetic tools, like metaphor, be developed for building mathematics? Does anyone know how to do that? I definitely do not. I don't know who builds mathematics without dismembering everything into parts and making construction kits for conjectures out of them."

Yes, thoughts for conjecture. I never did much like construction kits.

It has been put forth that infants first perceive everything as one continuous whole without differentiation. As senses begin to click in the brain using these perceptions to grow connections necessary to function as it does. If this is so our brains are programmed to recognize differences and form individualized meaning. We construct in math and all understanding, learning from part-to-part, supposedly towards understanding something larger we call whole. The problem is that unless we start with the Whole we will never get there because of the infinite nature, scale, and complexity of parts on a magnitude far beyond human comprehension. We all start out knowing nothing of differences. At some point we must acknowledge both building knowledge through experience and beyond experience by having faith in what already is. Faith is the bases for any axiom, agreed upon as first principle, thus requires no proof and becomes foundational for the theoretical development of constructing understanding.

I would like to compare and contrast this with Alan Kay's notion of similarities over differences http://learningevolves.wikispaces.com/nonUniversals#simDiff

Alan will host a webinar in our series on August 7th, by the way. A quote:

"Similarities over Differences" was to contrast with the standard processes of most nervous systems of most species to be more interested in "differences over similarities" (which is listed on the universal side). At most levels from reflexes to quite a bit of cognition, most similarities are accommodated and normalized while differences to the normalizations have a heightened significance (of "danger" or "pay attention").

Paying attention to differences is good for simple survival but makes it hard to think in many ways because it leads to so many cases, categories and distinctions -- and because some of the most important things may have disappeared into "normal" (in particular, things about oneself and one's own culture). So we unfortunately are much more interested in even superficial differences between humans and cultures and have a very hard time thinking of "the other" as being in the same value space as we are..."

An axiom is a higher-level similarity. Looks like there is a certain synthesis-analysis cycle between differences and similarities, at each level in the development of understanding. And then "folding back" (Pirie-Kieren) to previously understood things, we can notice new similarities and new differences.

Jul 5, 2010, 10:31:35 AM7/5/10

to mathf...@googlegroups.com

Commenting on Maria's points, "Can synthetic tools, like metaphor, be developed for building
mathematics? Does anyone know how to do that? I definitely do not. I
don't know who builds mathematics without dismembering everything into
parts and making construction kits for conjectures out of them." Several classes of modern mathematicians intuitively define number, and proceed to use synthetic tools.These well meaning mathematicians tend to avoid pitfalls of improperly parsing ancient numeration and ancient arithmetic systems(used by an appropriate ancient culture). One class of modern mathematician that intuitively defines number, skipping over ancient abstract details, can be called Platonic, as discussed by: http://planetmath.org/encyclopedia/PlatosMathematics.html Neo-Pythagoreans are a second class of modern mathematicians that love aspects of our Western Tradition, but begin serious math discussions with axioms. Where is the beef .. meaning where are the fundamental definitions of number and arithmetic operations? One view is that intuitive numeration and arithmetic systems must be made specific at an early point. In my high school days set theory was used to provide specific numeration background for algebra, geometry, trig and solid geometry problems. Is set theory passe these days? Milo Gardner --- On Mon, 7/5/10, Bradford Hansen-Smith wrote:<wholem...@sbcglobal.net> |

Jul 5, 2010, 10:55:35 AM7/5/10

to mathf...@googlegroups.com

Maria,

Thanks for reference to Alen Kay

Further down he says,

"One way to think of my chart is that a lot of things we correlate with "enlightenment" and "civilization" are rather un-natural and rare inventions whose skills require us to learn how to go against many of our built in thought patterns. I think this is one of the main reasons to have an organized education (to learn the skills of being better thinkers than our nervous systems are directly set up for)."

"History suggests that we not lose these powerful ideas. They are not easy to get back."

"The non-built-in nature of the powerful ideas on the right hand list implies they are generally more difficult to learn -- and this seems to be the case. This difficulty makes educational reform very hard because a very large number of the gatekeepers in education do not realize these simple ideas and tend to perceive and react (not think) using the universal left hand list ..... "

Right hand and left hand are of the same body. If any part is left out it is not complete in differences, therefore not Whole. "...to pay attention" to differences is to understand there are only differences and that similarities are generalizations that give comfort to predictions about not knowing. Fear of differences is the animal brain, acceptance of differences is development of the human brain and finding value in differences is recognition of mind potential. Differences that generate fear are not valued; similarities that come from not paying attention are often driven by comfort and often laziness.

Thanks for reference to Alen Kay

Further down he says,

"One way to think of my chart is that a lot of things we correlate with "enlightenment" and "civilization" are rather un-natural and rare inventions whose skills require us to learn how to go against many of our built in thought patterns. I think this is one of the main reasons to have an organized education (to learn the skills of being better thinkers than our nervous systems are directly set up for)."

"History suggests that we not lose these powerful ideas. They are not easy to get back."

"The non-built-in nature of the powerful ideas on the right hand list implies they are generally more difficult to learn -- and this seems to be the case. This difficulty makes educational reform very hard because a very large number of the gatekeepers in education do not realize these simple ideas and tend to perceive and react (not think) using the universal left hand list ..... "

Right hand and left hand are of the same body. If any part is left out it is not complete in differences, therefore not Whole. "...to pay attention" to differences is to understand there are only differences and that similarities are generalizations that give comfort to predictions about not knowing. Fear of differences is the animal brain, acceptance of differences is development of the human brain and finding value in differences is recognition of mind potential. Differences that generate fear are not valued; similarities that come from not paying attention are often driven by comfort and often laziness.

--

Jul 6, 2010, 12:29:59 AM7/6/10

to mathf...@googlegroups.com

On Mon, Jul 5, 2010 at 5:59 AM, Maria Droujkova <drou...@gmail.com> wrote:

<< snip >>

>

> I would like to compare and contrast this with Alan Kay's notion of

> similarities over differences

> http://learningevolves.wikispaces.com/nonUniversals#simDiff

> Alan will host a webinar in our series on August 7th, by the way. A quote:

>

Cool.

I mention Alan Kay is in on some early planning meetings in this

recent post to math-teach. I was in on at least one of these

meetings myself, in London c/o The Shuttleworth Foundation.

Here's a link:

http://mathforum.org/kb/message.jspa?messageID=7116852&tstart=0

I maybe should have filed my recent posting under the Tessellations

heading rather than Circle-folding Axioms thread, but on second

thought there's a lot on making spherical polyhedra from folded

circles that might connect these two threads.

Example:

http://www.rwgrayprojects.com/synergetics/s09/figs/f3720.html

The mite / syte / kite stuff is a lot about nomenclature. Effective

pedagogy and andragogy has much to do with mnemonics

and well-organized charts.

Speaking of well-organized charts, this one by Guy Inchbald is

much in discussions on the Poly list these days:

http://www.steelpillow.com/polyhedra/AHD/chart_lg.gif

(good for homeschoolers)

http://www.steelpillow.com/polyhedra/AHD/AHD.htm

(good example of a math paper)

The shapes on that chart may all be identified with mites,

sytes and kites and/or assemblies thereof. For example,

the "tri-rectangular tetrahedron" in row two (the only

tetrahedron mentioned) is the Mite. The three arrows

therefrom (pointing to row three) point to the three Sytes,

left to right:

Skewed Trigonal Dipyramid: Lite

Mono-rectangular Tetrahedron: Bite

Disphenoid Tetrahedron: Rite

Note only is the nomenclature different, but on the right

side the volumes are easily set to 1/4, given every Syte

is made of two Mites of volume 1/8. Getting these easy

fractions is where "tetrahedral mensuration" comes in,

and that's cutting edge stuff (skip to last paragraph if

you like, if you wanna skip more nitty gritty details of

the debate).

The above chart is the Archimedean tradition, although

when it comes to constructing these "honeycomb duals"

it's OK to involve the Platonics as well (we have five

Platonic polyhedra and thirteen Archimedeans).

Euclidean geometry is in no way averse to Polyhedra by

the way. The Russian classics by Kiselev come as a set:

Book: Planimetry; Book 2: Stereometry, i.e first comes

planar, then comes spatial, with the former a special case

of the latter.

What are the criteria for a geometry to be non-Euclidean

then, if "planar versus spatial" is not one of them? That's

what Karl Menger's essay is about, the one entitled:

'Modern Geometry and the Theory of Relativity', in

Albert Einstein: Philosopher-Scientist ,

The Library of Living Philosophers VII,

edited by P. A. Schilpp, Evanston, Illinois,

pp. 459-474.

He suggest we might want to pioneer a new "geometry of

lumps" in which points, lines, planes, polyhedra, are

distinguished not by "dimension" but by aspect ratios.

This was the genesis of "4D geometry" with every "lump"

a tetrahedron basically (but maybe roller pinned out to

make a "plane" or what have you -- a claymation geometry).

Anyway, in doing our dissections or disassemblies of

the Platonic and/or Archimedean shapes, we're sometimes

relying on Euclidean proofs, the results of which are

included in the essay below. Example: given a rhombus,

its two diagonals bisect one another and define four

triangles that are either congruent or mirror images of

one another. Euclid's theorems suffice.

http://groups.yahoo.com/group/synergeo/message/61533

As an example of where I think a lot of our teachers

are at (including home schooling parents), I commend

this page at the Math Forum.

NSF funding is involved, as is Zome (a building tool

I sometimes mention, good for budding architects).

An Amazing, Space Filling, Non-regular Tetrahedron

by Joyce Frost and Peg Cagle

http://mathforum.org/pcmi/hstp/resources/dodeca/paper.html

(you might be surprised how often this article is cited,

including from Wikipedia, and Facebook where Tetrahedron

has a Facebook page).

The authors write about a "puzzling tetrahedron" but

do not have the vocabulary to connect it to the

disphenoid tetrahedron or tetragonal disphenoid as

it's sometimes called.

Or, in newer nomenclature, they are writing about

the Rite (one of three Sytes).

They're missing the surrounding discussion of

tetrahedral space-fillers, and the controversial fact

that Math World doesn't mention any, but only

by a hair.

The only missing ingredient is more a inclusive

nomenclature that covers more of the pattern. Combine

their paper with Guy Inchbald's chart, and you've got

the basis for some great lesson plans (especially if

you have Zome -- or maybe vZome by Scott Vorthmann).

If you've gotten this far, congratulations. I think what's

intimidating and frustrating to young mathematicians

today is a sense that the frontier must be impossibly

far and that valuable contributions might only come

from those holding advanced degrees in exchange for

many years of study. Yet here we are on the front

lines in some of the currently ongoing discussions,

participating in contemporary debates, in a way that's

accessible to high school aged students and younger.

Kirby

Jul 7, 2010, 3:03:45 PM7/7/10

to mathf...@googlegroups.com

On Mon, Jul 5, 2010 at 9:29 PM, kirby urner <kirby...@gmail.com> wrote:

<< snip >>

> I maybe should have filed my recent posting under the Tessellations

> heading rather than Circle-folding Axioms thread, but on second

> thought there's a lot on making spherical polyhedra from folded

> circles that might connect these two threads.

>

> Example:

> http://www.rwgrayprojects.com/synergetics/s09/figs/f3720.html

>

This Youtube is one of many in a genre, relating explorations of

polyhedral constructions to paper folding and origami:

http://www.youtube.com/watch?v=298Ffj7xTYo&feature=related

Heartening to see this growing collection, where we develop

manual dexterity skills as an aspect of studying mathematics.

[ Watching these kinds of video clips on Youtube is a legitimate

math-learning activity. One way to jump into the future is to

be able to give Youtubes as part of the recommended viewing.

Not every school is lucky enough to have access. Some have

Internet, but block Youtube, even for teachers (an anti-thinking

reflex). ]

Modular dissections of polyhedra into constituent tetrahedra

(a kind of tessellation or mosaic-making) increases appreciation

for the difference between left and right handed. Computer

games like Tetris accomplish this in the planar context, e.g.

when you have a falling left handed L shape, there's nothing

you can do, rotationally, to have it slot into a right handed L's

slot.

One feature I find frustrating about Euclidean geometry is

how the notion of "congruence" is allowed to sweep this under

the rug. I'm reading Kiselev's Planimetry (in translation, I

can't read Russian) and noticing how "congruence" gets

off the ground with "superimposition" i.e. if you can move

A over B such that everything lines up, then A and B

are congruent.

However, right at the outset an important caveat is introduced

such that "flipping over" a plane figure, by "temporarily

taking it out of the plane" is a permitted transformation

and considered "congruence preserving" (as are translation

and rotation). Left and right L shapes are considered

congruent, even though superimposition through translation

and rotation is not in the cards with such a difference

(as we learn playing Tetris).

In the case of spatial geometry (Kiselev's volume 2 is

Stereometry), turning a left shoe into a right shoe is

akin to inside-outing, i.e. unfold to the plane-net and

then crease the folds the other way, making convex

be concave. That's a pretty significant transformation.

Why should we accept that a left and right handed glove

are "congruent"? If we decide to break ranks and change

the meaning of congruence, then do we have another

non-Euclidean geometry then?

In my vision of future math, there's fairly routine

familiarity with breaking a regular tetrahedron into

component parts, irregular tetrahedra obtained

by symmetric slicing and dicing. We get 12 left

handed and 12 right handed. One may start with

the same plane-net and fold either way. Here's

a picture:

http://www.rwgrayprojects.com/synergetics/s09/figs/f1301.html

(includes plane net).

http://www.rwgrayprojects.com/synergetics/s09/figs/f87230.html

(exploded diagram of a regular tetrahedron)

This so-called A-module, when set to a volume

of 1/24, participates in set of constructions wherein

polyhedra come out with simple whole number

volumes:

http://www.rwgrayprojects.com/synergetics/plates/figs/plate03z.html

(note "A module" in last row)

Learning spatial geometry in this way is not status

quo I think we'd all agree, so I project it into a possible future.

Consider this science fiction for the time being --

although I did manage to set up an all-sixth-grade assembly

in a local public school where we performed this assembly

as an exercise, after taking in more information about

Alexander Graham Bell's "kite" assemblies (also

known as the "octet truss" in architecture).

http://worldgame.blogspot.com/2006/02/octet-truss.html

Kirby

Jul 8, 2010, 5:34:01 PM7/8/10

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Kirby,

You are right about the importance of polyhedra to understanding geometry and the importance of geometry to understanding mathematical functions and the importance of hands-on to understanding polyhedra. Yes, it seems a bit circular.

Youtube helps stimulate people by seeing what other people are doing. Until one does the actual work themselves there is no real understanding that comes from looking at pictures, whether on Youtube or in a book. Virtual education is only virtual understanding.

You are right about the importance of polyhedra to understanding geometry and the importance of geometry to understanding mathematical functions and the importance of hands-on to understanding polyhedra. Yes, it seems a bit circular.

Youtube helps stimulate people by seeing what other people are doing. Until one does the actual work themselves there is no real understanding that comes from looking at pictures, whether on Youtube or in a book. Virtual education is only virtual understanding.

"If we decide to break ranks and change

the meaning of congruence, then do we have another

non-Euclidean geometry then?"

Maybe not another no-Euclidean geometry, rather an expanding of the old one, which I think we can agree is fundamentally sound, but not "solid". We
now know nothing is solid, yet all remains consistency to pattern that is also reflected in polyhedral forms.

How few hands-on geometry teachers there are. Fortunately through a process of information division they are multiplying .

How few hands-on geometry teachers there are. Fortunately through a process of information division they are multiplying .

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Jul 8, 2010, 7:31:42 PM7/8/10

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On Thu, Jul 8, 2010 at 2:34 PM, Bradford Hansen-Smith

<wholem...@sbcglobal.net> wrote:

> Kirby,

>

> You are right about the importance of polyhedra to understanding geometry

> and the importance of geometry to understanding mathematical functions and

> the importance of hands-on to understanding polyhedra. Yes, it seems a bit

> circular.

>

<wholem...@sbcglobal.net> wrote:

> Kirby,

>

> You are right about the importance of polyhedra to understanding geometry

> and the importance of geometry to understanding mathematical functions and

> the importance of hands-on to understanding polyhedra. Yes, it seems a bit

> circular.

>

I've got this Geometry + Geography thing going, where the former is the

Platonic world of "abstractions" (includes algebraic / symbolic encodings

thereof) and the latter is the "real world" of experience (energy in action).

Geometric metaphors become somewhat all inclusive in this schema.

> Youtube helps stimulate people by seeing what other people are doing. Until

> one does the actual work themselves there is no real understanding that

> comes from looking at pictures, whether on Youtube or in a book. Virtual

> education is only virtual understanding.

>

It's also an outlet for people to share their doings, which supplies some

motivation. To have your work "world viewable" even in principle just

adds some pizazz.

Sometimes it's not the hands-on aspect that excites me, but the

bandwidth and the fact that it's more like television.

Students get used to getting their information through TV, then find

their schooling is resolutely anti-TV.

How much better if there's some overlap between the two cultures.

Making Youtubes, such as cooking shows (which might also be

math shows) as school projects, for credit (even for compensation):

how excellent. Takes teamwork too. All the better.

http://controlroom.blogspot.com/2006/02/boosting-bandwidth.html

> "If we decide to break ranks and change

> the meaning of congruence, then do we have another

> non-Euclidean geometry then?"

>

> Maybe not another no-Euclidean geometry, rather an expanding of the old one,

> which I think we can agree is fundamentally sound, but not "solid". We now

> know nothing is solid, yet all remains consistency to pattern that is also

> reflected in polyhedral forms.

>

I go into more detail regarding this congruence business in this

recent posting (essay) to the Math Forum:

http://mathforum.org/kb/thread.jspa?threadID=2093280&tstart=0

I'm suspicious of a notion of congruence that de-sensitizes us

to "handedness".

When assembling with polyhedra, it's critical to have the right

number of left handed and right handed pieces. To call all these

"congruent" is just goofy.

Here's a short skit by one of the radical math teachers I work

with, designed to help make this point:

So, I am ordering material to have a cube built. I order 48

Charateristic Tetrahedron.

They salesman says that they are all the same according to Projective

Geometry and by gum, it is the most fundamental of geometries.

A pallet arrives and work is started and the head mason comes to me and says

"Ain't gonna work"

"Why not?"

"They don't fit together, right. Or left, for that matter"

As project engineer, I go over to the pallet of blocks and the

delivery sign says,

"48 Char Tets - R"

I call up the salesperson.

"Guy, what is up with this pile of all right handed blocks? I thought

they were all the same"

"Well, gee, ahh, ya know a lot of really smart people came up with

this and a...I'll fix this right away"

An hour later a two pallets arrive and are identified 24 char Tet - R

and 24 Char Tet - L.

The mason has been watching all of this and comes over and says

"Down at Fuller Block, you get blocks twice as big, they do not get

mixed up and you use half the mortar"

Another reason why Rome wasn't built in a day.

> How few hands-on geometry teachers there are. Fortunately through a process

> of information division they are multiplying .

>

> Brad

I'd like to get more welding into the picture. Make a rhombic

triacontahedron out of metal.

I'd take this course myself if I could, as I'm no good at welding.

The idea that math is just something you do in a chair, with no tools

-- pernicious, classist and ridiculous no?

Kirby

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> For more options, visit this group at

> http://groups.google.com/group/mathfuture?hl=en.

>

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Jul 8, 2010, 7:59:13 PM7/8/10

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On Thu, Jul 8, 2010 at 7:31 PM, kirby urner <kirby...@gmail.com> wrote:

It's also an outlet for people to share their doings, which supplies some

> Youtube helps stimulate people by seeing what other people are doing. Until

> one does the actual work themselves there is no real understanding that

> comes from looking at pictures, whether on Youtube or in a book. Virtual

> education is only virtual understanding.

>

motivation. To have your work "world viewable" even in principle just

adds some pizazz.

I like to think of short online videos as rather powerful social objects in blended environments. They can help a network grow by capturing many aspects of a situation and succinctly, quickly making it available to others - who want to do, surely, but don't know how.

For example, I am pretty sure that the numbers of people who fold origami and participate in local origami clubs is growing explosively due to video availability. Look at going between Step 27 to Step 28 of this 75-step dragon pattern I am figuring out at the moment: http://dev.origami.com/images_pdf/dragon_ce_25.pdf It is shown around 9:40 of this video: http://www.youtube.com/watch?v=RCqCT_QZYIo Without the video, I would be hopelessly stuck at this point, abandoning the project. It has been viewed about more than two hundred thousand times.

Look at how the GeoGebra community is using videos to support people and groups: http://www.youtube.com/geogebrachannel Another excellent example of the community role of videos: seeing how people use interactive software!

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