Circle-folding axioms
Maria Droujkova
Original axioms:
- Given two points p1 and p2, there is a unique fold that passes through both of them.
- Given two points p1 and p2, there is a unique fold that places p1 onto p2.
- Given two lines l1 and l2, there is a fold that places l1 onto l2.
- Given a point p1 and a line l1, there is a unique fold perpendicular to l1 that passes through point p1.
- Given two points p1 and p2 and a line l1, there is a fold that places p1 onto l1 and passes through p2.
- 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.
- Given one point p and two lines l1 and l2, there is a fold that places p onto l1 and is perpendicular to l2.
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.
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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Alexander Bogomolny
Actually, now that you mention it, won't we have to change all the axioms to include "line or arc or circle"?
Bradford Hansen-Smith
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
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
Maria Droujkova
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 p1 and p2 on the circle, there is a unique fold that places p1 onto p2 and places the half-circles l1 and l2 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 p1, there is a unique fold that passes through point p1 and places the half-circles l1 and l2 defined by the fold onto one another.
This is equivalent to finding a diameter through p1.
Axiom 5-C
Given two points p1 and p2, there is a fold that places p1 onto the circle and passes through p2. 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 p1 and p2, 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 l1, there is a fold that places p onto the circle and is perpendicular to l1. 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.
Bradford Hansen-Smith
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.
Sent: Wed, June 23, 2010 3:22:13 PM
Subject: Re: [Math 2.0] Circle-folding axioms
Patrick Vennebush
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
Maria Droujkova
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!
Maria Droujkova
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 the diameter through p1." instead of "This is equivalent to finding a diameter through p1." You are only discussing one point in this axiom, so you could call it simply "p" instead of "p1".
5c
This sentence is false:
"Given two points p1 and p2, there is a fold that places p1 onto the circle and passes through p2."
So suggest that you rewrite 5c. It would be more accurate to say:
"Given two points p1 and p2, consider the possibility of a fold (or folds) through p2 that place(s) p1 onto the circle."
~*~*~*~*~*
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
Alexander Bogomolny
6-C
Alexander Bogomolny
Bradford Hansen-Smith
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.
Subject: Re: [Math 2.0] Circle-folding axioms
Maria Droujkova
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.
ckuh...@gmail.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.
kirby urner
> 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.
Linda Whipker
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 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
�
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 the diameter through p1." instead of� "This is equivalent to finding a diameter through p1." 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 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.
"Given two points p1 and p2, there is a fold that places p1 onto the circle and passes through p2."
So suggest that you rewrite 5c. It would be more accurate to say:
"Given two points p1 and p2, consider the possibility of a 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
S. Ali Ghasempouri
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 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
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 the diameter through p1." instead of "This is equivalent to finding a diameter through p1." 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 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.
"Given two points p1 and p2, there is a fold that places p1 onto the circle and passes through p2."
So suggest that you rewrite 5c. It would be more accurate to say:
"Given two points p1 and p2, consider the possibility of a 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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Maria Droujkova
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.
Alexander Bogomolny
--
kirby urner
> 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
LFS
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.
S. Ali Ghasempouri
--
Bradford Hansen-Smith
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.
From: "ckuh...@gmail.com" <ckuh...@gmail.com>
To: mathf...@googlegroups.com
Sent: Sun, July 4, 2010 10:02:11 AM
Subject: Re: [Math 2.0] Circle-folding axioms
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.
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
Maria Droujkova
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.
milo gardner
| 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 <wholem...@sbcglobal.net> wrote: |
Bradford Hansen-Smith
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.
Subject: Re: [Math 2.0] Circle-folding axioms
kirby urner
<< 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
kirby urner
<< 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
Bradford Hansen-Smith
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?"
How few hands-on geometry teachers there are. Fortunately through a process of information division they are multiplying .
From: kirby urner <kirby...@gmail.com>
To: mathf...@googlegroups.com
Sent: Wed, July 7, 2010 12:03:45 PM
Subject: Re: [Math 2.0] Circle-folding axioms
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kirby urner
<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
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>
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Maria Droujkova
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!