celebrating Pi Day
kirby urner
Patrick Vennebush
My contribution on 3.14 isn’t nearly as heady as Kirby’s, but here’s what I have to offer:
http://mathjokes4mathyfolks.wordpress.com/2011/03/13/a-cool-quick-trick-for-pi-day/
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Ihor Charischak
CLIME
Council for Technology in Math Education
Website: http://clime.org
Blog: http://climeconnections.blogspot.com
My contribution on 3.14 isn’t nearly as heady as Kirby’s, but here’s what I have to offer:
From: kirby urner [mailto:kirby...@gmail.com]
Sent: Sunday, March 13, 2011 7:03 PM
To: mathf...@googlegroups.com
Subject: [Math 2.0] celebrating Pi DayThe Cult of Pi is ramping up big time for celebrations March 14.In calling it a cult, I'm not casting aspersions or throwing stones at glass houses.I'm a proud member of a cult myself (of Athena).Here's some of the pro Pi PR (much more if you watch "related videos").What am I contributing?I'm challenging coders to verify Ramanujan's wild formula for 1/pi, by going out to 1000 digits and then comparing to a published source.
<image001.png>
Even if you're not doing this in Python (Java, PHP, Perl, Ocaml, Haskell...?), you can use this unit test below to grab the digits in question.http://on.fb.me/ic8gZy (sorry, gotta be in Facebook to follow this one)http://mail.python.org/pipermail/edu-sig/2011-March/010224.html (a link to the unit test)Yes, I know that many libraries have it built in, including gmpy and no doubt Mathematica.More fun to follow along with Ramanujan though.An epic ballad: http://www.archive.org/details/RamanujanI'm not sure there's any proof of why his thing works, nor any clear information on how he derived it.A gift from Athena perhaps?Kirby--
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kirby urner
Cooper Macbeth
--
shaun
Shaun Errichiello
Salk School of Science
212-614-8786
http://sites.google.com/site/shaunteaches/
http://shaunteaches.blogspot.com/
http://www.youtube.com/user/shaunteaches
Cooper Macbeth
Bradford Hansen-Smith
|
Vi is right that pi is wrong, but not for the reasons she gave. First, pi day is just another day to celebration the
irrational; a confusion found in mathematics and reflected in other areas of
human thinking. That radii or diameters, or any arrangements of endless chords will
ever equal the circumference is nonsense. To say it does is similar to saying any
given part or combination of parts will equal the whole. Parts accumulate, endlessly
approaching but never finding resolve, except I guess in mathematical concepts. Much like fractals, or a set of infinite
sets of infinity to the power of itself; we do not know what all that means, yet
somehow feel the need to prove it. So we have imagined pi as a number that gives standing to not understanding. I suppose that is reason enough to celebrate. --- On Tue, 3/15/11, Cooper Macbeth <cooper...@gmail.com> wrote: |
kirby urner
Bradford Hansen-Smith
|
|
Murray
suggest an alternative symbol for the new circle constant.
http://www.squarecirclez.com/blog/lets-drop-pi/5665
Maria Droujkova
Is doubling Pi worth the worldwide confusion it would cause for at least two generations - until the kids who grew up with educators who grew up with the new Pi become educators? I don't think the change is systemic or revolutionary enough to justify it.
Therefore, for the sake of more radical simplification and therefore larger computational benefits, I propose making Pi=4, as per this proof: http://qntm.org/trollpi
Trollingly yours,
Maria Droujkova
Make math your own, to make your own math.
Linda Fahlberg-Stojanovska
I haven’t been following this very closely, but this last remark reminds me of my problems with physics (I am nothing if not single-tracked J).
Why couldn’t they have made 1m the length so that g=-10m/s2? Or some other length that was useful?
And whether or not countries use metric, they all still do pipes and bikes in inches. Argh.
Linda
From: mathf...@googlegroups.com [mailto:mathf...@googlegroups.com] On Behalf Of Maria Droujkova
Sent: Wednesday, March 16, 2011 11:49 AM
To: mathf...@googlegroups.com
Subject: Re: [Math 2.0] Re: celebrating Pi Day
There is social and cognitive cost of unit changes. In the case of switching to the metric system, most countries (other than US, Liberia and Burma) felt the cost justified by significant computational benefits.
Is doubling Pi worth the worldwide confusion it would cause for at least two generations - until the kids who grew up with educators who grew up with the new Pi become educators? I don't think the change is systemic or revolutionary enough to justify it.
Therefore, for the sake of more radical simplification and therefore larger computational benefits, I propose making Pi=4, as per this proof: http://qntm.org/trollpi
Trollingly yours,
Maria Droujkova
Make math your own, to make your own math.
--
Maria Droujkova
I haven’t been following this very closely, but this last remark reminds me of my problems with physics (I am nothing if not single-tracked J).
Why couldn’t they have made 1m the length so that g=-10m/s2?
I blame intergalactic political correctness. They did not want to be too earthcentric.
Cheers,
kirby urner
Kirby, '"I often have trouble understanding you but I'm guessing, may be wrong, that you're one of those in the discrete camp, philosophically."
Sorry about not being understandable, I'm sure there are others that must feel similar. Written language is difficult, I do what I can. I'm not sure what you mean since I do not know the philosophical implications of being in the discrete camp. Philosophically I hold to my own experiential understanding because I do not know enough about what others believe to declare one over the other.
Brad
Andrius Kulikauskas
Thank you for alerting me to the anti-Pi movement. I add my vote that it
makes more sense to think in terms of 2-pi than pi. Murray, I like your
repurposing of the trademark symbol as the new symbol for 2-pi.
I intend to write my book "Gospel Math" using such novelties.
http://www.gospelmath.com/Math/DeepIdeas
If a book like mine catches on, then that's a natural way to shift over.
I'm writing for people who want to know and are willing to re-learn if
it means they will master the deep ideas. So why should I use pi, except
as a footnote? Maybe I'll use the trademark symbol but call it 2-pi.
That way people can always remember how to compare it to traditional math.
One thing I noticed from this discussion is that, whatever 2-pi means,
the roundness that it stands for is one-dimensional. Note that the
formulas for:
Circumference of a circle ® R
Area of a circle 1/2 ® R**2
Area of both sides of a disc ® R**2
Surface area of a sphere 2 ® R**2
Surface area of cone (without the base) 1/2 ® R * H
Volume of a sphere 2/3 ® R**3
Volume of a cone 1/6 ® R**3
All of these formulas use the same ® as opposed to ®-squared or ®-cubed.
I find this very strange. I suppose it means that roundness is a
one-dimensional property. The circle, disc, sphere, cone are all round
in the same way, even though they have different dimensions. It suggests
that ® relates a shape in a relative way, perhaps to the geometry that
circumscribes or inscribes it, but not in an absolute way.
This means to me that I don't know what ® really means. If we knew deep
ideas behind it, then we could decide whether to use ® or pi.
I share with my group Living by Truth.
Andrius
Andrius Kulikauskas
http://www.selflearners.net
m...@ms.lt
(773) 306-3807
Twitter: @selflearners
Bradford Hansen-Smith
| Here is a another look at pi. Attached is maybe a reason to keep the pi symbol beyond the issue of legibility and arbitrary choice. Maybe we should look towards a broader understanding of this most fundamental relationship. Brad |
Bradford Hansen-Smith Wholemovement 4606 N. Elston #3 Chicago Il 60630 www.wholemovement.com wholemovement.blogspot.com/ facebook.com/wholemovement |
| --- On Wed, 3/16/11, Murray <murray...@gmail.com> wrote: |
|
|
|
|
Andrius Kulikauskas
I read what you sent, but I didn't understand how it related to the
number pi.
I calculated some of the lengths and areas of the shape that you folded.
The equilateral triangle has sides of length square root of 3 times the
radius. It is made of 3 isosceles triangles, which is to say 6 right
triangles, each of which is 30-60-90 degrees and has lengths 1/2,
square-root-of-3 / 2 and 1 times the radius. The area of the right
triangle is (square-root-of-3 / 8 )* r**2, and the equilateral triangle
is 6 times that and the hexagon is 12 times that. That is all I understood.
Your picture shows to me that a hexagon is a nice approximation of a
circle, which may be simply to say, that 6 is a nice approximation of 2-pi.
Andrius
Andrius Kulikauskas
http://www.selflearners.net
m...@ms.lt
(773) 306-3807
@selflearners
2011.03.16 13:53, Bradford Hansen-Smith rašė:
> Here is a another look at pi.
> Attached is maybe a reason to keep the pi symbol beyond the issue of
> legibility and arbitrary choice. Maybe we should look towards a
> broader understanding of this most fundamental relationship.
>
> Brad
>
> Bradford Hansen-Smith
> Wholemovement
> 4606 N. Elston #3
> Chicago Il 60630
> www.wholemovement.com
> wholemovement.blogspot.com/
> facebook.com/wholemovement
>
> --- On *Wed, 3/16/11, Murray /<murray...@gmail.com>/* wrote:
>
>
> From: Murray <murray...@gmail.com>
> Subject: [Math 2.0] Re: celebrating Pi Day
> To: "MathFuture" <mathf...@googlegroups.com>
> Date: Wednesday, March 16, 2011, 4:59 AM
>
> You may be interested in my arguments against Tau. In this article, I
> suggest an alternative symbol for the new circle constant.
>
> http://www.squarecirclez.com/blog/lets-drop-pi/5665
>
>
>
> > On Mar 15, 8:36 pm, Cooper Macbeth <coopermacb...@gmail.com
> </mc/compose?to=coopermacb...@gmail.com>> wrote:
> > Totally in agreement. Tau is correct, the derivatives make more
> since,
> > everything is more eloquent AND Tau radians does not need to be
> an even
> > distance around the circle like 2 pi radians does. Heehee
> >
> >
>
> --
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> http://groups.google.com/group/mathfuture?hl=en.
>
> --
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kirby urner
Bradford Hansen-Smith
|
Andrius, You calculated lengths, areas and some relationships of what
I folded. Did you actually do the folding that was suggested? Did you engage in
the process or just think about it looking at the image based on what you know? We are educated to
recognize abstracted symbols that represent ideas and concepts about
relationships, but rarely are we asked to become engaged in discovering
something outside those boundaries, to become involved in what we do not
know; to put our hands on it, mind around it and spirit into it. Had you folded
the circle you would have understood that that equilateral triangle is the circle,
having only been reformed through a sequence of movements and through movement
can again be reformed without losing any circle properties. The circle is
constant, only the form has changed. Is not this another form of transformation
done with a paper circle instead of numbers and symbols? Through folding the
circle we have transformed it to a triangle and a hexagon without losing
anything of the circle. There are in the first fold over 120 different math
functions, relationships and concepts that can be identified if we observed
what we are doing in folding. We do not know because we are taught the circle
is a unit of nothing, an empty set: we do not expect anything to be there until we make a construction.
If you were in one of my workshops I would ask you to count again how many isosceles triangles, and to count all the right triangles, and not leaving it until all have been identified . You have indicated that the triangle is 180 ° by identifying the 30-60-90 right triangle. In folding the circumference behind to form the triangle you might have wondered, maybe the circle has been folded in half. The triangle is 180° and the circle folded forms two equilateral triangles. This makes no sense unless you have the folded circle in hand and see the overlap of circumference with one triangle: more than 180°. That overlap area is the difference between the length of three diameters, (hexagon) and the circumference. The hexagon is a primary since six radii (3 diameters) fit exactly around the circumference. The difference is seen in the three small vesicas centered to the triangle and when spread out around the circumference or inverted to the inside show six half vesicas. Pi number represent three diameters plus the difference between the hexagon and circumference. An infinite number of sides to an inscribed polygon will always be a polygon. That given I do not understand “approximation of a circle.” It is a circle or it is a polygon.
It is sloppy thinking to change a straight line to a curved line arbitrarily at some point of higher frequency development. The diameter is root function to the circle. One is a realized whole the other is an infinite progression. Traditionally we learn about the circle from a static image, truncating it to study the separated “dead” parts that are all numbered and classified. The nature of the circle is dynamic and comprehensive and that understanding is in the folding.
Maybe this will help, but explanations are always a poor second to our own experience. |
|
Brad Bradford Hansen-Smith Wholemovement 4606 N. Elston #3 Chicago Il 60630 www.wholemovement.com wholemovement.blogspot.com/ facebook.com/wholemovement |
| --- On Wed, 3/16/11, Andrius Kulikauskas <m...@ms.lt> wrote: |
|
|
|
|
|
Andrius Kulikauskas
Thank you for explaining further and for sharing your intuition of many
years.
I did the folding as you encouraged me. Indeed, I noticed some things
along the way. You fold the circle three times. I'm not sure how you
get the folds evenly spaced, although mine came out close enough,
perhaps. I noticed that, after folding, if I pinch together opposite
sides, then I get two three-sided pyramids. (I think you mention this
in your letter). In folding, I also noticed the relevance of whether
the folds are single-sided or double-sided, which I didn't realize from
your picture. I tried to make them double-sided. It's interesting that
the petals fold almost automatically by folding together the equilateral
triangle.
Still, I thought that this all may derive, as I wrote, from the fact
that the number 6 is close to 2*pi, which is to say, the perimeter of
the hexagon is close to the circumference of the circle. So I tried to
make an eight-sided octogon to see what happened. I ended up with a
square in the middle instead of an equaliteral triangle. And then I
realized a key difference, which is that the equilateral triangle is
exactly half of the hexagon. This means that when you make the folds
behind the equilateral triangle, you are using up all of the hexagon,
and what remains, what gets folded automatically, are the little petals,
the difference between the hexagon and the circle. Whereas in the case
of the octogon and larger polygons they can't fold up this way because,
for example, the square is much more than half of the octogon and so you
can't fold it over like that. So that suggests that the equilateral
triangle and hexagon are related to the circle in a way that the larger
polygons are not. And perhaps it's not simply a circle but a disc.
Brad, I'm curious what you think about the following thoughts... I'm
trying to understand what's basic about a circle or simply roundness.
One fact that stands out for me is that if I have a line segment, and if
I think of it as the hypotenuse of a right triangle, and I look at all
the possibilities for the "third point" of that triangle, then they form
a circle (minus the two end points). That can be thought of as a folded
circle, except that the fold comes first, and the circle comes later. I
noticed further that a sphere can be defined likewise. I start with an
axis and, in three dimensions, consider all the possibilities for the
"third point" of the triangle, and they will form a sphere. And for
some reason that's seeming to me a more intuitive way to define a circle
and a sphere then simply "the set of points that are an equal distance
from a center". Perhaps it's because "distance" only makes sense
defined globally, whereas "right angle" makes sense locally. A term
that insists on being defined everywhere seems less resilient and less
fundamental than a term that can be defined on an as needed, as relevant
basis. I'm thinking that "distance" comes into play later, after right
triangles are well established.
If I define a circle as the set of all possible "third points", then how
do I know that they are equally far away from the center? The deep idea
is: "A right triangle is 4 copies of itself." This you can see if you
bisect the hypotenuse. You can draw a rectangle inside of the right
triangle, and draw the rectangle's diagonal from the hypotenuse's
midpoint to the right angle. That will make for four little right
triangles, all the same shape as the big one. Each little hypotenuse
will have length "radius".
I wondered, where does pi come from? I noticed that if I start with the
right triangle 45-45-90, then its hypotenuse is 2*R but the distance by
way of its two legs is 2*sqr-root(2)*R. Suppose I insist that it not
take any "short-cut" and I require that at the midpoint of its leg it be
R away from the center. Then instead of cos(45) = sqrt(2)/2 the cos
half-angle formula gives sqrt of the mean of that and 1, which is
sqrt((1 + sqrt(2)/2)/2) and with each half-angle there is a progression
of these means and square roots. That progression relates to pi. So I
will look at that. But what it suggested to me is that pi is the
"anti-shortcut" number, it is the distance for going "the long way
around", always staying at least R away from a point.
Maybe in some way I am doing with my mind what you are doing with your
hands. I'm looking for the vantage point from which it all unfolds in a
way that makes intuitive sense. I'm looking for a big picture.
You wrote that it is "sloppy" to think of a circle as the limit of
polygons. Indeed, it is that sloppiness that intrigues me. More and
more, I'm thinking that we have very different faculties for working
with 2 or 3, with 7 or 8, with 20 or 30, with 150, with 10,000 etc.,
with discrete as opposed to continuous. Our counting system glosses
over so neatly all such distinctions. But actually it would be helpful
to stop and think and realize, for example, that we MUST use commas or
spaces between every third number or so (we think in 2s or 3s), that we
CAN'T make sense of numbers that are a string of 20 digits, and so on.
Such limitations, such assumptions are lurking throughout our number
system, but it's so successful that we don't bother to think about
them. I look forward to studying all the many paradoxes and I
hypothesize that they all arise from such dissonance, from such break
downs in the patch work. In my thoughts above, I like the idea that if
we look at a single "third point", then we have a right triangle, but if
we look at ALL the "third points", then we have a "folded circle" or
"sphere with axis", and if we consider ALL the compatible axes, then we
have our "usual circle" or "usual sphere". And that genealogy, if
sensible, can show which concept is more fundamental, which is less.
I'm working on that here:
http://www.gospelmath.com/Math/DeepIdeas
Thank you for ideas!
Andrius
Andrius Kulikauskas
http://www.selflearners.net
m...@ms.lt
(773) 306-3807
@selflearners
2011.03.17 08:23, Bradford Hansen-Smith rašė:
>
> Andrius,
>
> You calculated lengths, areas and some relationships of what I folded.
> Did you actually do the folding that was suggested? Did you engage in
> the process or just think about it looking at the image based on what
> you know?We are educated to recognize abstracted symbols that
> represent ideas and concepts about relationships, but rarely are we
> asked to become engaged in discovering something outside those
> boundaries, to become involved in what we do not know; to put our
> hands on it, mind around it and spirit into it. Had you folded the
> circle you would have understood that that equilateral triangle is the
> circle, having only been reformed through a sequence of movements and
> through movement can again be reformed without losing any circle
> properties. The circle is constant, only the form has changed. Is not
> this another form of transformation done with a paper circle instead
> of numbers and symbols? Through folding the circle we have transformed
> it to a triangle and a hexagon without losing anything of the circle.
> There are in the first fold over 120 different math functions,
> relationships and concepts that can be identified if we observed what
> we are doing in folding. We do not know because we are taught the
> circle is a unit of nothing, an empty set: we do not expect anything
> to be there until we make a construction.
>
> If you were in one of my workshops I would ask you to count again how
> many isosceles triangles, and to count _all_ the right triangles, and
> --- On *Wed, 3/16/11, Andrius Kulikauskas /<m...@ms.lt>/* wrote:
>
>
> From: Andrius Kulikauskas <m...@ms.lt>
> Subject: Re: [Math 2.0] Re: celebrating Pi Day
> To: mathf...@googlegroups.com
> Date: Wednesday, March 16, 2011, 2:14 PM
>
> Hi Brad,
>
> I read what you sent, but I didn't understand how it related to
> the number pi.
>
> I calculated some of the lengths and areas of the shape that you
> folded. The equilateral triangle has sides of length square root
> of 3 times the radius. It is made of 3 isosceles triangles, which
> is to say 6 right triangles, each of which is 30-60-90 degrees and
> has lengths 1/2, square-root-of-3 / 2 and 1 times the radius. The
> area of the right triangle is (square-root-of-3 / 8 )* r**2, and
> the equilateral triangle is 6 times that and the hexagon is 12
> times that. That is all I understood.
>
> Your picture shows to me that a hexagon is a nice approximation of
> a circle, which may be simply to say, that 6 is a nice
> approximation of 2-pi.
>
> Andrius
>
> Andrius Kulikauskas
> http://www.selflearners.net
> m...@ms.lt </mc/compose?to=m...@ms.lt>
> (773) 306-3807
> @selflearners
>
> 2011.03.16 13:53, Bradford Hansen-Smith rašė:
> > Here is a another look at pi.
> > Attached is maybe a reason to keep the pi symbol beyond the
> issue of legibility and arbitrary choice. Maybe we should look
> towards a broader understanding of this most fundamental relationship.
> >
> > Brad
> >
> > Bradford Hansen-Smith
> > Wholemovement
> > 4606 N. Elston #3
> > Chicago Il 60630
> > www.wholemovement.com
> > wholemovement.blogspot.com/
> > facebook.com/wholemovement
> >
> > --- On *Wed, 3/16/11, Murray /<murray...@gmail.com
> </mc/compose?to=murray...@gmail.com>>/* wrote:
> >
> >
> > From: Murray <murray...@gmail.com
> </mc/compose?to=murray...@gmail.com>>
> > Subject: [Math 2.0] Re: celebrating Pi Day
> > To: "MathFuture" <mathf...@googlegroups.com
> </mc/compose?to=mathf...@googlegroups.com>>
> > Date: Wednesday, March 16, 2011, 4:59 AM
> >
> > You may be interested in my arguments against Tau. In this
> article, I
> > suggest an alternative symbol for the new circle constant.
> >
> > http://www.squarecirclez.com/blog/lets-drop-pi/5665
> >
> >
> >
> > > On Mar 15, 8:36 pm, Cooper Macbeth <coopermacb...@gmail.com
> </mc/compose?to=coopermacb...@gmail.com>
> > </mc/compose?to=coopermacb...@gmail.com
> </mc/compose?to=coopermacb...@gmail.com>>> wrote:
> > > Totally in agreement. Tau is correct, the derivatives make more
> > since,
> > > everything is more eloquent AND Tau radians does not need to be
> > an even
> > > distance around the circle like 2 pi radians does. Heehee
> > >
> > >
> >
> > -- You received this message because you are subscribed
> to the Google
> > Groups "MathFuture" group.
> > To post to this group, send email to
> mathf...@googlegroups.com
> </mc/compose?to=mathf...@googlegroups.com>
> > </mc/compose?to=mathf...@googlegroups.com
> </mc/compose?to=mathf...@googlegroups.com>>.
> > To unsubscribe from this group, send email to
> > mathfuture+...@googlegroups.com
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> > </mc/compose?to=unsub...@googlegroups.com
> </mc/compose?to=unsub...@googlegroups.com>>.
> > For more options, visit this group at
> > http://groups.google.com/group/mathfuture?hl=en.
> >
> > -- You received this message because you are subscribed to the
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Bradford Hansen-Smith
Thank you Andrius for folding.the circle. Often when I make suggestions to fold circles people will find it safer to talk about than to actually do it. |
I did the folding as you encouraged me. Indeed, I noticed some things along the way. You fold the circle three times. I'm not sure how you get the folds evenly spaced, although mine came out close enough, perhaps. I noticed that, after folding, if I pinch together opposite sides, then I get two three-sided pyramids. (I think you mention this in your letter). In folding, I also noticed the relevance of whether the folds are single-sided or double-sided, which I didn't realize from your picture. I tried to make them double-sided. It's interesting that the petals fold almost automatically by folding together the equilateral triangle. |
When trying to fold the half folded circle into thirds we do not realize that our eyes were made to see proportionally so we try to measure. Because we are taught to measure very early that is our standard. Most students are surprised to see how accurate they are when they stop thinking about it and just do it. This is eye/hand coordination exercise. Accuracy is not as important as understanding proportions. Error naturally compounds when joining separate parts together; the unity of the circle absorbs error. Accuracy is relative to the tools we use, proportions are not. |
|
Still, I thought that this all may derive, as I wrote, from
the fact that the number 6 is close to 2*pi, which is to say, the perimeter of
the hexagon is close to the circumference of the circle. So I tried to make
an eight-sided octogon to see what happened. I ended up with a square in
the middle instead of an equaliteral triangle. And then I realized a key
difference, which is that the equilateral triangle is exactly half of the
hexagon. This means that when you make the folds behind the equilateral
triangle, you are using up all of the hexagon, and what remains, what gets
folded automatically, are the little petals, the difference between the hexagon
and the circle. Whereas in the case of the octogon and larger polygons
they can't fold up this way because, for example, the square is much more than
half of the octogon and so you can't fold it over like that. So that
suggests that the equilateral triangle and hexagon are related to the circle in
a way that the larger polygons are not. And perhaps it's not simply a
circle but a disc. |
You have discovered by doing, observing, and thinking about what you are doing. Just thinking and drawing pictures is not enough, experience is necessary for understanding. There is nothing similar to curiosity that will move us forward into doing something. It is difficult to be curious when we are trained in a particular way of thinking, without questioning and when the larger context is missing. This is along the line of what Linda has brought up and others are discussing in “math and physics – a common language?” |
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One fact that stands out for me is that if I have a line
segment, and if I think of it as the hypotenuse of a right triangle, and I look
at all the possibilities for the "third point" of that triangle, then
they form a circle (minus the two end points). That can be thought of as
a folded circle, except that the fold comes first, and the circle comes
later. I noticed further that a sphere can be defined likewise. I
start with an axis and, in three dimensions, consider all the possibilities for
the "third point" of the triangle, and they will form a sphere.
And for some reason that's seeming to me a more intuitive way to define a
circle and a sphere then simply "the set of points that are an equal
distance from a center". Perhaps it's because "distance"
only makes sense defined globally, whereas "right angle" makes sense
locally. A term that insists on being defined everywhere seems less
resilient and less fundamental than a term that can be defined on an as needed,
as relevant basis. I'm thinking that "distance" comes into play
later, after right triangles are well established. |
Of
course we can construct almost anything from a line segment, we are trained to
do just that. Do circles and sphere really come from the accumulated
construction of lines and planes? Where does the axis come from, the axis of what? As you have observed the circle must be
folded before you have a line, it is only in the abstraction of drawing that we
can have a line first. The point/distant definition of a circle is a
constructed way to talk about the 2-D image coming from using a compass. What
is the first fold of the circle in half? It has nothing to do with the center;
it is about alignment through right angle movement by touching two points on
the circumference. Every line starts
with a point, ends with a point, an three part system. A point
is a small circle that happens to be at that boundary where we would
not see it if it were any smaller. To enlarge each end point to the radial
length of the line you would have an image of two intersecting circles sharing one radius with information that correlates closely to the first fold of the
circle. |
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If I define a circle as the set of all possible "third points", then how
do I know that they are equally far away from the center? The deep
idea is: "A right triangle is 4 copies of itself." This you can see if
you bisect the hypotenuse. You can draw a rectangle inside of the right
triangle, and draw the rectangle's diagonal from the hypotenuse's
midpoint to the right angle. That will make for four little right
triangles, all the same shape as the big one. Each little hypotenuse
will have length "radius". |
| The right angle division is the proportional organization of folds within
the circle. You are talking about drawing it, constructing in 2-D what appears
by folding the circle. We are so conditioned to thinking in flat
concepts that we move from 3-D into 2-D without realizing it. I catch myself
doing it a lot because so much of our information, our concepts are based in 2-D thinking. If these
constructions were not inherently in the circle we would not be able to draw
them or fold them. |
Maybe in some way I am doing with my mind what you are doing with your hands. I'm looking for the vantage point from which it all unfolds in a way that makes intuitive sense. I'm looking for a big picture. |
No matter how large our "big picture" is, there is a larger frame. I also keep in mind that pictures are only images of something else. I try to always trace back to origin, where the picture comes from, the reality of what is being expressed. I find where revelation occurs and discoveries are made is beyond the frame. |
You wrote that it is "sloppy" to think of a circle as the limit of polygons. Indeed, it is that sloppiness that intrigues me. More and more, I'm thinking that we have very different faculties for working with 2 or 3, with 7 or 8, with 20 or 30, with 150, with 10,000 etc., with discrete as opposed to continuous. Our counting system glosses over so neatly all such distinctions. But actually it would be helpful to stop and think and realize, for example, that we MUST use commas or spaces between every third number or so (we think in 2s or 3s), that we CAN'T make sense of numbers that are a string of 20 digits, and so on. Such limitations, such assumptions are lurking throughout our number system, but it's so successful that we don't bother to think about them. I look forward to studying all the many paradoxes and I hypothesize that they all arise from such dissonance, from such break downs in the patch work. In my thoughts above, I like the idea that if we look at a single "third point", then we have a right triangle, but if we look at ALL the "third points", then we have a "folded circle" or "sphere with axis", and if we consider ALL the compatible axes, then we have our "usual circle" or "usual sphere". And that genealogy, if sensible, can show which concept is more fundamental, which is less.
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It is not so much the sloppiness
that intrigues me, rather how, why we have missed and misused the
inconsistencies and can tolerate the irregularities to the extent that they are
accepted without question. This seems a major part in letting this planet and
all animal life deteriorate to our present condition. Logic comes from observation of what we do and rational
thinking is a construct, and I see more of the rational in math and human action than I do logic. I like your question about a space between every set of three numbers in sequence. It is a convenient reminder about the structural nature of three ibeing primary to all forms of sustainable relationships. This is in line with your observation about the right triangle as first movement in the circle. |
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Brad Bradford Hansen-Smith Wholemovement 4606 N. Elston #3 Chicago Il 60630 www.wholemovement.com wholemovement.blogspot.com/ facebook.com/wholemovement |
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