duocircles

Higher-dimensional geometry (previously "Polyshapes").

duocircles

Postby alkaline » Wed Dec 17, 2003 7:13 pm

This question is mostly a question to Polyhedron Dude, but may also be of interest to Aale.

Today I read an article about Fermat's Last Theorem and the Fourth Dimension. On page 5 it described a shape that i realized was the duocircle. In the article, the equation they gave was s[sup]2[/sup] + t[sup]2[/sup] = 1, u[sup]2[/sup] + v[sup]2[/sup] = 1. They describe the construction of the shape as taking a square, folding two sides together to make a cylinder, then folding the other two sides into the fourth dimension to make the four-dimensional shape that they call a "circle of circles". If you perform this procedure on a deformable square, you can create the torus in the third dimension. Does this mean that the torus and the duocircle are topologically equivalent? Mathworld has a page of Non-Orientable Surfaces. The square they show for the torus (which is an orientable shape) is basically the same that i would imagine would create a duocircle. Is any path on a duocircle a circle?
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Postby Aale de Winkel » Wed Dec 17, 2003 7:41 pm

so 11 --> 21 --> 22 indeed the lathing pocess twice, thus forming the duocircle.

the torus is 11 -> 21 -> 2R ringing the lathed square.

Indeed I do think there is some isomorphism between the torus and the duocircle R[sup]-1[/sup]L <=> (L[sup]-1[/sup]R)[sup]-1[/sup] (denoting lathing by L and ringing by R)

(unfortunately I lack the mathematicle skills to understand the proof of fermats theorem, so I won't dig into that article shortly)

circle of circles is I do think a short description of the duocircle, a point on one x-y circle is point on an z-w circle:
the duo circle in polar coordinates is: R [ cos(α) , sin(α) , cos(β) , sin(β) ]

in fact Lathing, Rotating, Twisting are all isomorphisms and sofar the objects desscribed by the methods given in the polytope section are isomorphic to the square :?: :!:

The Mobius strip is: 1R[sub]180[/sub], and the Klein bottle 2R[sub]180[/sub] so a simple lathed Mobius strip. Which in also depicted in the square pictures Eric uploaded. :lol: his http://mathworld.wolfram.com/NonorientableSurface.html showed me that R[sub]180[/sub]R[sub]180[/sub] is called "real projective plane".
Last edited by Aale de Winkel on Thu Dec 18, 2003 10:06 am, edited 3 times in total.
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Postby alkaline » Wed Dec 17, 2003 10:53 pm

it's an introductory article, you don't have to know much if any math at all to understand it.
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Re: duocircles

Postby Aale de Winkel » Thu Dec 18, 2003 7:45 am

alkaline wrote:Today I read an article about Fermat's Last Theorem and the Fourth Dimension.


That darn article, is uploaded in unreadable postscript format. I'm certainly not in the business of buying / installing software that can read that! :twisted:
( :cry: no mention of what "me" to contact when one has problems with the file :lol: )
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Re: duocircles

Postby Polyhedron Dude » Fri Dec 19, 2003 7:51 am

alkaline wrote: They describe the construction of the shape as taking a square, folding two sides together to make a cylinder, then folding the other two sides into the fourth dimension to make the four-dimensional shape that they call a "circle of circles". If you perform this procedure on a deformable square, you can create the torus in the third dimension. Does this mean that the torus and the duocircle are topologically equivalent? Mathworld has a page of Non-Orientable Surfaces. The square they show for the torus (which is an orientable shape) is basically the same that i would imagine would create a duocircle. Is any path on a duocircle a circle?


The duocircle has only one face which separates the two cells - this face is the result of taking a square and curving it on both axises (I usually call this face a duoring - it is the product of two circles (hollow) - where the duocircle is the product of two disks). The duoring is topologically equivalent to the torus. Any path on the duoring part of the duocircle is a circle, but if we picked a path elsewhere on the duocircle, you could get squares!

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Re: duocircles

Postby Aale de Winkel » Fri Dec 19, 2003 10:10 am

Polyhedron Dude wrote:The duocircle has only one face which separates the two cells - this face is the result of taking a square and curving it on both axises (I usually call this face a duoring - it is the product of two circles (hollow) - where the duocircle is the product of two disks). The duoring is topologically equivalent to the torus. Any path on the duoring part of the duocircle is a circle, but if we picked a path elsewhere on the duocircle, you could get squares!


I thought I caught up with you, but this confuses the hell out of me.
The duocircle looks to me indeed like a circle of circles :R [ cos(α) , sin(α) , cos(β) , sin(β) ], hence the name.
the duoring likewise a ring of rings or (R[sub]1[/sub],R[sub]2[/sub]) [ cos(α) , sin(α) , cos(β) , sin(β) ]
using the disc this way I get the duoball (r = 0..R) [ cos(α) , sin(α) , cos(β) , sin(β) ]
and the duodoball (r = R[sub]1[/sub] .. R[sub]2[/sub]) [ cos(α) , sin(α) , cos(β) , sin(β) ] :!: :?:

:cry: a really confused trionian :?
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Re: duocircles

Postby Polyhedron Dude » Fri Dec 19, 2003 9:25 pm

Aale de Winkel wrote:I thought I caught up with you, but this confuses the hell out of me.
The duocircle looks to me indeed like a circle of circles :R [ cos(α) , sin(α) , cos(β) , sin(β) ], hence the name.
the duoring likewise a ring of rings or (R[sub]1[/sub],R[sub]2[/sub]) [ cos(α) , sin(α) , cos(β) , sin(β) ]
using the disc this way I get the duoball (r = 0..R) [ cos(α) , sin(α) , cos(β) , sin(β) ]
and the duodoball (r = R[sub]1[/sub] .. R[sub]2[/sub]) [ cos(α) , sin(α) , cos(β) , sin(β) ] :!: :?:

:cry: a really confused trionian :?


A duoring is the product of two circles (hollow circles that is).

The duocircle is the solid counterpart - product of two disks (solid circles) - the name may be confusing, we could call the solid thingy a duodisk instead and let duocircle be the duoring. This duodisk is what I refer to as 22 , the one I originally called a duocylinder.

The duoball is 33 = product of two balls. The duosphere would be the hollow version.
The duodoball would be the product of two doballs (this is 8-D). The doball is the 4-D donut that looks like a sphere minus small sphere, and has circular bulge.

The tridisk (222) would be the product of three disks, the hollow version is a tricircle - which can be formed by taking a cube and curving all three axises into rings.

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Postby Aale de Winkel » Sat Dec 20, 2003 8:19 am

Ah, problem is then at my first entry of the polyshape forum.
uptil now I saw the 2,3,.. for the open unfilled circle, sphere etc. to be filled into the disc, ball etc
for me figired like 11, 21 are unfilled squares, cylinders, are they also there filled counterparts :?:
then "Unfilling" is the more fundamental transformation and the figures I had in mind are 2U, 3U, 11U, 21U :wink:

I found filling something would be more natural, it's just how you define the things :!:

I look forward to a new polyshape thread where you define these things more precisely :wink: :lol: (to replace our discussion (or was it my monolog :lol: ))
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Postby Keiji » Sat Dec 20, 2003 5:08 pm

ring = hollow
circle = solid
disk is something completely different, it is a horizontal cross section of a solid torus

sphere = solid
? = hollow (let's think of a name, "sphun" anyone?)

glome = solid
gongyl = hollow

as for this business of signatures... i really don't think you guys get it...
a disk/torus is not a rotatope so cannot be given a signature of this form.
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Postby alkaline » Sat Dec 20, 2003 5:26 pm

bobxp: a number of those are wrong, check out the glossary.

circle: hollow = same as ring
disk: solid (meaning it's a rotatope)

sphere: hollow
ball: solid

glome: hollow
gongyl: solid
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Postby Keiji » Sat Dec 20, 2003 6:04 pm

well i'm sorry, but i have to contradict you here.

if you cut open a football, you will find it is hollow, so it is best to use ball = hollow. that makes sphere a solid.

also, a disc relates to a cd, which as we all know is a solid circle with a smaller solid circle taken out. which leaves ring and circle - a ring relates to one such that you would wear on your finger, so it is hollow, which leaves circle to be solid.

therefore, if a glome is analogous to a sphere, it must be solid. which leaves gongyl to be hollow.
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Postby Aale de Winkel » Sat Dec 20, 2003 6:26 pm

Bobxp, alkaline is right here,
In every mathematicle textbook I've ever seen the circle is: R [ cos(α) , sin(α) ] and so is the hollow object.
The "football" is nothing more then an inflateble balloon, and is certainly NOT the mathematical "ball";
the disc: (r = 0..R) [ cos(α) , sin(α) ] (my adopted shorthand for the filled circle)
the mathematical "ball" is simular defined as: x[sup]2[/sup] + y[sup]2[/sup] + z[sup]2[/sup] < R and is thus by definition the filled sphere.
sphere: x[sup]2[/sup] + y[sup]2[/sup] + z[sup]2[/sup] = R

(Hope to see a polyshape thread on this from Polyhedron Dude when I'm back from my holliday)
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Postby Keiji » Sat Dec 20, 2003 6:46 pm

well it is easier and more practical to think of a ball as hollow and a sphere as solid.
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Postby Aale de Winkel » Sat Dec 20, 2003 6:53 pm

bobxp wrote:well it is easier and more practical to think of a ball as hollow and a sphere as solid.


You get a lot of mathematicians against this point of view, everyone sees the sphere as hollow. To change t your way of thinking is far too IMpracticle.

You cought me just prior to editing out the mishap in my previous post, The tubes just airs the "sikaris episode" of voyager (mentioned elsewhere)
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Postby Keiji » Sat Dec 20, 2003 6:57 pm

Aale de Winkel wrote:
bobxp wrote:well it is easier and more practical to think of a ball as hollow and a sphere as solid.


You get a lot of mathematicians against this point of view, everyone sees the sphere as hollow. To change t your way of thinking is far too IMpracticle.


Then why did the mathematicians decide on these definitions then... :roll:

You cought me just prior to editing out the mishap in my previous post, The tubes just airs the "sikaris episode" of voyager (mentioned elsewhere)


What?
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Postby Aale de Winkel » Sat Dec 20, 2003 7:39 pm

bobxp wrote:Then why did the mathematicians decide on these definitions then... :roll:
:cry:

Unfortunately I can't travel faster then light, and ask the ancient Greek for the reason why the define the circle as a circle and not as the disk.
(I haven't the foggiest wheter this can be found anywhere) All I know is that the objects "cicle", "sphere" are defined with an '=' sign and not with an '<' sign.
These things are simply solidly defined for centuries, and should therefore be defined as they are :!: :twisted:
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Postby alkaline » Sun Dec 21, 2003 2:54 pm

well, the Greeks spoke Greek, and not english. The english terms for these concepts are more recent.
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Postby Aale de Winkel » Mon Dec 22, 2003 11:33 am

So(?) the concepts Pythagoras had already are translated through the centuries, with as far as my knowledge goes onto the terms I use.
But I'm NOT a mathematical historian, as I just uploaded onto the polyshape forum I uploaded my formalae onto a private page:
http://home.wanadoo.nl/aaledewinkel/Enc ... hapes.html
to be augmented when I'm back from a little holliday.
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