undefined fundamental surface in hyperbolic geometry

28/05/2010 - 16:55 von gudi | Report spam
Two of the 5 Platonic solids can be formed thus:

A unit circle has angle 360 deg at center divided into 4/6 parts of
90/60 degrees each.

Gauss Curvature positive.

One fraction is cut and _removed_ so that 3/5 sectors are respy left.
Other corners are made up into vertices of regular polygons ( flat
squares and triangles) and when continuation surface is assembled
repetitively, form a cube/ icosahedron, inscribable in a sphere as
platonic solids with rotational symmerties.

Gauss Curvature is negative.

Now one extra fraction is added so that 5/7 sectors respy come
together at a vertex/center in warped assembly. These are made up into
regular polygons ( flat squares and triangles) and when continuation
surface is assembled repetitively, form a new hyperbolic surface,
embeddable on a beautifully warped surface that has not been described
anywhere, imho, upto this point of time. Its existence has also never
been investigated.

Am I right?

I have left out other platonic bodies in the above, but the similar
comments apply.

With high regards

Narasimham
 

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#1 Tim Golden BandTech.com
28/05/2010 - 17:19 | Warnen spam
On May 28, 10:55 am, gudi wrote:
Two of the 5 Platonic solids can be formed thus:

A unit circle has angle 360 deg at center divided into 4/6 parts of
90/60 degrees each.

Gauss Curvature positive.

One fraction is cut and _removed_ so that 3/5 sectors are respy left.
Other corners are made up into vertices of regular polygons ( flat
squares and triangles) and when continuation surface is assembled
repetitively, form a cube/ icosahedron, inscribable in a sphere as
platonic solids with rotational symmerties.

Gauss Curvature is negative.

Now one extra fraction is added so that 5/7 sectors respy come
together at a vertex/center in warped assembly. These are made up into
regular polygons ( flat squares and triangles) and when continuation
surface is assembled repetitively, form a new hyperbolic surface,
embeddable on a beautifully warped surface that has not been described
anywhere, imho, upto this point of time. Its existence has also never
been investigated.

Am I right?



Well, partially you are right, but without the folds I have done this:
http://bandtechnology.com/ConicalStudy/conic.html
I beleive that the Gaussian curvature is zero, except at the
singularity, where it is infinitely negative when inserting area. I
wish that I could name these 'superplane' and 'subplane', for that is
the most pure naming possible. We can build a cone as, say, a 0.523
plane. Likewise we can build a 1.523 plane, or even a 4 plane.

Didn't we discuss this a long time ago and you put up some graphics in
Mathematica? Maybe that was somebody else. Anyway, I have not
attempted to formally publish this work. I congratulate you if you
found this on your own.

- Tim


I have left out other platonic bodies in the above, but the similar
comments apply.

With high regards

Narasimham

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