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Volumes of Hyperspace Enclosed by Regular Polygons


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Friends,

I just saw video on "U" Tune discussing "N"-Dimensional polytopes. Each polytope is constructed using polytopes from  the N-1 Dimension.

A Hyper-Tetrahedron is a volume bounded by Tetrahedra. A 5-D Hyper-Tetrahedra would  be bounded by 4-D Hyper-Tetrahedra. A 6-D Hyper-Tetrahedra would be constructed of 5-D Hyper-Tetrahedra, und so weiter.

BUT what if I want to enclose a volume of Hyperspace using only 2-Dimensional Polygons—all of the same size and shape—instead of using Polytopes?

Can I completely enclosed a volume of Hyperspace using ONLY, Triangles; OR ONLY Squares OR ONLY Pentagons—or whatever?

If I can create a regular, non-concave volume that more or less meets the definition of a Polyhedron, but in higher dimensional space—WHAT do I call the Damned thing?

What are some of the more interesting—or fundamental—of these higher-dimensional "Polyhedrons"?

Are there regular and fundamental "N-Hedrons" that might use two or more polygons to uniquely contain a space?

 

Thanks!

 

…..Saxon Violence

Edited by SaxonViolence
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I don’t know if this is any help to you, but I absolutely love this essay!

“Just as a polyhedron can be constructed by assembling together suitable polygons, so a higher polytope may be constructed by assembling together suitable polytopes of one lower dimension”

It seems reasonable to me then, that a volume of Hyperspace can be enclosed by assembling together suitable polytopes, which can all be decomposed as much as you like, all the way down to nothing but points, line segments, plane regions and so on.

In fact, that is exactly what the essay says:

“Meanwhile topology relied on the decomposition of a surface into polygons (or a hypersurface into polytopes) in order to study its overall form. This gave rise to the idea of incidence complexes as sets of geometric elements such as points, line segments, plane regions and so on. An n-polytope (a polytope immersed in some n-space) was just such a decomposition of some unbounded (n−1)-manifold or (n−1)-space. The decompositions equivalent to polytopes became known as CW complexes. Significantly, such polytopes need not be convex

The best part comes right at the end: “Whether this affects the criteria you care about is up to you”

As for what to call it, if you make it, you get to call it whatever you want, although I do like the sound of the “SaxonViolence hypermonstertope”

 

 

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On 3/5/2022 at 1:12 AM, SaxonViolence said:

Friends,

I just saw video on "U" Tune discussing "N"-Dimensional polytopes. Each polytope is constructed using polytopes from  the N-1 Dimension.

A Hyper-Tetrahedron is a volume bounded by Tetrahedra. A 5-D Hyper-Tetrahedra would  be bounded by 4-D Hyper-Tetrahedra. A 6-D Hyper-Tetrahedra would be constructed of 5-D Hyper-Tetrahedra, und so weiter.

BUT what if I want to enclose a volume of Hyperspace using only 2-Dimensional Polygons—all of the same size and shape—instead of using Polytopes?

Can I completely enclosed a volume of Hyperspace using ONLY, Triangles; OR ONLY Squares OR ONLY Pentagons—or whatever?

 

The answer to your question is contained in your post.  With 2D polygons (all the same size and shape, if you want) you can construct 3D polyhedra.  Then from those 3D polyhedra you can construct 4D polytopes, and from there 5D, etc.

 

One of the more interesting features of higher dimensions is the fact that surface areas and corners become much larger than bulk volumes.  Of course you may say that's nonsense, since areas and volumes have different units, so how can I compare them.  But it is true nevertheless in many different senses.  For instance, the ratio of the surface area of a unit sphere to its volume starts to decrease above 7 (if I remember correctly) dimensions.  Nearly all the volume of a higher dimensional cube is in its corners.  The number of corners grows much faster than the number of faces, so most vectors point towards corners rather than towards faces, etc.  This is particularly apparent when you study polytopes in high numbers of dimensions.

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On 3/6/2022 at 11:57 PM, OceanBreeze said:

I don’t know if this is any help to you, but I absolutely love this essay!

“Just as a polyhedron can be constructed by assembling together suitable polygons, so a higher polytope may be constructed by assembling together suitable polytopes of one lower dimension”

It seems reasonable to me then, that a volume of Hyperspace can be enclosed by assembling together suitable polytopes, which can all be decomposed as much as you like, all the way down to nothing but points, line segments, plane regions and so on.

In fact, that is exactly what the essay says:

“Meanwhile topology relied on the decomposition of a surface into polygons (or a hypersurface into polytopes) in order to study its overall form. This gave rise to the idea of incidence complexes as sets of geometric elements such as points, line segments, plane regions and so on. An n-polytope (a polytope immersed in some n-space) was just such a decomposition of some unbounded (n−1)-manifold or (n−1)-space. The decompositions equivalent to polytopes became known as CW complexes. Significantly, such polytopes need not be convex

The best part comes right at the end: “Whether this affects the criteria you care about is up to you”

As for what to call it, if you make it, you get to call it whatever you want, although I do like the sound of the “SaxonViolence hypermonstertope”

 

 

So, IF I understand you correctly, the polygons are already there and you could call grouping them into polytopes, simply an easy way to keep track of them.

How does one even begin to understand things like stellated polyhedra; polyhedral compounds—or what you get if you rotate stellated polyhedra in the 4th Dimension comparable to a 4-D Torus?

It is like WOW MAN!

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