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AN EXPERIMENTAL STUDY OF SELECTED REGULAR POLYTOPES IN HYPER-SPACE AND THEIR REPRESENTATIONS IN TWO AND THREE DIMENSIONS

By

December, 1972

TABLE OF CONTENTS

I. THE PROBLEM AND DEFINITIONS OF TERMS USED

II. THE HYPER-CUBE

III. THE SIMPLEX

IV. THE SPHERE

V. NEW METHOD OF REPRESENTING HYPER-CUBES IN TWO DIMENSIONS

VI. CONCLUSION

**CHAPTER I**

**THE PROBLEM AND DEFINITIONS OF TERMS USED**

In recent years much study has gone into the math and geometry of dimensions
greater than three.^{(1)} Despite this flood
of interest, though, no easy and comprehensive way of geometrically representing
regular hyper-polytopes in two or three dimensions has yet been laid out.

**The Problem**

__Statement of the Problem__

It is the purpose of this study (1) to observe the mathematical properties of a few selected regular polytopes in hyper-space; (2) to note the existing methods of representing these selected regular polytopes in two and three dimensions; (3) to attempt to find new and more comprehensive methods of representing regular polytopes in two and three dimensions.

__Importance of the Study__

The development of a system for easily representing regular polytopes
in two and three dimensions would be of considerable help to mathematicians
and theoretical physicists dealing with the differential aspect of relativity.
In spite, however, of the recent flood of interest in this subject, there
are very few recent publications on the topic in English.^{(2)}
This puts a limitation on the study as did the fact that there were no
three dimensional models of hyper-solids or descriptions thereof available.

**Definitions of Terms Used**

__"Hyper-"__

The prefix, "hyper-", is used to indicate that the geometrical term is being expanded into dimensions that are greater than its usually implied dimensions. One example is "hyper-cube". This term is used to describe a polytope which has the qualities of a cube such as equal sides and all right angles, but it exists in more than three dimensions as is usually implied. (Note: A hyper-cube is not to be confused with a tesseract which is a four-dimensional hyper-cube.)

__Regular Polytopes__

A regular polytope is a finite convex region on n-dimensional space
enclosed by a finite number of lines. This provides for the hyper-volume
to be equal to the finite integral Ç_{0}^{xn}Ç_{0}^{xn-1}...Ç_{0}^{x1}
dx_{1} dx_{2}...dx_{n}.

__Simplex__

The simplex is the simplest possible polytope as is implied by its name. It consists of one vertex for every dimension, and each vertex is connected by line segments to every other vertex.

__Sphere__

The conventional definition of a sphere is not used in this study, rather it is defined as the set of points equidistant from a given point no matter what the dimension.

**CHAPTER II**

**THE HYPER-CUBE**

**Basic Formulas**

The basic formulas for a hyper-cube are very simple. The hyper-volume
of the n-dimensional hyper-cube is x^{n} where x is equal to a
one-dimensional measurement. The formula for N_{j} where N_{o}
= number of vertices, N_{1} = number of edges, N_{2} =
number of faces, etc. is 2^{n - j}(n/j).^{(3)}

**Dimensional Representations**

It is theorized that if a three dimensional cube can be represented in two dimensions, then a tesseract can be represented in three dimensions in a similar manner. The process of making the representation of a cube in two dimensions is shown in Figure II.1. The process of constructing a three-dimensional representation of a tesseract is similar. Two cubes of wire frame are intersected and pieces of wire of appropriate length are connected to the corresponding vertices. This representation has the right number of vertices, line segments, square faces, and cubes.

The next step was to draw a good representation of a tesseract in two dimensions. Figures II.2, II.3, and II.4 all show different methods of representing a tesseract in two dimensions. All of these methods are faulty in their visualization in that they have overlapping lines. Figure II.5 was produced by changing slightly the point of projection of Figure II.4 This produces a slightly warped but good representation of a tesseract in two dimensions.

It follows that if a tesseract can be represented in two dimensions, then a five-dimensional hyper-cube can be represented in three dimensions. This can easily be done by taking two of the wire frame representations of a tesseract, intersecting them, and adding all of the lines connecting the vertices. This representation has to be warped in a similar manner to Figure II.5 to eliminate the overlapping lines.

Figure II.1 - Process of making a two dimensional representation of a cube.

1. A simple square with a center point.

2. Intersect another square.

3. Segments are added between corresponding vertices.

Figure II.2 - A two-dimensional representation of a four-dimensional hyper-cube.

Figure II.3 - A two-dimensional representation of a four-dimensional hyper-cube.

Figure II.4 - A two-dimensional representation of a four-dimensional hyper-cube.

Figure II.5 - A warped two-dimensional representation of a four-dimensional hyper-cube.

**CHAPTER III**

**THE SIMPLEX**

**Basic Formula**

The simplex consists of an extension of one point for an extension of
each dimension. The formula for N_{j} where N_{o} = number
of vertices, N_{1} = number of edges, N_{2} = number of
faces, etc. is ((n+1)/(j+1)), for the regular simplex of n dimensions.^{(4)}

**Dimensional Representations**

The method of making a representation of a four-dimensional simplex in three dimensions is similar to that of representing a pyramid in two dimensions, which is shown in Figure III.1. To make a wire frame representation of a four dimensional simplex, a regular pyramid is first assembled. A center is then determined and four pieces of wire are attached between this point and each of the vertices of the pyramid.

Figures III.2^{(5)} and III.3^{(6)}
are two-dimensional representations of four-dimensional simplexes. The
problem with Figure III.3 is again the overlapping lines.

Figure III.1 - A regular pyramid represented in two dimensions.

1. Triangle with a center point.

2. Line segments are used to connect the center point to the vertices.

Figure III.2 - A two-dimensional representation of a four-dimensional simplex.

Figure III.3 - A two-dimensional representation of a four-dimensional simplex.

**CHAPTER IV**

**THE SPHERE**

**Basic Formulas**

The general formula for a sphere is taken from __Regular Polytopes__
by Coxeter and is as follows:

Let S_{n} denote the (n-1)-dimensional content or "surface"
of an n-dimensional sphere of unit radius; eg., S_{1} = 2, S_{2}
= 2_{n}. Then the "surface" of a sphere of radius r is, of course,
S_{n}r^{n-1}, and the n-dimensional content or "volume"
of a sphere of radius r is:

Equation IV.1Ç_{0}^{R}
S_{n}r^{n-1} dr = (S_{n}/n)R^{n}

An expression for S_{n} (as a function of n) can be obtained
very neatly by comparing these two methods of integration as applied to
the special function e^{-r2}. We have:

Ç_{0}^{v}e^{-r2}
S_{n} r^{n-1} dr = Ç_{-v}^{v}Ç_{-v}^{v}...Ç_{-v}^{v}
e^{-x12-x22} ^{. .
. -xn2} dx_{1} dx_{2} ...dx_{n}
= ( Ç_{-v}^{v}e
^{-x2}
dx)^{n}.

But the integrals involved are gamma functions: in fact,

2Ç_{0}^{v}
e^{-r2} r^{2m-1} dr = Ç_{0}^{v}e^{-t}
t^{m-1} dt = G(m)

and

Ç_{-v}^{v}e^{-x2}
dx = 2Ç_{0}^{v}e^{-x2}
dx = G(½).

Hence

½S_{n}G((½)n) = [G(½)]^{n}.

Since S_{2} = 2p, the case where
n=2 yields the well-known value G(½)
= p^{½}. Thus

Equation IV.2 S_{n} = 2p ^{½}
/G ((½)n);

e.g., S_{4} = 2p^{2}. Since
G(m+1)
= mG(m), it follows from Equation IV.1 that
the n-dimensional content (for radius R) is:

Equation IV.3 S_{n}(R^{n})/n = p^{(½)n}R^{n}/G((½)n+1).

The particular values of S_{n} are very easily computed with
the aid of the recurrence formula S_{n+2} = 2pS_{n}/n,
which states that the (n+1)-dimensional sphere is 2p
times the n-dimensional content of the n-dimensional unit sphere; e.g.,
S_{2}/2 =p and S_{3}/3 =(4/3)p
, so S_{4} = 2p^{2} and S_{5}
= (8/3)p^{2}.^{(7)}

A similar formula for volume could be obtained by the operation;

Ç_{0}^{xn}Ç_{0}^{xn-1}Ç_{0}^{(r2-x22-x32...xn2)1/2}
... dx_{2} dx_{3} ... dx_{n} .^{(8)}

**Dimensional Representation**

The representation of a four-dimensional sphere in three dimensions can be likened to the method of representing a three-dimensional sphere in two dimensions as shown in Figure IV.1. To build a three-dimensional representation of a four-dimensional sphere, a sphere similar to that in Figure IV.1 should be assembled using a wire frame. Then two or three ellipsoids of lessening minor radii should be assembled inside this sphere with their tops and bottoms connected to the top and bottom of the sphere.

Figure IV.1 - A two-dimensional representation of a three-dimensional sphere.

CHAPTER V

NEW METHOD OF REPRESENTING HYPER-CUBES IN TWO DIMENSIONS

A new method of representing any hyper-cube is based upon the observation
of a two-dimensional representation of a cube (Figure V.1) by Dr. Laatsch
of Miami University.^{(9)} A particular relationship
is apparent among the base points A, B, and C. It is that they are not
directly connected to each other, however, they are all directly connected
to point q. It is apparent that a tesseract
can be represented in a similar manner as shown in Figure V.2. Figure V.3
goes one step further and shows a five-dimensional hyper-cube in two dimensions.
Any n-dimensional hyper-cube can be represented in this manner by using
n number of basepoints on a straight line, connecting them all to a sub-point
q
and building the representation up from there.

Figure V.1 - A two-dimensional representation of a three-dimensional hyper-cube.

Figure V.2 - A two-dimensional representation of a tesseract.

Figure V.3 - A two-dimensional representation of a five-dimensional hyper-cube.

**CHAPTER VI**

**CONCLUSION**

**New Developments**

__The Sphere__

The method of representing a four-dimensional sphere in three dimensions was not found in any of the literature and might provide a new tool for mathematicians working geometry or algebra in greater than three dimensions.

__New Method of Representing Hyper-Cubes in Two Dimensions__

The new method of representing hyper-cubes of any dimension in two dimensions is mathematically correct. It too could provide mathematicians with a new tool for work with n-dimensions.

**A Remaining Problem**

Although this paper establishes some new methods of representing regular polytopes in two and three dimensions, it fails to solve the problem of designing a method where regular polytopes can be represented in two and three dimensions and the shape and properties of these polytopes in their dimensions can be readily visualized.

1. H.S.M. Coxeter, __Regular Polytopes__ (New
York: Macmillan Company, 1963), Preface.

2. D.M.Y. Sommerville, __An Introduction to the
Geometry of N Dimensions__ (New York: Dover Publications, Inc., 1958),
Preface.

3. H.S.M. Coxeter, __Regular Polytopes__ (New
York: Macmillan Company, 1963), pp. 290, 294.

4. H.S.M. Coxeter, __Regular Polytopes__ (New
York: Macmillan Company, 1963), pp. 290, 294.

7. Coxeter, __op. cit.__, p.p. 125-126.

8. Discussion with Dr. Richard Conklin and Dr. Darryl Steinert of Hanover College, Hanover, Indiana, November 11, 1972.

9. Letter from Dr. Laatsch of Miami University, Oxford, Ohio, October 24, 1972.

Bibliography

1. H.S.M. Coxeter, __Regular Polytopes__. New York: Macmillan Company,
1963.

2. D. Hilbert and S. Cohn-Vossen, __Geometry and the Imagination__.
New York: Chelsa Publishing Company, 1952, (translated by P. Nememyi).

3. D.M.Y. Sommerville, __An Introduction to the Geometry of N-dimensions__.
New York: Dover Publications, Inc., 1958.

Other Sources

1. Personal Correspondence of the Author:

Letter from Dr. Laatsch of Miami University, Oxford, Ohio, October 24, 1972.

Discussion with Dr. Richard Conklin and Dr. Darryl Steinert of Hanover College, Hanover, Indiana, November 11, 1972.