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AN EXPERIMENTAL STUDY OF SELECTED REGULAR POLYTOPES IN HYPER-SPACE AND THEIR REPRESENTATIONS IN TWO AND THREE DIMENSIONS
David G. Gossman
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
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.
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
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.)
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 Ç0xnÇ0xn-1...Ç0x1 dx1 dx2...dxn.
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.
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.
The basic formulas for a hyper-cube are very simple. The hyper-volume of the n-dimensional hyper-cube is xn where x is equal to a one-dimensional measurement. The formula for Nj where No = number of vertices, N1 = number of edges, N2 = number of faces, etc. is 2n - j(n/j).(3)
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.
The simplex consists of an extension of one point for an extension of each dimension. The formula for Nj where No = number of vertices, N1 = number of edges, N2 = number of faces, etc. is ((n+1)/(j+1)), for the regular simplex of n dimensions.(4)
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.
The general formula for a sphere is taken from Regular Polytopes by Coxeter and is as follows:
Let Sn denote the (n-1)-dimensional content or "surface" of an n-dimensional sphere of unit radius; eg., S1 = 2, S2 = 2n. Then the "surface" of a sphere of radius r is, of course, Snrn-1, and the n-dimensional content or "volume" of a sphere of radius r is:
Equation IV.1Ç0R Snrn-1 dr = (Sn/n)Rn
An expression for Sn (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:
Ç0ve-r2 Sn rn-1 dr = Ç-vvÇ-vv...Ç-vv e-x12-x22 . . . -xn2 dx1 dx2 ...dxn = ( Ç-vve -x2 dx)n.
But the integrals involved are gamma functions: in fact,
2Ç0v e-r2 r2m-1 dr = Ç0ve-t tm-1 dt = G(m)
Ç-vve-x2 dx = 2Ç0ve-x2 dx = G(½).
½SnG((½)n) = [G(½)]n.
Since S2 = 2p, the case where n=2 yields the well-known value G(½) = p½. Thus
Equation IV.2 Sn = 2p ½ /G ((½)n);
e.g., S4 = 2p2. Since G(m+1) = mG(m), it follows from Equation IV.1 that the n-dimensional content (for radius R) is:
Equation IV.3 Sn(Rn)/n = p(½)nRn/G((½)n+1).
The particular values of Sn are very easily computed with the aid of the recurrence formula Sn+2 = 2pSn/n, which states that the (n+1)-dimensional sphere is 2p times the n-dimensional content of the n-dimensional unit sphere; e.g., S2/2 =p and S3/3 =(4/3)p , so S4 = 2p2 and S5 = (8/3)p2.(7)
A similar formula for volume could be obtained by the operation;
Ç0xnÇ0xn-1Ç0(r2-x22-x32...xn2)1/2 ... dx2 dx3 ... dxn .(8)
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.
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.
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.
5. Ibid, p. 120.
6. Ibid, Plate VI, Figure 5.
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.
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.
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.