N-D Geometrical Figures

by Bill McKeeman

Using MATLAB to generate and display 3 and 4 dimensional geometric figures.

Updated March 7, 2006


Polyhedra and Polytopes

Coxeter's Regular Polytopes describes the analog of Polyhedron for all dimensions of Euclidian N-Space. The generic names as N increases from 0 to 4 are

There are only three trivial cases for each N>4, so there is neither much interest nor terminology. The three are called the n-simplex, the n-cross and the n-measure.

The main trick in computing the figures is using permutations to generate the vertex sets and then post-analysis of the vertex sets to extract other features.

I looked up some formulae. Let C stand for all Circular shifts of a vector, P for all permutations of a vector and E for all even permutations. The "names" {p,q} are the Schlafli symbols for the five regular figures in 3-D and {p,q,r} are the same for the six regular figures in 4-D.

Vertex Sets

There are many related kinds of figures (concave, expansions, prisms) in the Mathematica pages but they are not treated here. You could also look at some of these other things in The Derived Polytopes in Euclidian N-Space, William Marshall McKeeman, Master of Science thesis, The Columbian College of The George Washington University, (June 7, 1961). Anyway, back to the main story:

Computing Vertices

Using a permutation generator, we can generate the vertex sets for regular polyhedra with edge length 1. For instance, the cube {4,3} with unit edge centered on the origin has 8 vertices and the analogous hypercube {4,3,3} has 16. They could be generated as follows

s43  = perms([1 1 1]/2, 'signs');             % cube
s433 = perms([1 1 1 1]/2, 'signs');           % hyper cube

The details are tucked away in my MATLAB function vertices. Here are the calls.

s33 = vertices('s33');                           % tetrahedron
s34 = vertices('s34');                           % octahedron
s43 = vertices('s43');                           % cube
s35 = vertices('s35');                           % icosahedron
s53 = vertices('s53');                           % dodecahedron

Feature Count for 3-D Regular Polyhedra

The vertex sets can be examined to count the edges and faces.

format compact
fprintf ' tetra   octa   cube  icosa dodeca\n'
fprintf('%6d %6d %6d %6d %6d   vertices\n', ...
fprintf('%6d %6d %6d %6d %6d   edges\n', ...
fprintf('%6d %6d %6d %6d %6d   faces\n', ...
 tetra   octa   cube  icosa dodeca
     4      6      8     12     20   vertices
     6     12     12     30     30   edges
     4      8      6     20     12   faces

Draw the Tetrahedron, Octahedron, Cube, Icosahedron, Dodecahedron

The regular polyhedra and polytopes can be displayed using MATLAB. The displays are "shadows of edges" (technically projections onto the xy plane). In the case of 3-D figures, the view is immediately intuitive to us 3-D people. Taking the viewpoint of a Flatlander (2-D person) living in the xy plane gives some flavor of the mental gyrations you will have to go through to make sense of the 4-D figures.

The edges are color-coded by distance from the viewer so that connected edges tend to have the same shade. In the 3-D case only black is used. In the 4-D case, blue and red are used to indicate the two missing dimensions. The plots have to be offset from the origin to keep them from falling on top of each other. The z dimension is represented by intensity (imagine viewing the figures in fog).

rot3 = ndrotate([0 .2 .4; 1 0 .4]);    % 3D rotation matrix
angles = [...
   0 .1 .2 .3;
  .3  0 .4 .6;
  .5 .6  0 .8;
  .1 .1 .1  0];
rot4 = ndrotate(angles);               % 4D rotation matrix
offset = @(M,v) M+repmat(v, size(M,1),1);

hold on
%               vertices   offsets
plotpoly(offset(s33*rot4, [1.7 0.7 0 0]));
plotpoly(offset(s34*rot3, [-.5 0 0]));
plotpoly(offset(s43*rot3, [4 0 0]));
plotpoly(offset(s35*rot3, [4 3 0]));
plotpoly(offset(s53*rot3, [0 3 0]));
hold off

Bucky Ball

The Buckminster Fuller ball is made of pentagons and hexagons. It is constructed by truncating the 1/3 tip of the 20 vertices of the icosahedron s35. The vertices function (above) does not "know" bucky, so here is the vertex computation using the permutation primitives.

d = @(a,b) a + b*(1+sqrt(5))/2;
buckyball = perms(...
  [d(0,0), d(0,3), d(1,0)
   d(1,0), d(0,2), d(2,1)
   d(2,0), d(0,1), d(1,2)]/2, 'cycles', 'signs', 'unique');

fprintf('v=%d e=%d f=%d\n', ...
  size(buckyball,1), nedge(buckyball), nface(buckyball));
v=60 e=90 f=32

4-D Polytopes

Using the Coxeter formulae for the six 4-D polytopes:

s333 = vertices('s333');                         % 4d simplex
s334 = vertices('s334');                         % 4d cross
s343 = vertices('s343');                         % 24 cell
s433 = vertices('s433');                         % 4d measure
s335 = vertices('s335');                         % hyper icosahedron
s533 = vertices('s533');                         % hyper dodecahedron

Feature Count for 4-D Regular Polyhedra

The vertex sets can be examined to locate the edges.

fprintf '  s333   s334   s433   s343   s335   s533\n'
fprintf('%6d %6d %6d %6d %6d %6d   %s\n', ...
  size(s333,1),  size(s334,1),  size(s433,1), ...
  size(s343,1),  size(s335,1),  size(s533,1),...
fprintf('%6d %6d %6d %6d %6d %6d   %s\n', ...
  nedge(s333),   nedge(s334),   nedge(s433), ...
  nedge(s343),   nedge(s335),   nedge(s533),...
  s333   s334   s433   s343   s335   s533
     5      8     16     24    120    600   vertices
    10     24     32     96    720   1200   edges

Edge Plot of 4-D Cross Polytope (analog of octahedron)

set(gcf, 'color', 'white');

Edge Plot of Hypercube (tesseract)

set(gcf, 'color', 'white');

Edge Plot of 4-D 24-cell (unique to 4-D)

set(gcf, 'color', 'white');

Edge Plot of 4-D 600 Cell (analog of icosahedron)

set(gcf, 'color', 'white');

Edge Plot of 4-D 120 Cell (analog of dodecahedron)

set(gcf, 'color', 'white');
angles = [...                            very small angles
   0  .1  .1  .03;
  .01  0  .04 .06;
  .13 .06  0  .12;
  .13 .1  .1   0];
rot4 = ndrotate(angles);               % a better viewpoint

N-D for N>4

As it turns out, there are only three regular figures from N=5 on up. I could put some here, but they get really busy. So I stop at 4.