%% N-D Geometrical Figures
% by Bill McKeeman
%
% Using MATLAB to generate and display 3 and 4 dimensional geometric figures.
%
% Updated February 26, 2008
%
% 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
%
% * point
% * segment
% * polygon
% * polyhedron
% * polytope (or polychoron)
%
% 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
%
% $$ \tau = \frac{1+\sqrt 5}{2} $$
%
% $$ \begin{array}{rcl}
% \{3,3\} &:\ & C(\frac{\sqrt 2}{2}, 0, 0, 0) \\
% \{3,4\} &:\ & C(\pm \frac{\sqrt 2}{2}, 0, 0) \\
% \{4,3\} &:\ & \pm\frac{1}{2}, \pm\frac{1}{2}, \pm\frac{1}{2} \\
% \{3,5\} &:\ & C(0, \pm\frac{1}{2}, \pm\frac{\tau}{2}) \\
% \{5,3\} &:\ & C(0, \pm\frac{1}{2}, \pm\frac{\tau^2}{2})
% \cup \pm\frac{\tau}{2}, \pm\frac{\tau}{2}, \pm\frac{\tau}{2}
% \end{array} $$
%
% $$ \begin{array}{rcl}
% \{3,3,3\} \ &:\ & C(\frac{\sqrt 2}{2}, 0, 0, 0, 0) \\
% \{3,3,4\} \ &:\ & C(\pm\frac{\sqrt 2}{2}, 0, 0, 0) \\
% \{3,4,3\} \ &:\ & P(\pm\frac{\sqrt 2}{2}, \pm\frac{\sqrt 2}{2}, 0, 0) \\
% \{4,3,3\} \ &:\ & \pm\frac{1}{2},\pm\frac{1}{2},\pm\frac{1}{2},
% \pm\frac{1}{2}
% \end{array} $$
%
%
% $$ \begin{array}{rcl}
% \{3,3,5\} &:& C(\pm\tau,0,0,0) \\
% &\cup\ &
% C(\pm\frac{\tau}{2},\pm\frac{\tau}{2},\pm\frac{\tau}{2},
% \pm\frac{\tau}{2})\\
% &\cup\ &
% E(\pm\frac{\tau^2}{2},\pm\frac{\tau}{2},\pm\frac{1}{2}, 0)
% \end{array} $$
%
%
% $$ \begin{array}{rcl}
% \{5,3,3\} &:&
% P(0,0,\pm\tau^2,\pm\tau^2) \\
% &\cup\ &
% P(\pm\frac{\tau^2}{2},\pm\frac{\tau^2}{2},\pm\frac{\tau^2}{2},
% \pm\sqrt 5\frac{\tau^2}{2})\\
% &\cup\ &
% P(\pm\frac{\tau^3}{2},\pm\frac{\tau^3}{2},\pm\frac{\tau^3}{2},
% \pm\frac{1}{2}) \\
% &\cup\ &
% P(\pm\frac{\tau}{2},\pm\frac{\tau}{2},
% \pm\frac{\tau}{2},\pm\frac{\tau^{-2}}{2}) \\
% &\cup\ &
% E(\pm\frac{1}{2},\pm\frac{1}{2},\pm\frac{\tau^2}{2},0)\\
% &\cup\ &
% E(\pm\sqrt 5\frac{\tau^2}{2},\pm\frac{\tau}{2},
% \pm\frac{\tau^3}{2},0) \\
% &\cup\ &
% E(\pm\tau^2,\pm\frac{\tau^2}{2},
% \pm\frac{\tau^3}{2},\pm\frac{\tau}{2})
% \end{array} $$
%%
% 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', ...
size(s33,1),size(s34,1),size(s43,1),size(s35,1),size(s53,1));
fprintf('%6d %6d %6d %6d %6d edges\n', ...
nedge(s33),nedge(s34),nedge(s43),nedge(s35),nedge(s53));
fprintf('%6d %6d %6d %6d %6d faces\n', ...
nface(s33),nface(s34),nface(s43),nface(s35),nface(s53));
%% 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);
cla
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));
close(gcf)
figure
plotpoly(buckyball*rot3);
%% 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),...
'vertices');
fprintf('%6d %6d %6d %6d %6d %6d %s\n', ...
nedge(s333), nedge(s334), nedge(s433), ...
nedge(s343), nedge(s335), nedge(s533),...
'edges');
%% Edge Plot of 4-D Cross Polytope (analog of octahedron)
close(gcf)
figure
set(gcf, 'color', 'white');
plotpoly(s334*rot4);
%% Edge Plot of Hypercube (tesseract)
close(gcf)
figure
set(gcf, 'color', 'white');
plotpoly(s433*rot4);
%% Edge Plot of 4-D 24-cell (unique to 4-D)
close(gcf)
figure
set(gcf, 'color', 'white');
plotpoly(s343*rot4);
%% Edge Plot of 4-D 600 Cell (analog of icosahedron)
close(gcf)
figure
set(gcf, 'color', 'white');
plotpoly(s335*rot4);
%% Edge Plot of 4-D 120 Cell (analog of dodecahedron)
close(gcf)
figure
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
plotpoly(s533*rot4);
%% 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.
close(gcf)
