Although this section is intended primarily for those already familiar with the broad outlines of the theory of regular polytopes, let us recall a few salient points:
In n-dimensional space, there are always the three primitive polytopes: the simplex, the cross polytope, and the hypercube.
The simplex (generalization of the triangle and tetrahedron) is the simplest polytope (with n-dimensional content) which can be constructed in an n-space. An n-simplex is bounded by (n-1)-simplexes, and has an (n-1)-simplex for its vertex figure. Regular simplexes are self-dual.
The cross polytope (generalization of the square and Platonic octahedron) is named after the "cross" or frame of mutually perpendicular Cartesian coordinate axes which can be constructed in an n-space. There are just n such axes, and points equidistant from the origin along each axis, in both directions, are the vertices of a cross polytope. Cross polytopes are bounded by regular (n-1)-simplexes and have for vertex figures (n-1)-cross polytopes. The dual of an n-dimensional cross polytope is an n-cube.
Hypercubes (generalizations of the square and cube) are specially simple zonotopes. They are bounded by (n-1)-cubes and have for vertex figure an (n-1)-simplex. Taken as zonotopes, they are "determined" by a star of n mutually perpendicular equal vectors. Thst is, they are "hyper-solids of translation" along those n vectors, "traced out" by a series of orthogonal translations. The dual of an n-cube is an n-dimensional cross polytope. They close-pack to fill the n-space, and moreover, when of unit edge length, the hypercube is called a measure polytope, and provides the measure for the unit of n-dimensional content.
These three primitive figures have easily defined numbers of vertices, edges, faces, polyhedra, and cells of all kinds. For instance, the n-simplex has (n+1) vertices and (n+1) cells. The cross polytope has 2n vertices, while the dual hypercube has 2n cells. The n-cube has 2^n vertices, and the dual cross polytope has 2^n cells. Similarly, the symmetry groups of these polytopes follow rigid patterns. See Chapter 7 of Coxeter's Regular Polytopes.
Were these the only regular figures, there would be little to attract us to the subject. In space of two dimensions we can construct an infinite number of regular convex polygons, and most of these have various regular star-polygon relatives.
In space of three dimensions (aside from the three universal primitives) we have the Platonic icosahedron and its dual, the pentagonal dodecahedron. These two polyhedra (and their star-polyhedral relatives, the four Kepler-Poinsot polyhedra) have what is called icosahedral symmetry. It is a bit of a miracle, although easily explained in arithmetic terms. Briefly, there are axes of 2-fold rotation (joining the midpoints of opposite edges), 3-fold rotation (joining opposite vertices of the dodecahedron, opposite face-centers of the icosahedron), and 5-fold rotation (joining opposite vertices of the icosahedron, opposite face-centers of the dodecahedron).
In space of four dimensions, we have the 24-cell {3,4,3}, without any perfect analogue in higher or lower spaces. Then we have the 120-cell {5,3,3} and the 600-cell {3,3,5}, and their ten star-polytope relatives, all exhibiting the "hyper-icosahedral" symmetry group.
Archimedes discovered thirteen polyhedra bounded by regular polygons of more than one type, and possessing symmetries such that any vertex can be carried into any other vertex, under operations of the symmetry group. These are the Archimedean solids. When we say "bounded by regular polygons, and possessing symmetries such that any vertex may be transformed into any other," we define a class of polyhedra called the uniform polyhedra; these include the Platonic and Archimedean solids, along with infinite families of prisms and antiprisms, and some fifty other non-convex polyhedra. If we apply a similar criterion to polytopes in 4-space (allowing the polytopes to be bounded, not just by regular polyhedra, but also by uniform polyhedra), and look for the uniform 4-dimensional polytopes, thousands exist. Their complete enumeration is a work in progress, pursued most diligently by George Olshevsky and J. Bowers.
So, in the critical spaces of two, three, and
four dimensions, we have found many beautiful exceptions to the
inexorable rule of the three primitive n-polytopes. Duality engenders
an even broader palette of polytopes; for instance, since the
uniform polyhedra have polygons of one or more types, but vertices
all surrounded alike, their duals invariably have but one type
of (usually irregular) polygonal face. In four dimensions, the
duals of the uniform polytopes have only one type of polyhedral
cell.
Kepler's Rhombic Triacontahedron (left)
and its uniform dual, the Icosidodecahedron. Note that both have
the full icosahedral symmetry. The triacontahedron is determined
by the six vectors (neglecting opposites) from the center to the
vertices of the Platonic icosahedron. It is an orthogonal, isometric
shadow of a 6-cube, cast into 3-space.
The palette of polytopes can be extended much further by considering the stellations of the duals of the uniform polytopes. For instance, the dual of the Archimedean icosidodecahedron is Kepler's rhombic triacontahedron (a zonohedron), and its stellations number into the many millions. Little wonder, that so much remains to be discovered and rediscovered.
Going beyond the uniform polytopes, their duals, and the stellations of their duals, there is an important class of polytopes, many highly symmetrical, which have engaged my attention for years. These are the zonotopes: zonogons, zonohedra, and, well, zonotopes. Let us call an n-zonotope determined by just n vectors which span the n-space, a "primitive" zonotope. Hypercubes are primitive zonotopes.
Zonotopes are always convex. They are "polytopes of translation," for they may be "traced out" by a series of translations. Any set of k vectors (k>=n) which span an n-space, determine an n-zonotope, which is traced out by k translations. Let the k vectors be of equal magnitude, and the determined zonotope is equilateral. If they are equal, orthogonal, and there are but n of them, and a hypercube is traced out. To further develop this idea of "tracing out" I quote from Chapter 7 of H.S.M. Coxeter's Regular Polytopes; after describing constructions for the simplexes and cross polytopes, he continues with the hypercube, a primitive zonotope. I use P(n) for his more elegant Greek Pi-sub-n, denoting an n-dimensional polytope:
"A third series of figures may be constructed as follows. When a point P(0) is moved along a line from an initial to a final position, it traces out a segment P(1). When P(1) is translated (not along its own line) from an initial to a final position, it traces out a parallelogram. Similarly a parallelogram traces out a parallelepiped. The n-dimensional generalization is known as a parallelotope. It has 2^n vertices. The remaining elements are k-dimensional parallelotopes."
Coxeter goes on to remark that the n translations
used on constructing the parallelotope define n vectors, represented
by the n edges that meet at any one vertex. And, if the vectors
are mutually perpendicular, the parallelotope is an orthotope,
the generalization of the rectangle and the "box." If
the n perpendicular vectors all have the same magnitude, the orthotope
is a hypercube.
The ability of equal hypercubes to close-pack and fill an n-space is significant. It is not at all a commonplace, for a regular polytope to fill space. Consider the plane: we may tile it with regular triangles, with squares, or with hexagons. By mixing different types of regular polygon, we may obtain a few more "Archimedean" tilings; however, our options are rather limited. There is no tiling of regular polygons which includes a pentagon, for instance, not to speak of a heptagon (7-gon), or a 17-gon.
In three dimensions, we have the space-filling of cubes, and that is that. There are a few Archimedean space-fillings by mixtures of regular and Archimedean polyhedra. For instance, one may fill 3-space with Platonic tetrahedra and octahedra.
In four dimensions, surprisingly, there are three space-fillings by regular polytopes (the hypercube, the cross polytope, and the 24-cell may all be used to fill the 4-space). But in all higher spaces there is only the hypercube.
Within the context of space-filling, then,
arises a very interesting property of zonotopes: they may always
be dissected into primitive zonotopes, i.e., into zonotopes determined
by just n vectors. Zonogons may always be dissected into parallelograms,
zonohedra into parallelepipeds, and 4-zonotopes into "squished
tesseracts." Of course, if the zonotope is primitive to begin
with, it is "dissected" into itself.
An n=5 polar zonohedron, and one of its
dissections into 10 rhombic hexahedra (of two types). It has only
twenty faces, and thus, is a rhombic icosahedron.
Usually, there is more than one way to dissect a zonotope into its component primitive zonotopes. The complete enumeration of all possible ways to so dissect a zonotope of arbitrary complexity has proved a baffling problem, although we do know unequivocally how many primitive zonotopes compose an n-zonotope determined by k vectors (k>=n): there are exactly "k things taken n at a time" primitive zonotopes (in point of fact, some of the primitive zonotopes may be degenerate, have have zero n-dimensional content).
When n=2 and we dissect zonogons into parallelograms, there are k things taken two at a time, or a "triangular" number of parallelograms. When n=3 and we dissect zonohedra into parallelepipeds, there are k things taken three at a time, or a "tetrahedral" number of parallelepipeds. I looked into these patterns very closely around fifteen years ago, while struggling to find an expression for the volume of the general polar zonohedron. My approach was to sum the volumes of the component rhombic hexahedra, into which any polar zonohedron can be dissected. Incidentally, a rhombic hexahedron is an equilateral parallelepiped.
At this point in the discussion, it is important, principally, to recognize that while generally zonotopes are not space-fillers, they always may be dissected into primitive zonotopes, and thus, comprise a perfect filling of a limited space. Furthermore, the dissections need not split the zonotope down to atomic level; it is possible to dissect most zonotopes into a mixture of primitive zonotopes and "higher" zonotopes. For examples of such dissections, see my Quasicrystal pages.
In Chapter 13 of Regular Polytopes, Coxeter develops an important theme: the orthogonal projection of polytopes into sub-spaces. For instance, a cube has an infinitude of different plane projections, the most regular projections being face-first (a square) and vertex-first (bounded by a regular hexagon). We may obtain "wireframe" projections, in which only the edges and vertices are projected, as in the figure from Regular Polytopes reproduced above; or hidden-detail-removed projections, in the image depicting a polar zonohedron, above. There are many types of projection. One useful variant is the orthogonal shadow: the boundary of a projection, together with its interior. Then its is proper to say that the orthogonal shadow of a cube, taken vertex-first (i.e., projected onto a plane perpendicular to an axis connecting opposite vertices), is a regular hexagon. Note that the regular hexagon can be dissected into three congruent rhombs, in two ways.
We have discovered an instance in which one regular polytope projects into another, or to be more precise, an instance of one regular polytope (the hexagon) being the shadow of another (the cube). It turns out that this is a commonplace. For instance, Kepler's rhombic dodecahedron is an orthogonal shadow of a 4-cube (tesseract) cast vertex-first into a 3-space! Now, it happens that this rhombic dodecahedron can be dissected, in exactly two ways, into four congruent rhombic hexahedra.
If it were an orthogonal projection of the tesseract into the 3-space, it would still be bounded by Kepler's dodecahedron, but the dodecahedron would be dissected into eight interpenetrating congruent rhombic hexahedra (projections of the eight 3-cubes bounding the tesseract).
A significant attribute of both these projections--3-cube into hexagon, 4-cube into dodecahedron--is that they are isometric. The edges of the 3-cube are equally foreshortened under projection into the hexagon; and the edges of the tesseract are equally foreshortened under projection into the dodecahedron. Not only that, but this forces also their dissections into primitive zonotopes, to involve equal, congruent zonotopes.
Consider all possible plane shadows of the cube. Which has greatest area? Consider all possible solid shadows of the tesseract. Which has greatest volume? It turns out that the isometric projection of an n-cube into an (n-1)-space always induces the shadow-of-greatest-content. In this projection, the edges of the n-cube are equally foreshortened. So also are its cells: that is, the squares bounding a cube foreshorten into equal rhombs, the cubes bounding a tesseract foreshorten into equal rhombic hexahedra. And so on.
Here it is worth noting that one of the more symmetrical solid shadows of a tesseract is a right regular hexagonal prism. This may be dissected into three equal parallelepipeds, and one degenerate parallelepiped of zero volume, four altogether, which is equal to "four things taken three at a time."
We have verged upon questions which relate to the classic isoperimetric problems of geometry. Of all closed plane curves of equal length, which contains greatest area? The circle. Of all closed surfaces in 3-space, of equal area, which contains greatest volume? The sphere. And similarly: of all polygons with equal perimeter, which contain greatest area? Regular polygons.
In this last, we see for a moment, and through a glass darkly, that somehow symmetry is related to content. This is really my principal theme, here. Many a time I have attempted to explain my ideas about symmetry, and the contents of shadows, and isometric projections of hypercubes into lower spaces. I can't say I have had much success. To me it seems of supreme importance, to link the symmetry groups of the regular polytopes, to the solution of isoperimetric problems, and the isometric projections of hypercubes. To me, this is a kind of "unified field theory" of polytopes.
Let me remark here that the dissection of the regular hexagon into three equal rhombs, and the dissection of Kepler's rhombic dodecahedron into four equal rhombic hexahedra, both represent "hidden-detail-removed" projections. That is, were one in a 4-space, looking vertex-first at a tesseract, only four of the eight bounding cubes would be visible. The hidden-detail-removed projections of 4-polytopes into a 3-space always induce a close-packing of polyhedra to fill the bounding polyhedron of the projection. The dissection of zonotopes into primitive zonotopes, or into mixtures of primitive zonotopes and higher zonotopes, is related to hidden-detail-removed projections; however, it is significant that not all dissections of a zonotope are equivalent to hidden-detail-removed projections.
To prepare for subsequent developments, recall that a zonohedron is determined by a set of three or more vectors, which span the 3-space. Among the infinitude of different parallelepipeds or "primitive" zonohedra, the cube is both equilateral, and its three determining vectors are mutually perpendicular. These three vectors and their opposites form a Cartesian cross, that is, six points equidistant from the origin on the orthogonal frame of Cartesian coordinate axes. Let us call any set of k vectors in an n-space, with their opposites, a "star" of vectors, while recalling that of all the 2k vectors in the star, really only k are needed to determine and trace out a zonotope (if one used all 2k vectors, the same zonotope would be traced out, but of twice the edge length). The Cartesian cross is itself a star, but a very special one: we shall call it a cross. In summary: every star of vectors which span an n-space determines an n-zonotope.
Returning to isometric projections of hypercubes into lower spaces, and symmetry, and the contents of shadows (how much area, volume, content?), some historical remarks are in order.
The discovery of the regular polytopes is rightly credited to one Ludwig Schlafli, a Swiss, about 1850. He considered the vectors to the vertices of regular polytopes, and he named these eutactic stars. ("Eutactic" means, well-ordered; Schlafli did not consider what zonotopes might be determined by such stars). Almost a century later, Hadwiger showed that all eutactic stars are the orthogonal projections of crosses!
Now, since an n-cross comprises the vectors which determine an n-cube; and since the vectors to the vertices of all regular polytopes are eutactic stars (and of equal magnitude); and since all eutactic stars are orthogonal projections of crosses ...
The vectors to the vertices of any regular polytope form a star, which determines a zonotope which is an orthogonal isometric shadow of a hypercube.
And we might as well spell it out:
All regular polytopes arise from the projection of Cartesian crosses into lower spaces.
We noted the projection (actually, shadowing) of a 3-cube into regular hexagon, and that this is the shadow of greatest area. Now, the 3-cube has octahedral symmetry, and this symmetry group is of order 48; while the regular hexagon has cyclic symmetry and reflective symmetry, and its symmetry group is of order 12. No other shadow of the cube has as much symmetry, as the hexagon; no other shadow has as much area. The same may be said for the rhombic-dodecahedral shadow of the tesseract: no other solid shadow has as much symmetry, no other solid shadow has as much volume.
These considerations occurred to me while pursuing the question of the volume of the general polar zonohedron. These zonohedra have an axis of k-fold cyclic symmetry for all k>=3, but for any given k, an infinite number of different k-polar zonohedra may be constructed, of varying proportions. A polar zonohedron may be oblate and flying-saucer-like, or prolate and spindle-like. These changes in shape may be parameterized by a single angle, which I call "pitch."
To explain, consider first a right regular n-gonal pyramid. It has 2n edges: n basal edges, n lateral edges. Fix your attention on the n lateral edges, radiating from the apex of the pyramid. These may be equated to n vectors of equal magnitude; and their "pitch" is the angle between any one edge, and the basal n-gon. This angle ranges from 0 to 90 degrees. Suppose, first, that the basal polygon is of fixed size. If the lateral edges have a pitch of 0 degrees, they lie in the base, and the pyramid is degenerate, and has no volume. No raise them slowly. Volume appears instantly and increases. At 90 degrees, the pyramid becomes infinitely high, and has infinite volume.
This is not a very fruitful way to look at the stuation, though. Better is to hold the lateral edges to some fixed length, and let the base vary as pitch changes. At pitch=0 the lateral edges again lie in the base and volume is zero. But, at pitch=90 the lateral edges coincide and the basal polygons shrinks to a point, and again volume is zero. Somewhere between the two extremes, a maximum volume occurs; and it turns out that it occurs when pitch is arc tan Sqrt[.5], or 35.2643+ degrees.
Let the n lateral edges of fixed and equal magnitude determine a zonohedron; it is an n-polar zonohedron. Its volume behaves exactly as does that of the related right regular pyramid: zero at pitch=0 and pitch=90, maximum when pitch=35.2643+ degrees.
Recalling that the lateral edges of a right regular pyramid are equally-spaced generators of a cone, then, from Chapter 13 of Regular Polytopes we find that
"Thus the cone of semi-vertical angle arc tan Sqrt[2] has the remarkable property that vectors of equal magnitude, taken in both directions along n symmetrically spaced generators, form a eutactic star, for all values of n."
In other words, whenever a polar zonohedron is determined by a set of n such vectors--when its "pitch" is 35.2643+ degrees and its volume is at a maximum--it is an isometric orthogonal shadow of an n-cube.
Adding more mystique and complexity to the subject, more than one isometric shadow of an n-cube can exist. For instance, both Kepler's rhombic triacontahedron, and an n=6 polar zonohedron of pitch=35.2643+ degrees, are orthogonal, isometric shadows of 6-cubes cast into a 3-space. I have tentatively concluded that in a case such as this, the shaodw with the higher order of symmetry, has the greater volume (or content, in the general case); for Kepler's triacontahedron does have greater volume than the n=6 polar zonohedron, at equal edge lengths. That two different isometric shadows can arise points to the degrees of freedom which obtain, between six and three dimensions.
Regular n-polytopes may be inscribed in n-spheres; the largest is called the circumsphere. In fact, there is an n-sphere touching the centers of all the k-dimensional elements bounding the n-polytope: vertices, edges, faces, polyhedra, and so on. Finally there is the in-sphere, which touches the centers of all the (n-1)-dimensional cells of the polytope. We have noted that circles and spheres are isoperimetric figures. Furthermore, their orthogonal shadows are isoperimetric, for the shadow of an n-sphere cast into a k-space, k<n, is just a k-sphere. We might say that there is a special kind of roundness, which is isoperimetric in quality, and then cast about, searching for the roundest possible shadows of hypercubes, in lower spaces.
What I am saying is that, having found such roundest-possible-shadows, we shall also have found the most symmetrical shadows. Furthermore, having found the roundest, most-symmetrical shadows, we shall also have found the isometric shadows.
The vectors of an n-cross project into the vertices of a regular polytope. Such a set of vertices can only only represent vectors of equal magnitude, since all regular polytopes have circumspheres. Hence the vectors of the cross are equally foreshortened under orthogonal projection; hence the projection is isometric.
Having reiterated the same theme so many times, let's consider a few discrete cases. Regular polygons have circumcircles, and a regular k-gon gives a eutactic star of k equal vectors. Actually, the star has 2k vectors, if we include the opposites. We shall only use k of these. Beginning with the regular 2-simplex, the equilateral triangle, we have a star of three vectors. Translations along these three trace out a regular hexagon, which we have already identified as an orthogonal, isometric shadow of a 3-cube, cast into a plane.
Next, the square's four vertices give us only two distinct vectors (we discard opposites), and translations along these two directions trace out a square.
The pentagon gives a star of five vectors, which trace out a decagon. This is an orthogonal, isometric plane shadow of a 5-cube.
The hexagon gives a star of six vectors, but, discarding opposites, we have only three, and these trace out a regular hexagon.
Continuing with the five Platonic solids:
The tetrahedron gives star of four vectors, which trace out Kepler's rhombic dodecahedron. This is an isometric solid shadow of a 4-cube, or tesseract.
The octahedron gives a star of three vectors (actually, before discarding opposites, a cross), which traces out a cube.
The cube gives a star of four vectors, after discarding opposites, which trace out Kepler's rhombic dodecahedron, again.
The icosahedron gives a star of six vectors, after discarding opposites, which trace out Kepler's rhombic triacontahedron, an isometric solid shadow of a 6-cube.
The pentagonal dodecahedron fives a star of
ten vectors, after discarding opposites, which trace out a rhombic
enneacontahedron, an isometric solid shadow of a 10-cube.
Top row: the Platonic tetrahedron, octahedron, cube, icosahedron, dodecahedron.
Bottom row: the respective zonohedra, determined
by the Platonic stars.
There are more than just the vectors to the vertices of the regular polytopes, but symmetry vectors of all kinds give eutactic stars. For instance, the Platonic icosahedron and dodecahedron are dual: the vectors to the vertices of one are in the same directions as those to the face-centers of its dual. The two stars may be combined and given equal magnitudes to yield a eutactic star of ten plus six, or sixteen, vectors (discarding opposites), and these trace out an isometric shadow of a 16-cube. To these sixteen may be added the fifteen vectors to the edge-centers of either polyhedron, giving a star of thirty-one vectors, and yielding a zonohedron which is an isometric shadow of a 31-cube.
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