The Regular Polyhedra

Above: Photo of my models of the nine regular polyhedra, five convex and four nonconvex. All nine are painted with enamels in various colors rather than constructed out of colored card stock; unfortunately, the highlights off the glossy enamels do not photograph particularly well. First row, left to right: cube, octahedron, tetrahedron; second row, left to right: great dodecahedron, icosahedron, dodecahedron, small stellated dodecahedron; third row, left to right: great icosahedron, great stellated dodecahedron. The largest of these is about 23 cm in diameter. I built these models on a model-making binge during a hot 1981 San Diego summer.

EGULAR POLYHEDRA have regular polygons as their faces, all alike (congruent) and meeting the same way at each vertex. The five different convex regular polyhedra, called the Platonic Solids because the Greek philosopher Plato employed them in a universal “theory of everything” in his book Timaeus, were well known to the Pythagoreans of classical Greece and to other philosophers dating as far back as 500 BC. The Pythagoreans, a secret society of mathematicians and musicians, developed among themselves the logical foundations of algebra, number theory, and geometry. The existence of five Platonic solids is a fundamental property of any locally Euclidean three-dimensional space, like the value of pi: the ratio of the circumference to the diameter of any circle. A set of Platonic solids is usually among the first polyhedra that a novice polyhedron model-maker builds, to hone model-making skills before attempting more challenging polyhedra.

Existence

It is easy to see why there can be only five convex regular polyhedra. In any convex polyhedron, the sum of the face angles at a vertex is always less than 360°; otherwise the polyhedron cannot be convex. And there must be at least three polygons at each vertex for the figure to be a polyhedron and not just a pair of coplanar polygons. (This is related to the reason that the simplest polygon is a triangle.) A quick check of interior angles reveals only three regular polygons that can serve as faces of convex regular polyhedra under these very stringent conditions: (1) an equilateral triangle, with an interior angle of 60°; (2) a square, with an interior angle of 90°; and (3) a regular pentagon, with an interior angle of 108°. The next polygon in line, a regular hexagon, has an interior angle of 120°. The sum of the interior angles of three regular hexagons at one vertex equals 360°, which is not less than 360°, so no convex polyhedron can have three regular hexagons at a corner: Three hexagons sharing a corner are constrained to lie in the same plane. The situation gets worse for regular polygons with more than six sides, since their interior angles are even larger than 120°, and one cannot even fit three together at a corner in Euclidean space.

So: if the faces of a regular polyhedron are equilateral triangles, there can only be three, four, or five of them at each vertex; six or more add up to 360° or more, and the polyhedron cannot be convex. If the faces are squares, there can only be three at each vertex; four or more add up to 360° or more, and polyhedron again cannot be convex. And if the faces are pentagons, there can likewise be only three at each vertex. Altogether, five possibilities.

Just because there are at most five possibilities doesn’t mean that all five must exist as real polyhedra. To show that all five possibilities do indeed exist, one must mathematically (or physically!) construct the polyhedra face by face and show that the faces do come together to bound closed figures with no free edges. For example, to construct the regular polyhedron with regular pentagons for faces, start with a group of three pentagons fitted together at a corner. Three of the free corners in this group can each accommodate one more pentagon for the full complement of three, so affix those three pentagons into this configuration. This creates three new free vertices that will each accommodate one additional pentagon, so add those next three pentagons in. This finally leaves a gap into which three more pentagons exactly fit, and the construction is finished. The total number of pentagons in the closed figure is twelve.

The other four possibilities each lead via this kind of construction to four other regular polyhedra, which establishes the existence of the five convex regular polyhedra. (There are two distinct ways to assemble a set of equilateral triangles into a polyhedron with five triangles at every vertex, but only one of these ways yields a convex polyhedron.) Models of the five Platonic solids appear in the photograph at the top of this page.

The regular polyhedron with three equilateral triangles at every vertex has four faces altogether and is called a regular tetrahedron. The one with three squares at every vertex has six faces and is the familiar solid called a cube, and the one with four equilateral triangles at every corner has eight faces and is called a regular octahedron. The one with three pentagons at every vertex has twelve faces and is known as a regular dodecahedron, and the one with five equilateral triangles at every vertex has 20 faces and is called a regular icosahedron.

Duality

As soon as one learns about the five regular solids, one sees patterns among them. Four of them fall into groups of two that may be called dual pairs: the icosahedron and the dodecahedron make up a dual pair, the cube and the octahedron make up another. The tetrahedron stands alone as self-dual. In each pair, the corners of one polyhedron correspond to the faces of the other: The cube, with six faces, has three squares meeting at each of its eight corners, whereas the octahedron, with eight faces, has four triangles meeting at each of its six corners. And the dodecahedron, with twelve faces, has three pentagons meeting at each of its 20 corners, whereas the icosahedron, with 20 faces, has five triangles meeting at each of its twelve corners. One can always number the corners of one polyhedron in a dual pair and the faces of the other so that any two corners connected by an edge in one have the same numbers as the faces that adjoin along an edge in the other. Among other things, this shows that the polyhedra in a dual pair have the same number of edges.

Duality is a deeply fundamental feature of the geometry of polytopes (figures in n dimensions; polyhedra are three-dimensional polytopes, and polygons are two-dimensional polytopes). It arises from the abstract definition of a polytope as a particular kind of partially ordered set of k-dimensional facets, -2 < k < n+1. Whatever the partial ordering function might be, one can always define the complementary ordering function that goes the other way (e.g., “less than” becomes “greater than#148; or vice versa), and this automatically creates the dual polytope. Among regular polyhedra, reversing the abstract partial ordering function converts the dodecahedron into the icosahedron, the cube into the octahedron, and the tetrahedron into itself.

In the middle of the 19^th century, Swiss mathematician Ludwig Schläfli invented a simple but extraordinarily useful notation for regular polytopes. Beginning with polygons in two dimensions, let {p} denote the regular polygon with p sides. Then denote by {p,q} the regular polyhedron that has {p}’s as faces, q around each vertex. For example, the regular dodecahedron is denoted by {5,3}: its faces are pentagons {5}, three around each vertex. The dual of {p,q} then becomes the polyhedron {q,p}. Higher-dimensional regular polytopes require longer strings of numbers. All kinds of numerical properties of regular polytopes become expressible in terms of the strings of p’s and q’s.

For example, it is easy to calculate the number of vertices, edges, and faces that a convex regular polyhedron {p,q} has, just from the numbers p and q. Begin with the wonderful formula that Russian-German-Swiss mathematician Leonhard Euler discovered concerning polyhedra that can be mapped onto a sphere. If V is the number of vertices, E is the number of edges, and F is the number of faces, then it is always true that

V – E + F = 2.

In the regular polyhedron {p,q}, every vertex is surrounded by q regular p-gons, so the total number of vertices in the polyhedron must be F times p divided by q, or V = pF/q. And the total number of edges is F times p divided by 2, because each edge belongs to two faces, so E = pF/2. Therefore, plugging these values into Euler’s formula gives us

pF/q – pF/2 + F = 2.

Multiplying both sides of the equation by 2q and factoring out the F gives us

F(2p – pq + 2q) = 4q.

So the number of faces of the regular polyhedron {p,q} is

F = 4q/[2(p+q)–pq].

The same kind of algebra gives us the formula for the number of vertices of {p,q} as

V = 4p/[2(p+q)–pq].

And if we plug these numbers back into Euler’s formula, we can solve for the number of edges of {p,q} as

E = 2pq/[2(p+q)–pq].

The interesting thing about these formulas is that the actual shapes of the faces do not matter. Any polyhedron, regular or irregular, whose faces are pentagons, of any shape whatsoever as long as they don’t intersect one another and meet three at every vertex, must have 12 faces, 20 vertices, and 30 edges (just plug p=5 and q=3 into the formulas). Now suppose you think there may be a convex polyhedron whose faces are all seven-sided polygons (or heptagons) meeting four at every vertex. If you plug the numbers p=7 and q=4 into the formulas for V, E, and F, you get negative numbers back. Since a polyhedron can’t have a negative number of vertices, edges, or faces, one can only conclude that the polyhedron {7,4} does not exist, regular or not.

If you look at the denominators of these formulas for V, E, and F, you’ll see that they’re identical, namely, 2(p+q)–pq. The values become absurd when this denominator becomes zero or negative, because each numerator is always a positive number when p and q are integers greater than 2. Thus, in order for the regular polyhedron {p,q} to exist, we must have

2(p+q) – pq > 0.

That is,

2(p+q) > pq,

or, dividing through by pq and rearranging terms,

2/p + 2/q > 1.

The only combinations of integers p and q > 2 that satisfy this inequality are {3,3}, {4,3}, {3,4}, {5,3}, and {3,5}: the Schläfli symbols of the regular polyhedra.

Of course, duality extends to all polyhedra, not just to the regular polyhedra. For example, the duals of the Archimedean polyhedra are called the Catalan polyhedra; and the duals of the uniform star-polyhedra are described by Magnus Wenninger in his book Dual Models. All polyhedra have dual figures, although these figures may not be polyhedra in the traditional sense: The dual figures of some polyhedra may have vertices, edges, and even faces at infinity, while the dual figures of other polyhedra may have coincident vertices, edges, or faces. Such figures are sometimes or usually excluded from strict definitions of polyhedra. But most of the time, the dual of a polyhedron is another polyhedron.

Now bear with me; this gets a bit complicated! One method of dualizing a polyhedron is by inversion in a sphere: Position a sphere of suitable radius R with its center C at some significant point within the polyhedron, such as at its center of symmetry or its centroid. For any vertex V of the polyhedron, consider the line CV that passes through V and C. If V is at a distance r from C along CV, locate the point inv(V) at a distance R/r from C along CV, on the same side of C as V is. (Watch out: What happens if vertex V happens to lie exactly at the center C? It usually won’t, but there can be instances when it does. We are dealing with any polyhedron here.) Construct the plane perpendicular to the line CV at the point inv(V). The face of the dual figure that corresponds to vertex V will lie in this plane. Now perform this construction for every vertex of the polyhedron. This defines a set of as many intersecting planes as there are vertices.

Each plane is divided up into pieces we can call facelets by the planes that intersect it. Among these facelets will be some that make up the faces of the dual figure. But which facelets? The edges of the facelets lie on the lines of intersection between the planes; specifically, if vertex V is joined to vertex W by an edge E, then the dual figure has a corresponding edge somewhere on the line of intersection of the planes corresponding to vertices V and W. But where on this line? Each edge at vertex V is shared by two of the faces that meet at V (by definition of a polyhedron). So in each of the two faces sharing edge E, find the one other vertex that joins V by an edge (a polygon always has exactly two sides at each corner). The planes corresponding dually to these two vertices intersect the face plane dual to vertex V in two lines. These two lines intersect the line containing edge E in the endpoints of E. Proceeding in this fashion over all the different vertices of the polyhedron eventually yields its dual polyhedron (or figure).

Naturally, this construction is easiest to perform when the polyhedron is nice and symmetric, like a regular or uniform polyhedron. The two dual polyhedra in the compounds displayed in the atlas of regular polyhedra on this page, below, are related to each other by inversion in their common midsphere, the sphere whose center lies at the center of symmetry of either polyhedron and whose radius extends from the center to the midpoint of any edge (a midradius). When the two regular polyhedra are in dual position, then by construction either is the invert of the other in the midsphere. So the circumradius (distance from the center to a vertex) of either is the invert of the inradius (distance from the center to a face) of the other. More generally, and this works even for the regular star-polyhedra, we have a Cute Theorem: For any regular polyhedron and its dual, either being of any size, the product of the circumradius of one with the inradius of the other equals the product of their two midradii.

Symmetry

The constraints of convexity and having regular polygons for faces, arranged all alike around each vertex, force a regular polyhedron to be a highly symmetric figure. What, exactly, is symmetry? To begin at the beginning, an isometry is a transformation T of Euclidean space onto itself that preserves distances. That is, if P and Q are any two points of Euclidean space (of any number of dimensions greater than 0, not necessarily just three), the distance d[PQ] between them remains the same as the distance d[T(P)T(Q)] between their transformations under the isometry. The most obvious isometry is the identity I, which transforms every point into itself; clearly the distance between any two points before and after this transformation remains the same! Other isometries include reflections and inversions, which transform a figure into its mirror image; rotations, which turn a figure through an angle; translations, which move a figure to a different location in space without changing its orientation or left-right sense; and any combinations of these.

A famous theorem of geometry proves that all isometries can be expressed as combinations of reflections. For example, a translation is the combination of reflections in two parallel mirrors, a rotation is the combination of reflections in two intersecting mirrors, an inversion is the combination of reflections in three intersecting mirrors, and so on. If the number of reflections is odd, the isometry will reverse the sense of left and right, and it is called an opposite isometry. If the number of reflections is even, the isometry maintains the sense of left and right, and it is called a direct isometry. In general, combining isometries is not commutative; the order in which they are performed matters. Performing one isometry before another does not necessarily produce the same result as performing the other isometry first.

If an isometry is applied to a polyhedron and the resulting image lands right on top of the original, so that it doesn’t look as if anything has happened to the figure—it hasn’t rotated, or changed its position or orientation—the isometry is called a symmetry of the polyhedron. Of course, the identity I is a symmetry; it does nothing at all to the polyhedron. But nontrivial symmetries permute the elements of a polyhedron in some way. For example, reflecting a cube in a mirror plane that passes midway between two opposite faces and parallel to them interchanges the vertices on one side of the mirror with those on the other side. It also flips the edges that intersect the mirror, reflects the faces that intersect the mirror, and so on. The important thing is that after the reflection, one cannot tell that the cube has been transformed without numbering or coloring the elements to keep track of them. Because the reflection doesn’t move the cube somewhere else or turn it to a different orientation, it is a symmetry of the cube.

The set of all the symmetries of any polyhedron forms a mathematical entity called a group under the operation that combines symmetries; more specifically, it is called the polyhedron’s symmetry group. To form a group, symmetries have to satisfy a number of criteria. First, the group has to contain an identity element. This is, of course, the identity isometry I, which is trivially a symmetry of every figure. When the identity element is combined with any symmetry S, the result clearly remains the symmetry S. Second, every symmetry S in the group has to have an inverse S ^–1: a second symmetry that reverses the effects of the first. For example, any reflection R is its own inverse, since when applied twice, the result is the same as if nothing at all were done; that is, R² = I, or R ^–1 = R. Third, if two symmetries S and T are in the group, the combined symmetry ST must also be in the group. This is called closure. And fourth, if three symmetries are combined, it doesn’t matter whether the first and second are combined before the third, or the second and third before the first; the resulting symmetry is the same either way. That is, (ST)U = S(TU). The latter property is known as associativity.

In some symmetry groups, one can combine symmetries over and over, generating new symmetries endlessly. For example, the set of symmetries of a sphere is uncountably large, and they form a continuous group. The symmetries of a checkerboard that extends to infinity in all directions form a smaller (countable) but still infinite group: an infinite discrete group. But the set of symmetries of any polyhedron is finite; after a while, combining symmetries generates no new ones. The total number of different symmetries in the symmetry group of a polyhedron is called its order.

If a subset of the set of elements of a group is itself closed under combination, it becomes a subgroup of the symmetry group. The order of a subgroup always evenly divides the order of the whole group. The quotient of the order of the group by the order of a subgroup is called the index of the subgroup in the group. Any symmetry group that includes both opposite and direct isometries has a subgroup of index 2 that includes only the direct isometries. (This is because the product of two direct isometries is itself a direct isometry. The product of two opposite isometries, however, is a direct isometry, so the opposite isometries alone cannot form a subgroup.) This is called the rotational subgroup of the symmetry group.

The concept of symmetry is born of and intimately connected with the regular polyhedra. When we say that a regular polyhedron has its corners “surrounded all alike,” we mean that there is a symmetry of the polyhedron that carries any vertex and the faces that surround it into any other vertex and the faces that surround it. And when we say that the faces of a regular polyhedron must be regular polygons, we mean that each face has a symmetry group that cyclically permutes its vertices. That is, there is a symmetry that carries vertex i into vertex i+1 for each i, another that carries vertex i into vertex i+2 for each i, and so on—remembering that if i+j becomes greater than n, the number of corners, we subtract n to obtain the index number of the vertex that vertex i is carried into. The only way this can happen with a plane polygon is if its corners lie on a circle and its sides all have the same length, so that its internal angles are all the same.

Finally, when we say that the faces of a regular polyhedron must all be alike, we mean that there is a symmetry of polyhedron’s symmetry group that carries any face into any other face. This extends the symmetries of a particular face to all the faces of the polyhedron. It also allows us to calculate the order of the polyhedron’s symmetry group: Since a regular n-sided polygon has 2n symmetries (n reflections in lines passing through its center and n rotations about its center, which are combinations of any two of those reflections), the order of the symmetry group of a regular polyhedron with F faces is 2nF. The symmetry groups of the regular polyhedra and their orders are listed in the atlas of regular polyhedra below.

One of the most famous theorems of solid geometry is that all the symmetries in the symmetry group of any polyhedron are generated by combining various reflections in at most three different mirrors. These three mirrors intersect one another in a single point, the center of symmetry of the polyhedron. (If there are only two mirrors, the polyhedron has no center of symmetry but it still has an axis of symmetry, which is the line of intersection of the two mirrors; if only one mirror, the polyhedron has only it as a plane of symmetry; and if none, the polyhedron’s symmetry group consists solely of the identity I, and the polyhedron is asymmetric.) The angle between any two of the three planes must always be commensurable with 180°. Otherwise combined reflections in those two planes will never repeat, and the symmetry group will not be finite. Indeed, in Euclidean space the possible angles among these planes are highly restricted. It should not be surprising that the angles are limited to just those that generate the symmetry groups of the regular polyhedra, the prisms and antiprisms, and their various subgroups.

In particular, for a convex regular polyhedron {p,q}, the angle between two of the three planes must be some integral multiple of 180°/p, the angle between two other planes must be some integral multiple of 180°/q, and the angle between the third pair of planes must be a right angle. A set of mirror planes that generates all the symmetries in a symmetry group is called a kaleidoscope.

For example, in a cuboctahedral kaleidoscope, the three mirror planes that generate its 48 symmetries are (1) a plane midway between two faces of a cube and parallel to them; (2) a plane passing through two opposite edges of the cube that are located on either side of the first plane; and (3) a plane passing through two opposite edges of the cube that intersect the first plane. The angle between planes (1) and (2) is 45° = 180°/4; the angle between planes (1) and (3) is 90°; and the angle between planes (2) and (3) is 60° = 180°/3. This set of mirrors generates the symmetry group of a cube and a regular octahedron; we use all possible reflections in the three mirrors. The three lines of intersection between the three different pairs of these planes are prototype axes of symmetry of the cube and octahedron. They pass through a vertex, the midpoint of an edge, and the center of a face of either polyhedron, respectively. Of course these three lines, and the three planes, all intersect in the center of symmetry of the cube or octahedron. To make the icosidodecahedral kaleidoscope, we simply change the angle between planes (1) and (2) from 45° to 36° = 180°/5; and to make the tetrahedral kaleidoscope, we make this angle 60°, like the angle between planes (2) and (3).

If a sufficiently large subgroup of the symmetry group of a polyhedron is a subgroup of the symmetry group of another polyhedron, it becomes possible to construct a compound of several copies of the former polyhedron that has the symmetry of the latter polyhedron. A simple instance of this is the compound of two tetrahedra called a Stella Octangula (click on the link, or see the Atlas of Polyhedra below). The symmetry group of a regular tetrahedron is a subgroup of index 2 of the symmetry group of a cube, so two tetrahedra can be put together into a figure with cubic symmetry, which neither tetrahedron possesses by itself.

Anyway, this is merely a bare-bones introduction to the symmetry groups of the regular polyhedra. Much more information appears in H. S. M. Coxeter’s classic study Regular Polytopes, currently available as a Dover Books reprint, and in his newer Regular Complex Polytopes (Cambridge University Press), now available in a revised and expanded second edition. The theory of regular polyhedra extends naturally into spaces of any number of dimensions, perhaps most interestingly to four. In four-dimensional space there are six convex regular polychora (the analogues of the Platonic solids) and ten regular star-polychora (the analogues of the Kepler-Poinsot solids). For more information about polychora, visit my Four Dimensional Figures website. In spaces of five and more dimensions, there are only three regular polytopes: the analogues of the regular tetrahedron, cube, and regular octahedron. And no regular star-polytopes at all.

The Kepler-Poinsot Regular Star-Polyhedra

For nearly 2000 years following Plato’s description of the universe in terms of the five convex regular polyhedra, no one realized that there actually were four more regular polyhedra. Then, during the Renaissance, artisans and craftsmen started playing around with the Platonic solids—symmetrically adding pyramids to their faces or gouging chunks out of them, for example. The high order of symmetry of the regular polyhedra made them particularly decorative and artistically interesting to work with.

Early in the 17^th century, mathematician and mystic Johannes Kepler noticed that one can add pyramids of a particular height and shape to the faces of a regular octahedron, dodecahedron, and icosahedron so that the lateral faces of the pyramids fall into coplanar sets. In the case of the octahedron, the pyramids are ordinary regular tetrahedra. Three lateral faces of the tetrahedra become coplanar with the faces of the underlying octahedron, yielding eight coplanar sets. A close look reveals that each coplanar set of little triangles, together with the coplanar octahedral face, makes a complete equilateral triangle twice as large: the face of a regular tetrahedron. The resulting figure, taking the large triangles as faces, is not a polyhedron but a compound of two interpenetrating concentric regular tetrahedra, each placed in “dual position” to the other. It is the three-dimensional analogue of the two-dimensional Star of David, which is a pair of intersecting triangles in “dual position.” Kepler named this figure a Stella Octangula (“eight-angled star”). He had rediscovered a figure that Fra Luca Pacioli had about a hundred years previously, in 1509, described as an octaedron elevatum; but the idea of polyhedra and compounds with intersecting faces was something new.

When Kepler added pentagonal pyramids to the faces of a regular dodecahedron, he discovered that, when the lateral faces are “acute golden triangles”—acute isosceles triangles with interior angles of 72°, 72°, and 36°—they fall into twelve coplanar sets of five. Taken with the central pentagon of each set, they form twelve intersecting regular pentagrams. The lateral edges of the pyramids line up exactly with the edges of the underlying dodecahedron. Here was a brand-new kind of polyhedron: not a compound, like the Stella Octangula, but a figure whose twelve faces were identical five-pointed stars, coming together by fives at each of twelve corners. Kepler called it the echinus major icosaëdricus:— the “greater icosahedral hedgehog.” Note that the corners of the underlying dodecahedron are not corners of the figure; only the twelve outermost points are. The corners of the dodecahedron become simply the points where the edges of the greater hedgehog intersect one another, and the relationship of the hedgehog to its inner dodecahedron is exactly like the relationship of a pentagram to its central pentagon.

Kepler likewise added 20 triangular pyramids to the faces of a regular icosahedron, each pyramid having three “acute golden triangles” for its lateral faces. Again the lateral faces fall into twelve coplanar sets of five to make pentagrams, but here the icosahedral faces are not in those planes. Rather, we must imagine the centers of the pentagrams themselves as pentagons intersecting one another deep inside the icosahedron. Then it becomes clear that the figure is another kind of polyhedron, one whose faces are twelve identical pentagrams that come together three at each of 20 corners. Kepler called it the echinus minor icosaëdricus:— the “lesser icosahedral hedgehog.”

Kepler did not realize that these were regular polyhedra that were not convex, and he regarded the existence of these figures as a kind of capricious coincidence. He never finished his major treatise on geometry in which he would have described the two figures more fully. It was not until French mathematician Louis Poinsot in 1810 extended the idea of regularity to include nonconvex polyhedra as well as convex ones that Kepler’s two figures became accepted as regular. Poinsot not only rediscovered Kepler’s polyhedra; he also discovered their duals. The greater icosahedral hedgehog has five pentagrams at a vertex, and Poinsot described its dual polyhedron, whose faces are pentagons that meet five at each vertex. This number is greater than three because the pentagons interpenetrate one another, cycling around each corner twice, just as the edges of a pentagram circle twice about its center. Likewise, the dual of the lesser icosahedral hedgehog, which has has pentagrams meeting three at a point, has five interpenetrating triangles at each vertex, cycling around each vertex twice. The star-polyhedron with pentagonal faces is the internal polyhedron whose faces are the twelve interpenetrating central pentagons of the faces of the lesser icosahedral hedgehog.

Poinsot’s treatise conjectured, but did not prove, that there were no other nonconvex regular polyhedra. This rather difficult theorem was proved shortly afterward in 1812, by another French mathematician, the renowned Augustin Louis Cauchy. The key result in the proof is the theorem that every regular polyhedron has the face-planes of a convex regular polyhedron. In particular, Kepler’s hedgehogs have the face-planes of the dodecahedron deep inside them, as does Poinsot’s polyhedron with pentagonal faces. Poinsot’s other polyhedron has the face-planes of a very small central icosahedron. In 1859, Arthur Cayley gave these four polyhedra their formal English names, which are used today: the larger hedgehog became the small stellated dodecahedron; the lesser hedgehog became the great stellated dodecahedron; the dual of the small stellated dodecahedron became the great dodecahedron; and the dual of the great stellated dodecahedron became the great icosahedron. More about these beautiful star-polyhedra appears in the atlas of the regular polyhedra below.

An Atlas of Regular Polyhedra

THE ILLUSTRATIONS for this Atlas were generated using Mathematica 2.1 (an old version of this software). Lack of space prohibits more extensive illustrations to accompany the directions for building models of the nine regular polyhedra and their dual-position compounds, and I hope that the reader will be able to figure out how to build these polyhedra from the text I’ve provided. Experienced model-makers will find building the regular polyhedra to be “old hat”; this atlas is intended more to hook novice model-makers into acquiring an interest in these figures, so that they may later go on to build more elaborate figures of their own design.

Regular Tetrahedron {3,3}

Choose any four points in three-dimensional space that don’t all lie in the same plane and they will be the corners of a tetrahedron: a polyhedron with just four faces. This is the simplest possible kind of polyhedron, so it is often called a three-dimensional simplex. If the six edges, representing the distances between the six distinct pairs of the four points, are all of equal length, then the tetrahedron automatically is regular. Its faces, which correspond to the four different ways to select any three of the four points, become equilateral triangles; the regular tetrahedron is the particular regular polyhedron that has three equilateral triangles at every corner. (If the edges are not all equal, the tetrahedron is irregular. There are several different kinds of irregular tetrahedra, such as triangular pyramids, sphenoids, disphenoids, and orthoschemes, but they fall outside the scope of this page.) In Plato’s cosmology, the regular tetrahedron was associated with the element Fire.

To construct a regular tetrahedron out of paper or thin card, one must draw four identical equilateral triangles, cut them out with tabs on half the edges, and glue them together so that three of these triangles come together at every corner. What could be easier? I recommend using triangles with an edge length of four or more inches for a substantial model. Having all four faces colored differently means cutting four separate triangles out of four differently colored sheets, two tabs on two of them and one tab on the other two. But a monochrome model can be folded together from a pattern of four connected triangles (a tetriamond) with tabs on every other free edge. There are three different tetriamonds, and two of them will fold up into a regular tetrahedron. (The one that won’t has four triangles at one corner.)

Each equilateral triangle has six symmetries, and since there are four of them as faces, the whole polyhedron has 24 symmetries. Its symmetry group is denoted [3,3] and is called the complete tetrahedral group.

Stella Octangula {4,3}2{3,3}{3,4}

Self-dual polyhedra are few and far between, but even rarer are figures for which the duality operation is also a symmetry. The regular tetrahedron is self-dual, but when it is dualized, it inverts, so that the corners of the dual are located where the faces of the original were, and vice versa. The Stella Octangula, which is a compound of two regular tetrahedra in dual position, is not just self-dual but remains unchanged after dualization. Dualization simply exchanges the two tetrahedra; it is a central inversion.

The Stella Octangula is a regular compound: Its components are identical regular polyhedra that are permuted by some of its symmetries, and its corners are all alike. The internal convex polyhedron common to both components is an octahedron, and the corners of the compound belong to a cube; the Schläfli symbol for the compound includes the Schläfli symbol for the cube, {4,3}, and the octahedron, {3,4}, as well as for the two tetrahedra themselves, 2{3,3}. This figure shows quite vividly that a regular tetrahedron is an alternated cube: each tetrahedron in the compound uses four of the cube’s eight corners.

To build a paper model, we may ignore the portions of the faces of the tetrahedra that are hidden inside the figure and just model the exterior; the hidden parts are exactly the faces of the figure’s interior octahedron. The model then consists of eight identical trihedral corners made up of three equilateral triangles, each equilateral triangle having an edge ½ the length of the edge of either tetrahedron. To make a model in two colors, four trihedral corners of each color are needed. Each trihedral corner can be folded up from a triamond; place a tab on every outer edge except one of the two collinear edges. When the triamond is folded up, the two collinear edges will glue together, making a triangular pyramid with three tabs instead of a base. When all eight of these corner pyramids are ready, glue them together by their bottom tabs so that a pyramid of one color always adjoins three pyramids of the other color. It may take a little work to get the last pyramid into the assembly. The finished model is very sturdy.

Altogether, the Stella Octangula has twice the symmetry of a regular tetrahedron, namely, 48 symmetries. Its symmetry group is the same as that of a cube or octahedron: the complete octahedral group, denoted [3,4].

Cube {4,3}
Cube

Having seen the vertices of a cube in the Stella Octangula, we now meet the cube itself. This is by far the most familiar of the regular polyhedra, being the fundamental solid of all architecture and the only regular polyhedron that can be tiled by itself to fill three-space. A cube of unit edge is defined as the unit of volume, and all other volumes are measured by the number of unit cubes they can contain. For this reason, it is also known as the three-dimensional measure polytope. Three squares come together at each vertex, for a total of six squares. This gives us the cube’s formal Greek name, regular hexahedron. The cube is also the Archimedean square prism and, if stood on a corner, the triangular antibipyramid. In Plato’s cosmology, the cube was associated with the element Earth.

The cube is one of the simplest polyhedra to model. Simply cut six squares out of paper with tabs on alternate edges and glue them together. Three different colors are best, with two opposite squares having the same color. This places all three colors on the three squares at every corner. A monochrome cube can be constructed from a hexomino: a pattern of six edge-connected squares. There are 35 different hexominos,eleven of which will fold up into a cube.

Each square of the cube has eight symmetries, so the order of the cube’s symmetry group, the complete octahedral group [3,4], is 48.

Regular Octahedron {3,4}

The dual of the cube is the regular octahedron, which has four equilateral triangles meeting at each of its six vertices. One may think of this polyhedron as two square pyramids with equilateral triangles for lateral faces joined base to base, so that it is the Archimedean square bipyramid. Or one may think of it as a pair of equilateral triangles in parallel planes rotated 60° out of alignment with each other, connected by a band of six equilateral triangles between them. This makes it the Archimedean triangular antiprism. As the dual of the cube, or three-dimensional measure polytope and orthotope, it is also known as a three-dimensional cross polytope and as a three-dimensional orthoplex, respectively. There are many kinds of irregular octahedra—polyhedra with eight faces—some very pretty, but only one of them is regular. In Plato’s cosmology, the regular octahedron was associated with the element Air.

To build a model, one needs eight equilateral triangles. If these are cut to the same size as the triangles used for the regular tetrahedron, one can place the two models flush against each other face to face and demonstrate that the other three faces of the tetrahedron are coplanar with the three adjacent faces of the octahedron. One can fill space uniformly with regular octahedra and tetrahedra alternating in this manner. The dihedral angles of the regular tetrahedron and octahedron add up to 180°.

The octahedron is the only regular polyhedron that can be colored with just two colors so that no two adjoining faces have the same color. To show this with a model, cut out four equilateral triangles of each of two colors. Half the triangles will need two tabs, half will need only one; alternatively, put tabs on all the edges of all the triangles of one color, and none on the others. A monochrome model can be folded up from a connected pattern of eight triangles. One such pattern has a fan of four triangles in a C-shaped arrangement (the “C” tetriamond) connected to another such fan along a common outer edge; another has a strip of six triangles with the seventh sticking up from any triangle with an edge along the top of the strip and the eighth sticking down from any triangle with an edge along the bottom of the strip. Many other such octiamonds will also fold up into an octahedron. Place tabs on alternate free edges.

Each triangle of the octahedron has six symmetries, so the order of the octahedron’s symmetry group, the complete octahedral group [3,4], is 48, the same as that of the cube (because they’re duals).

Compound of Octahedron and Cube {3,4}+{4,3}
Octahedron + Cube

Just as two tetrahedra can be put together concentrically in “dual position” to produce a Stella Octangula, so can the octahedron and cube be compounded into a figure that vividly demonstrates their duality. In the compound, each corner of the octahedron is centered above a face of the cube, and vice versa. Because the two polyhedra have the same midradius (the distance from the center to the midpoint of any edge), each edge of the octahedron perpendicularly bisects a corresponding edge of the cube, and vice versa. The compound is not regular, because it comprises two different polyhedra and has two different kinds of corners; but it is nevertheless very symmetric. The polyhedron common to both solids has six squares and eight triangles for faces and is called a cuboctahedron. The smallest polyhedron that encloses the two polyhedra has twelve identical faces, each a rhombus whose diagonals are the perpendicularly bisecting edge-pairs. Not surprisingly, it is the dual of the cuboctahedron, and it is called a rhombic dodecahedron (to distinguish it from the regular dodecahedron, whose twelve faces are pentagons).

To make a paper model of this compound requires cutting out just the six octahedral corners and the eight cubic corners and gluing them together; the inner cuboctahedron can be ignored. An effective color scheme is to give each polyhedron its own monochrome color. Then the cube-corners will be one color and the octahedron-corners will be the other. If the cube is to have an edge length of four inches, then the eight cube-corners can be assemblies of three isosceles right triangles meeting at their right angles. Each triangle will have equal legs two inches long and a hypotenuse of length 2sqrt(2) = 2.828427... inches. The octahedron-corners are each made of four equilateral triangles that meet the cube-corners along their hypotenuses, so they have all their edges of length 2sqrt(2). This shows at once that the edge length of the octahedron in this figure is 4sqrt(2) = 5.656854... inches, given a cube with an edge of length 4. The pattern for the octahedron-corners is the “C” tetriamond: four triangles joined together in a fan around one vertex. The pattern for the cube-corners is three right triangles joined together in a fan at their right angles. It looks like a square with a right triangle bitten out of it. Make sure there are no gaps in the model at the places where the edges bisect each other; the confluence of eight facelets at those points weakens them. An extra drop of glue there, on the inside, wouldn’t hurt.

The symmetry group of this figure is the complete octahedral group [3,4], of order 48, as with the cube and the octahedron individually. The Stella Octangula has double the symmetries of either of its component tetrahedra because there is an extra symmetry that interchanges the two tetrahedra. This symmetry does not exist in the octahedron/cube compound.

Regular Dodecahedron {5,3}

The high order of symmetry and the appearance of fivefold rotational axes in the “pentagonal polyhedra” make them the most elegant of the regular solids. In the dodecahedron, three regular pentagons meet at each vertex, there being a total of twelve pentagons in the polyhedron. Plato had a difficult time fitting this figure into his cosmology. Having assigned the classical Four Elements to the other four regular polyhedra, he decided the dodecahedron should be associated with a “fifth element,” or quintessence: the universal Æther.

A regular pentagon of a particular edge length is best drawn with the aid of a protractor; ruler-and-compass methods sometimes allow errors to multiply. The interior angle of the pentagon is 108°. Lay out its angles with some care, and make sure its edges are as close to equal as possible. Double-check to see that its diagonals are equal, too. An edge-length of only two inches produces a dodecahedron of substantial size, since the dodecahedron has the largest circumradius-to-edge ratio among the regular polyhedra. Use the laid-out pentagon as a template and replicate it twelve times on the paper stock for the model, either as separate pentagons to be glued together or as a pattern of pentagons to be folded up and glued together. The dodecahedron requires four colors if any two adjoining faces must have different colors; there is a nice four-color scheme in which each color is used on three of the twelve faces. Six colors provide an interesting color scheme in which any two opposite faces are colored alike. Then each face of any color touches faces of all five other colors. A monochrome dodecahedron can be assembled from numerous connected patterns of twelve pentagons. For example, draw a pentagon with five more pentagons attached to its edges. Then draw the identical pattern of six more pentagons connected to it along an outer edge (be sure to choose the correct outer edge; five will work, five will not!).

Each pentagon has ten symmetries, and there are twelve pentagons. So the symmetry group of the regular dodecahedron is the complete icosahedral group, [3,5], of order 120.

Regular Icosahedron {3,5}

The regular polyhedron with five equilateral triangles at each corner requires 20 triangles for its assembly, so it is called an icosahedron. It is the dual of the regular dodecahedron, and it has twelve corners and 30 edges. Unlike the tetrahedron, cube, and dodecahedron, but like the octahedron, the icosahedron remains “floppy” until the last face is to be fitted in. This is because the first three polyhedra have all trivalent vertices (meaning three faces come together at each corner), which automatically makes the corners rigid, like any triangle. The octahedron and icosahedron, however, have more than three faces at a corner, and this allows them to flex until all but the very last face is in place. One may remove as many as four strategically placed triangles of the 20 of an icosahedron and the resulting structure will still remain rigid: choose the four so that only one of them is at each vertex. The icosahedron is just as pretty as the regular dodecahedron, by virtue of its threefold and fivefold symmetry axes. In Plato’s cosmology, the regular icosahedron was associated with the element Water.

To build a regular icosahedron requires cutting 20 equilateral triangles out of paper and gluing them togther. A good edge length is 3 inches; it yields an icosahedron of substantial size. Three colors are required if no two adjoining faces shall have the same color, and there is a balanced color pattern with two colors each appearing on seven faces and the third appearing on six. Every vertex then belongs to two faces of one color, two faces of another color, and one face of a third color. Another pretty color scheme uses five colors, four faces of each color. These may be arranged so that all five colors touch each corner. When this is done correctly, the four faces of any one color lie in the face planes of a regular tetrahedron, one of the five in either of the compounds of five tetrahedra. If the four faces having any one color are removed, the resulting structure of 16 triangles is still rigid, as described in the preceding paragraph. A monochrome icosahedron can be assembled from numerous connected patterns of 20 triangles. The most familiar of these has ten triangles connected together in a straight row, with the other ten triangles attached to the ten free edges along the top and bottom of the row.

Each triangle has six symmetries, and there are 20 triangles. So the symmetry group of the regular icosahedron is, like that of the regular dodecahedron, the complete icosahedral group, [3,5], of order 120.

Compound of Icosahedron and Dodecahedron {3,5}+{5,3}
Icosahedron + Dodecahedron

Like two tetrahedra and the cube and octahedron, the regular icosahedron and dodecahedron can be placed concentrically in “dual position.” This makes a pretty, symmetric compound in which the vertices of either polyhedron lie above the centers of the faces of the other, and the edges of either polyhedron perpendicularly bisect the edges of the other. The inner polyhedron common to both solids has twelve pentagons and 20 triangles for faces and is called an icosidodecahedron. The smallest polyhedron that encloses the two polyhedra has 30 identical faces, each a rhombus whose diagonals are the perpendicularly bisecting edge-pairs. It is the dual of the icosidodecahedron and is called a rhombic triacontahedron. These figures fall neatly into the pattern begun with the Stella Octangula and the compound of octahedron and cube.

To model the compound of the icosahedron and dodecahedron in paper requires cutting out the 20 trivalent corners of the dodecahedron and the twelve pentavalent corners of the icosahedron. A good-size model has a dodecahedron with a two-inch edge; this gives an icosahedron with an edge length of 2tau = sqrt(5)+1 = 3.236067... inches, where tau is the famous Golden Ratio, tau = [sqrt(5)+1]/2. A good effect is achieved by using just two colors, one for the dodecahedron, the other for the icosahedron. More elaborate color schemes tend to confuse the eye. Each trivalent corner consists of three obtuse isosceles triangles, called “obtuse golden triangles,” whose angles are 108°, 36°, and 36°. The short edges are 1 inch long (half the two-inch edge length of the dodecahedron). Lay them out so that they come together at their wide angle, and copy this pattern altogether 20 times. Put tabs on all the outer (long) free edges and a tab on one of the two short free edges, then fold them up into triangular-pyramidal corners.

Next, each pentavalent corner consists of five equilateral triangles whose edge length exactly equals that of the long edge of the obtuse golden triangles. These lay out as “C”-shaped pentiamonds (which resemble a regular hexagon with one triangle bitten out of it). Cut out twelve copies in the color chosen for the icosahedron, with tabs on all five free outer edges of these assemblies and a tab on one of the two free inner edges. Then fold them up into hollow (bottomless) pentagonal pyramids. Complete the model by gluing each pentagonal pyramid to five triangular pyramids, and each triangular pyramid to three pentagonal pyramids, tab-to-tab. The resulting model is “floppy” right until the last pyramid is glued in place. When finished, the model may have a disconcerting habit of “popping” inward at the points where the edges intersect. This can be prevented by gluing one or two cardboard braces across the tabs inside the model to keep the edges straight during assembly.

The symmetry group of this figure is the complete icosahedral group [3,5], of order 120, as with the dodecahedron and the icosahedron individually.

Small Stellated Dodecahedron {⁵/₂,5}
Small Stellated
Dodecahedron

The pentagram, or regular five-pointed star, was known to the Pythagoreans, who thought it had mystic medical significance but failed to see it as a regular polygon. The 14^th-century British mathematician Thomas Bradwardine (Bredwardin, or Bradwardinus) first studied star-polygons systematically and realized that there is a regular p-pointed star for every p > 4 and d such that 1 < d < p/2 and the fraction ^p/_d is in lowest terms. The pentagram is the simplest regular star-polygon, with p = 5 and d = 2. There is no regular six-pointed star, only a six-pointed compound of two equilateral triangles, the Star of David. There are two seven-pointed stars (or heptagrams), one eight-pointed star (or octagram), two nine-pointed stars (or enneagrams), one ten-pointed star (or decagram), and so forth. The Schläfli symbol for a p-gon, {p}, extends readily to star-polygons by using fractions: {^p/_d} denotes the star-polygon whose edges cycle d times around the center before returning to the starting vertex. It is when the fraction ^p/_d is not in lowest terms that a regular compound polygon results.

There are two systematic and symmetric ways to create regular star-polygons, each the dual process of the other. One can extend the edges of a convex regular polygon until they meet the extensions of other edges in new vertices; this process is called stellation. Or one can find all the diagonals of a particular length inside a convex regular polygon; these will comprise either a single star-polygon or a regular compound polygon. This process is called faceting. Both processes have n-dimensional analogues.

Regular star-polygons were not accepted as regular on a par with the convex regular polygons until Louis Poinsot published his study of regular polygons and polyhedra in 1810. He pointed out that an edge of a star-polygon is the line segment that connects two corners, regardless of how many other edges it might intersect in false vertices. And he noted that the three-dimensional analogue of a star-polygon is a figure whose faces pass through one another, much as the edges of a star-polygon pass through one another. As with convex regular polygons, a star-polygon is regular if its interior angles are all alike and its edges are all the same length. Having set up his definitions, Poinsot proceeded to describe the four regular star-polyhedra (without knowing that Kepler had already described two of them two centuries earlier).

If one stellates each pentagon of a regular dodecahedron into a pentagram, the resulting figure is a closed polyhedron whose faces are twelve congruent pentagrams and whose corners are each surrounded by five pentagrams. This makes it a regular polyhedron, the small stellated dodecahedron (or simply stellated dodecahedron). Another way to construct it is to find the pentagrams among the vertices and internal face-planes of a regular icosahedron; there is one pentagram below each icosahedral vertex. Thus, with its twelve vertices, the small stellated dodecahedron is also a faceted regular icosahedron. Whichever way it is constructed, it retains in extension the 30 edges of the dodecahedron inside it. The Schläfli symbol of a pentagram is {⁵/₂}, so, since there are five at a corner in a small stellated dodecahedron, the Schläfli symbol of the latter becomes {⁵/₂,5}.

In making a model of this figure, the parts of the faces that are hidden in the interior may be ignored; only the exterior facelets need to be cut out and assembled. As can be seen in the illustration, these facelets are the “acute golden triangles” that form the points of a pentagram. They are isosceles triangles whose interior angles are 36°, 72°, and 72°. Altogether 60 copies are required. A good color scheme is to have all five facelets that make up a pentagram the same color, and at the same time to have no two adjoining facelets the same color. This can be done easily with six colors: the two opposite (parallel) pentagrams are colored the same, since they cannot meet anywhere, but otherwise each pentagram adjoins five others that are all colored differently. There is also such a coloring using only four colors, with three non-intersecting pentagrams colored the same (an extension of the four-coloring of the dodecahedron described above).

Whatever coloring is used, 60 acute golden triangles must be cut out and glued together. This is best done by first gluing them into pyramids of five, which can then be glued together along their bases to make up the complete polyhedron. Good luck keeping the colors properly coordinated! A monochrome small stellated dodecahedron may be more easily assembled by cutting out a fan-shaped, semicircular pattern of five triangles all at once, then folding it up and gluing it into a pentagonal pyramid. Twelve such pyramids make up the whole model. In either case, the model will remain “floppy” until the last pyramid is glued into place.

Each pentagram has ten symmetries, and there are twelve pentagrams. So the symmetry group of the small stellated dodecahedron is the complete icosahedral group, [3,5], of order 120, as it is for the regular dodecahedron inside it and for the regular icosahedron whose vertices it possesses.

Great Dodecahedron {5,⁵/₂}

The twelve pentagrammatic faces of a small stellated dodecahedron may be extended until they meet the extensions of other faces, just as the edges of a dodecahedron are extended to meet other edge-extensions to create the small stellated dodecahedron. The polyhedron that results is the great dodecahedron. Although the twelve vertices of the original small stellated dodecahedron remain, the 30 original edges are lost; they simply become the lines of intersection between the faces of the great dodecahedron. These faces are twelve regular pentagons, specifically the pentagons that circumscribe the twelve pentagrams of the small stellated dodecahedron. The 30 new edges formed by these pentagons are also the edges of the circumscribing icosahedron, which shows that the great dodecahedron is also a faceted icosahedron.

The process whereby new polyhedra are created by extending the faces (but not the edges) of a convex core polyhedron until they meet one another in new edges is called greatening (or face-stellation); the great dodecahedron is a particular kind of “greatened dodecahedron” in which the faces of the internal dodecahedron are extended symmetrically and produce new faces (the pentagons of the great dodecahedron) that are exact enlargements of the original faces (the pentagons of the internal dodecahedron). In the great dodecahedron, the pentagonal faces meet five at a vertex, circling each vertex twice, so its Schläfli symbol is naturally {5,⁵/₂}. This is the reverse of the symbol for the small stellated dodecahedron, indicating that the small stellated dodecahedron and great dodecahedron are each other’s duals.

Among certain uniform polytopes there is a topological relationship called conjugacy, wherein regular star-faces may be exchanged for regular convex polygons with the same number of vertices and vice versa. Pentagrams and pentagons are conjugate, as are octagons and octagrams and decagons and decagrams. If a uniform polytope with any of these three kinds of faces exists, then there also exists a conjugate uniform polytope in which these faces are replaced by their conjugates while all other topological properties (such as the number of faces at a vertex) remain invariant. Conjugacy simultaneously exchanges pentagons for pentagrams in vertex figures. Since the small stellated dodecahedron has twelve pentagrammatic faces meeting five at each of twelve vertices, its conjugate must have twelve pentagonal faces meeting at twelve pentagrammatic vertices: This conjugate is the great dodecahedron.

To build a model requires cutting out the 60 “obtuse golden triangles” that are the visible portions of the great dodecahedron’s pentagonal faces. These form 20 trihedral dimples that may be glued together like the triangles of an icosahedron. These dimples are exactly the same shape as the pyramids that make up the vertices of the dodecahedron in the compound of an icosahedron and a dodecahedron. Each face contacts all ten other non-parallel faces, so six colors are required if no two adjoining facelets may have the same color. There will then be ten obtuse golden triangles of each color. The resulting model great dodecahedron appears to change colors with startling rapidity, since a rotation of only 36° brings a whole new face into view.

Each pentagon has ten symmetries, and there are twelve pentagons. So the symmetry group of the great dodecahedron is the complete icosahedral group, [3,5], of order 120, as it is for its dual and conjugate small stellated dodecahedron.

Compound of Small Stellated Dodecahedron
and Great Dodecahedron {⁵/₂,5}+{5,⁵/₂}
Small Stellated Dodecahedron +
Great Dodecahedron

It is possible to place a small stellated dodecahedron and a great dodecahedron in “dual position,” but the resulting figure hides the great dodecahedron inside the small stellated dodecahedron completely, so a paper model is not particularly enlightening! One might, however, construct a beautiful model of this compound with the small stellated dodecahedron in transparent plastic. The picture here shows the two polyhedra in proper position, with the small stellated dodecahedron rendered semitransparent. It was drawn by my polyhedron-pal Russell Towle using his POV-ray system, because Mathematica 2.1 cannot draw transparent faces. The polyhedron common to both star-polytopes in the compound is a dodecadodecahedron; its faces are twelve pentagons and twelve pentagrams. It is self-conjugate. Its dual is a star-polyhedron with 30 narrow rhombic faces called a medial rhombic triacontahedron; the diagonals of its rhombic faces are the edges of the two dually placed regular star-polyhedra. The larger diagonal (edge of the small stellated dodecahedron) is tau+1 = tau² times the length of the smaller diagonal (edge of the great dodecahedron), where tau is the golden ratio, [sqrt(5)+1]/2.

The symmetry group of this compound, of course, is the same as the symmetry group of either component, namely, the complete icosahedral group, [3,5]. The compound is self-conjugate, since either component is the conjugate of the other. It derives directly from the compound of the regular icosahedron and dodecahedron through stellation of the dodecahedron into a small stellated dodecahedron and faceting of the icosahedron into a great dodecahedron.

Great Stellated Dodecahedron {⁵/₂,3}
Great Stellated
Dodecahedron

If the edges of a great dodecahedron are extended until they meet other edges, the twelve pentagons stellate into pentagrams, and another regular star-polyhedron appears: the great stellated dodecahedron. This star-polyhedron has twelve interpenetrating pentagrammatic faces meeting three around a vertex, so it has 20 vertices. Its 30 edges are, of course, merely extensions of the 30 edges of the underlying great dodecahedron. Since each vertex lies directly above one of the 20 trihedral dimples of the underlying great dodecahedron, they must be the vertices of a circumscribing regular dodecahedron. The great stellated dodecahedron is the only regular polyhedron that is a faceting of the regular dodecahedron; the other three are all facetings of the regular icosahedron. It is the dual of the great icosahedron and the conjugate of the regular dodecahedron. Because of its derivation by stellation of the great dodecahedron, we can also call it a stellated great dodecahedron.

A paper model requires 20 triangular pyramids, each of which comprises three acute golden triangles whose bases remain free. Put tabs on all the bases. The pyramids fold up into sharp points: the great stellated dodecahedron has the most acute corners of any uniform polyhedron, with only three 36° angles at each vertex. Glue the points together by their base tabs the same way that the triangular dimples go together into a great dodecahedron, or equilateral triangles go together into an icosahedron. Like the great dodecahedron, the great stellated dodecahedron requires six colors, if the five facelets of each face are to have the same color but no two adjoining facelets are. There is no simple way to describe the color schemes for the 20 pyramids. It is easier for the modeler to simply work these out during assembly. As with the small stellated dodecahedron, there will be ten facelets of each of the six colors. In assembling the pyramids into a finished model, ten surfaces must be brought together at each of the twelve reentrant vertices. This tends to weaken the model at those points. A drop of glue right on each of those points helps to keep the model from bursting apart there.

Each pentagram has ten symmetries, and there are twelve pentagrams. So the symmetry group of the great stellated dodecahedron is the complete icosahedral group, [3,5], of order 120, as it is for all the regular pentagonal polyhedra. Further extension of the face-planes of the regular dodecahedron yields no more star-polyhedra, because in the great stellated dodecahedron every face maximally intersects every other face. It would be wonderful, for example, if there were a regular polyhedron that had both star faces and star vertices, but alas, we know now that that is impossible.

Great Icosahedron {3,⁵/₂}

The great icosahedron is the dual of the great stellated dodecahedron. Its faces are 20 triangles: large faces almost as big as the whole polyhedron, which pass through one another quite intricately. Deep inside the polyhedron, the face-planes bound a tiny central icosahedron. The great icosahedron is the conjugate of the regular icosahedron. Whereas the five triangles at every vertex of a regular icosahedron cycle around each vertex once in a pentagonal vertex figure, the five triangles at every vertex of a great icosahedron cycle around each vertex twice in a pentagrammatic vertex figure. Just as the vertices and edges of a regular icosahedron belong to an inscribed great dodecahedron, so do the edges and vertices of a great icosahedron belong to a circumscribed small stellated dodecahedron, which is the conjugate of a great dodecahedron.

This polyhedron is the most difficult of the regular polyhedra to model successfully. Not only are there three times as many pieces (180) to cut out as in any of the other regular star-polyhedra, there are 20 places where twelve facelets and their tabs come together, making the joints there particularly weak and prone to twisting. I have yet to build a paper model in which all the edges turn out as straight as possible. The one in the photograph at the beginning of this page is the most successful one I’ve built to date. It’s pretty nice; only a couple of edges are noticeably out of line, and none is unacceptably so (and they’re hidden on the back side, out of view of the camera!). Another well-built great icosahedron appears in Magnus Wenninger’s book Polyhedron Models, page 63. It is better to build a larger model than a smaller one, since the relative effects of the little modeling flaws and so forth are lessened.

A paper model requires two different kinds of facelets: an obtuse isosceles triangle, 60 copies of each, and a long, slender scalene triangle. The scalene triangle comes in left- and right-handed forms, 60 copies of each. Ten colors are required if no two adjoining facelets may have the same color; color schemes with fewer colors will inevitably produce some adjoining facelets with the same color, because each face intersects all 18 of the other non-parallel faces. (This, of course, assumes that one desires to have all the facelets that lie in the same face-plane to have the same color.) These two kinds of triangles appear in the complete face of the great icosahedron pictured below. The picture shows how each triangular face of the great icosahedron is cut up by the planes of the other faces; the blackened triangular facelets are the visible portions of the face. They are the parts that must be glued together to build a model.

Start by drawing the small isosceles triangle. Its long edge, the base, is the short middle section of a real edge of the figure. I recommend a length of at least two inches for this edgelet, which will result in a large model about 14.4 inches in diameter. The obtuse angle is 104.477512...° and the two acute angles are each 37.761243...°. Draw this triangle very carefully with a good protractor, and make sure the short edges are as close to equal in length as possible.

Great Icosahedron Face

Now use either short edge of the isosceles triangle as the short edge of the slender scalene triangle. The angles of the scalene triangle are 22.238756...°, 82.238756...°, and 75.522488...°. The short edge is opposite the smallest angle, of course, so lay out the triangle so that the larger two angles are at the short edge. Check the triangle by measuring the length of the edge opposite the 75+° angle: it should be very close to 3.236067... inches long for a 14.4-inch-diameter model. This edgelet is either of the two longer end sections of each true edge of the model and will, one hopes, line up straight with the base of an isosceles-triangular facelet when the model is assembled.

The model is constructed from 60 trihedral dimples made up of one isosceles triangle, one left-handed scalene triangle, and one right-handed scalene triangle. The two scalene triangles adjoin each other along their longest edges (the ones opposite the 82+° angles) and adjoin the isosceles triangle along its (and their) shortest edges. Five dimples cycle around each vertex, adjoining one another along the middle-size edges of the scalene triangles, to make a kind of fluted pentagonal pyramid. Each pyramid of five dimples adjoins five other such pyramids along its base edges, the long edges of the isosceles triangles; there are twelve dimpled pyramids altogether. The picture of the figure shown here can guide the assembly. There is no simple way to describe which colors go where in this model if it is assembled from ten different colored papers; you will have to work it out as you go along. Each triangular face has nine external facelets: three of the isosceles triangles and six of the scalene triangles (three of each handedness), so in the ten-color scheme there will be six isosceles triangles and twelve scalene triangles of each color. Be prepared to spend a day or two, or more, on this model. If you build the great icosahedron successfully, you pass the “final examination” of polyhedron model-making!

Each triangle has six symmetries, and there are 20 triangles. So the symmetry group of the great icosahedron is, like that of the other regular pentagonal polyhedra, the complete icosahedral group, [3,5], of order 120. The scalene-triangular facelets are completely asymmetrical (symmetry group is order 1), and this is why there are 120 of them.

Compound of Great Icosahedron and
Great Stellated Dodecahedron {3,⁵/₂}+{⁵/₂,3}
Great Icosahedron + Great
Stellated Dodecahedron

The compound of the great icosahedron and its dual great stellated dodecahedron in “dual position” has little fluted-pyramidal points—small pieces of the great icosahedron—emerging from each of the twelve reentrant vertices of a big great stellated dodecahedron: perhaps not a particularly interesting figure, since so much of the great icosahedron is hidden. But with its 32 points of two different sizes, it’s pretty to look at. The figure common to both star-polyhedra in the compound is the great icosidodecahedron; its dual is the great rhombic triacontahedron, whose 30 interpenetrating rhombic faces contain the edges of both regular star-polyhedra as their diagonals. The face of the great rhombic triacontahedron is a rhombus of the same shape as the face of the ordinary rhombic triacontahedron.

When the polyhedra are in this position, the edges of the great stellated dodecahedron are tau, the golden ratio, times as long as the edges of the great icosahedron. This compound is the conjugate of the compound of the regular icosahedron and regular dodecahedron. In {3,5}+{5,3}, the regular icosahedron has the greater circumradius, but in this compound, the great icosahedron has the smaller circumradius. The symmetry group of this compound, of course, is the same as the symmetry group of either component, namely, the complete icosahedral group, [3,5].

Polyhedral Properties

THE TABLE BELOW summarizes the key numerical and metric properties of the nine regular polyhedra. The first column provides the name, Schläfli symbol, and Wythoff symbol for each polyhedron. The second column provides the number of vertices, the third column the number of edges, and the fourth column the number and kind of faces. The fifth column provides the dihedral angle between two adjoining faces at an edge. The sixth column provides the designation and order of the polyhedron’s symmetry group. The seventh, eighth, and ninth columns provide, respectively, the circumradius, midradius, and inradius of the polyhedron, given an edge length of 2. The circumradius is the distance from the center to any vertex; the midradius is the distance from the center to the midpoint of any edge; and the inradius is the distance from the center to the center of any face. The letter T stands for tau, the golden ratio, which is (sqrt(5)+1)/2. Sqrt is, of course, the square root function; arctan is the inverse tangent function. The exact expressions for the values of the radii and dihedral angles are calculated to six decimal places (no rounding).

The Regular Polyhedra
Name & symbols	V	E	Faces	Dihedral angle	Symmetry group	Circumradius edge=2	Midradius edge=2	Inradius edge=2
Regular tetrahedron {3,3} 3 \| 2 3	4	6	4 triangles	2arctan[sqrt(2)/2] =70.528779...°	[3,3], of order 24	sqrt(3/2) =1.224744...	sqrt(2)/2 =0.707106...	sqrt(1/6) =0.408248...
Cube {4,3} 3 \| 2 4	8	12	6 squares	90°	[3,4], of order 48	sqrt(3) =1.732050...	sqrt(2) =1.414213...	1
Regular octahedron {3,4} 4 \| 2 3	6	12	8 triangles	2arctan[sqrt(2)] =109.471220...°	[3,4], of order 48	sqrt(2) =1.414213...	1	sqrt(2/3) =0.816496...
Regular dodecahedron {5,3} 3 \| 2 5	20	30	12 pentagons	2arctan(T) =116.565051...°	[3,5], of order 120	Tsqrt(3) =2.802517...	T+1 =2.618033...	(T+1)/sqrt(3-T) =2.227032...
Regular icosahedron {3,5} 5 \| 2 3	12	30	20 triangles	2arctan(T+1) =138.189685...°	[3,5], of order 120	sqrt(T+2) =1.902113...	T =1.618033...	sqrt(T+2/3) =1.511522...
Name & symbols	V	E	Faces	Dihedral angle	Symmetry group	Circumradius edge=2	Midradius edge=2	Inradius edge=2
Small stellated dodecahedron {⁵/₂,5} 5 \| 2 ⁵/₂	12	30	12 pentagrams	2arctan(T) =116.565051...°	[3,5], of order 120	sqrt(T+2)/T =1.175570...	T-1 =0.618033...	(T-1)/sqrt(3-T) =0.525731...
Great dodecahedron {5,⁵/₂} ⁵/₂ \| 2 5	12	30	12 pentagons	2arctan(T-1) =63.434948...°	[3,5], of order 120	sqrt(T+2) =1.902113...	T =1.618033...	1/sqrt(3-T) =0.850650...
Great stellated dodecahedron {⁵/₂,3} 3 \| 2 ⁵/₂	20	30	12 pentagrams	2arctan(T-1) =63.434948...°	[3,5], of order 120	sqrt(3)/T =1.070466...	1/(T+1) =0.381966...	1/[sqrt(3-T)]T³ =0.200811...
Great icosahedron {3,⁵/₂} ⁵/₂ \| 2 3	12	30	20 triangles	2arctan[1/(T+1)] =41.810314...°	[3,5], of order 120	sqrt(T+2)/T =1.175570...	T-1 =0.618033...	sqrt(T+2/3)/T⁴ =0.220528...

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Page 2: What Are Polyhedra? This page displays some more polyhedron models and introduces a working definition of a polyhedron. Use the chart of Greek Numerical Prefixes at the bottom of this page to create formalized names for all kinds of polyhedra.

Page 4: Specifications and Prices. Still more photos of models here, along with descriptions of the materials and methods I use in my craft. Find out how much I would charge to custom-build a polyhedron for you.