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# Polyhedra & Polytopes

This set of articles started with a simple question about hexahedra  (below)  whose original version can still be found on the main geometry page.

### Related Links (Outside this Site)

Dr. George W. Hart's site with its Encyclopedia and Pavilion of Polyhedrality.
Vladimir Bulatov has a large collection of polyhedra with pretty VRML models.
Detailed enumeration of polyhedra by Steven Dutch, Ph.D.
Polyhedra (and Five space-filling polyhedra) by Guy Inchbald.
Crystallographic Polyhedra (Java applets) in Dr. Steffen Weber's homepage.
Puzzles & Polyhedra by Jorge Rezende (University of Lisbon)
Platonic & Archimedean Solids  (Science U)   |   Glossary  by  Robert Webb
Isohedra   |   Fair Dice  by  Ed Pegg, Jr.   |   Dice (Klaus Æ. Mogensen)
Euler's polyhedral formula  by  Maurice Stark.
Gallery of Polyhedral Types  by  Boris Sprinborn  (2011).
Why the Rhombic Dodecahedron is a Shadow of the 4-Dimensional Hypercube (1997)
by  Russell Towle  (1949-2008)

Wikipedia :   Conway polyhedron notation   |   Disphenoid   |   Dodecahedron
Truncation   |   Omnitruncated polyhedra   |   (Snub) alternation
Euhedral, subhedral and anhedral minerals

DMOZ: Polytopes

### Video Lectures :

• Newton Polyhedra & Polynomial Equations, by  Nahum Zobin   | 1 | 3 | 4 |

## Polyhedra (3D), Polychora (4D), Polytopes (nD)

(Jerry of Nashville, TN. 2000-11-18)
What [polyhedron] has six faces?

A polyhedron with  6  faces is a  hexahedron. The cube is the best-known hexahedron, but it's not the only one:  Disregarding geometrical distortions and considering only the underlying  topology,  there are  7  distinct hexahedra:

Name of HexahedronEdgesNodes
Triangular Dipyramid95
Pentagonal Pyramid106
Tetragonal Antiwedge106
Hemiobelisk117
Hemicube117
Cube128
Pentagonal Wedge128

The above triangular dipyramid has 5 vertices and 9 edges.  It's the dual of a triangular prism, and looks like two tetrahedra "glued" on a common face.

The pentagonal pyramid has 6 vertices and 10 edges; it's a pyramid whose base is a pentagon. Like all pyramids, the pentagonal pyramid is self-dual.

The above three hexahedra are the only ones which exist in a version where all 6 faces are regular polygons.

The tetragonal antiwedge is the least symmetrical of all hexahedra; its only possible symmetry is a 180° rotation.  This skewed hexahedron has the same number of edges and vertices as the pentagonal pyramid. Its faces consist of 4 triangles and 2 quadrilaterals.  Such a solid is obtained from two quadrilaterals that share an edge [the hinge] but do not form a triangular prism.  After adding two edges to complete the two triangles whose sides are adjacent to the hinge, we are left with a nonplanar quadrilateral and must choose one of its 2 diagonals as the last edge of the polyhedron. Only one choice gives a convex polyhedron.

Loosely speaking, there are two types of tetragonal antiwedges which are mirror images of each other; each is called an enantiomer, or enantiomorph of the other. The tetragonal antiwedge is the simplest example of a chiral polyhedron.  Any other hexahedron can be distorted into a shape which is its own mirror image, and the tetragonal antiwedge may thus unambiguously be called the chiral hexahedron.  Each enantiomer is self-dual; a tetragonal antiwedge and its dual have the same chirality.

The other types of hexahedra are more symmetrical and simpler to visualize. One of them may be constructed by cutting off one of the 4 base corners of a square pyramid to create a new triangular face. This hexahedron has 7 vertices and 11 edges.  Its faces include 3 triangles, 2 quadrilaterals and 1 pentagon. It could also be obtained by cutting an elongated square pyramid (the technical name for an obelisk) along a bisecting plane through the apex of the pyramid and the diagonal of the base prism, as pictured at right.  For lack of a better term, we may therefore call this hexahedron an hemiobelisk.

Also with 7 vertices and 11 edges, there's a solid which we may call a hemicube (or square hemiprism), obtained by cutting a cube in half using a plane going through two opposite corners and the midpoints of two edges. Its 6 faces include 2 triangles and 4 quadrilaterals.

The cube  (possibly distorted into some kind of irregular prism or truncated tetragonal pyramid)  isn't the only hexahedron with 8 vertices and 12 edges:  Consider a tetrahedron, truncate two of its corners and you have a pentagonal wedge.  It has as many vertices, edges and faces as a cube, but its faces consist of 2 triangles, 2 quadrilaterals and 2 pentagons.

We can build a pentagonal wedge with 2 regular pentagons and 2 equilateral triangles, so that all edges but one are equal. The one "exceptional" edge is the longest side in the two trapezoidal faces. What's its length? Well, look at the wedge "from the side" (so both pentagons project into a line) you see two similar isosceles triangles. The base of the smaller is a regular edge seen perpendicularly (and therefore at its real size), whereas the base of the larger triangle is the length we're after. The ratio of similitude is simply the ratio of the height of a regular pentagon to the distance from a side to an adjacent vertex, namely 1+sin(p/5)/sin(2p/5) = (1+Ö5)/2.  A number known as the golden ratio, which happens to be the ratio of the diagonal to the side in a regular pentagon. The longest edge in our solid is thus 1.6180339887498948482... times the length of any other. In other words, both trapezoidal faces are congruent to the diagonal section of a pentagonal face (pictured at right).

(2007-09-19)   The  fattest  tetragonal antiwedge isn't a proper one !
Looking for the chiral hexahedron of unit volume and least surface area.
 Let's consider  convex  chiral hexahedra  (see above)  possessing the only allowed symmetry  (namely a  180°  rotation about an axis perpendicular to the hinge where the two tetragonal faces meet).  If the hinge AB has a length of  2 (arbitrary) units, the shape of such a  symmetrical tetragonal antiwedge  is determined by  5  positive parameters  (u,v,x,y,t  with  y>v  for convexity)  giving the coordinates of the  6  vertices in a suitable cartesian system:

 A  = -a00 B  = a00 C  = uv-vt D  = xyyt E  = -xy-yt F  = -uvvt

If we call  2q  the  dihedral  angle between the two tetragonal faces, then:

t   =   tan q       and       1 + t 2   =   1 / cos2 q

The volume  V  is the sum of  2  pairs of tetrahedra  (each having O as a vertex).

V   =   1/3 [ det (B,C,D)  +  det (D,E,F) ]   =   2/3 yt  ( av + uy + vx )

The above computation can be done mentally, using the preceeding layout which features the relevant triplets of columns adjacent to each other (and in the right order).  The first determinant  (2avyt)  is trivial.  The second one is quite simple too, when computed as  det (D,D+E,F).

Expressing the areas of triangles and quadrilaterals as cross-products  (and using pairwise equalities)  we obtain the total surface area  S  of the hexahedron:

 S   = || BC ´ BD ||  +  || DE ´ DF ||  +  || AD ´ BF || Where : || BC ´ BD || 2 = (1 + t 2 ) [ (u-a) y + (a-x) v ] 2  +  4 (vyt) 2 || DE ´ DF || 2 = 4 (y-v) 2 [ x 2  +  t 2 y 2 ]  +  4 t 2 (uy + xv) 2 || AD ´ BF || 2 = (1 + t 2 ) [ (u+a) y + (a+x) v ] 2

The maximum volume for a given surface  (or, equivalently, the minimum surface for a given volume)  is obtained when the differential forms  dV  amd  dS  are  proportional  (the coefficient of proportionality is the Lagrange multiplier associated to whichever quantity is considered a  constraint  under which the other is to be optimized).

Thus, the distance between A and B must vanish.  This is to say that the fattest  symmetrical  tetragonal antiwedge actually degenerates into a triangular dipyramid.  (There must be an easier way to reach that conclusion.  A tantalizing  conjecture  is that no chiral polyhedron can be the fattest of its own kind.)

(2000-11-28)   Polyhedral Duality
The faces of a polyhedron correspond to the vertices of its dual.

The duality of polyhedra is an  involutive  relationship  (the dual of the dual is the original polyhedron)  which can be defined either in abstract terms  (topologically)  or in more concrete geometrical terms.  When discussing two polyhedra that are duals of each other, it's convenient to identify one as the  primal  and the other as the  dual,  but the two rôles could be switched:

### Topological Duality :

The  dual  of a polyhedron is the polyhedron obtained by switching the rôles of vertices and faces:  Edges of the dual connects nodes associated with adjacent primal faces  (dual polyhedra have the same number of edges).

### Geometric Duality :   [ skip on first reading ]

Motivation :   The above  topological  relationship holds between any convex polyhedron and its  polar  (relative to any  center  O  inside it).  Therefore, the polar of a convex polyhedron is a proper geometric embodiement of its topological dual, called the  geometric  dual.

Elegant and fundamental as it may be, the polar transformation doesn't generalize immediately to  nonconvex  polyhedra.  However, the following elementary construction does  (which rely on the lemma proven below).  It's equivalent to the polar transform in the convex case and always defines the precise geometric characteristics of a polyhedron whose topology is the dual of that of the original  (i.e., primal)  polyhedron, in the above sense.  So, we may as well take this as the general definition of the  geometric dual  of a given polyhedron  (convex or not)  with respect to some  (arbitrary)  sphere of center  O  and radius  R  (the value of  R  merely provides a scaling factor which is often considered irrelevant):

Definition / Construction :   Consider the orthogonal projection  H  of  O  onto the plane of a primal face.  The dual vertex associated to that primal face is, by definition, the point  M  of the ray  OH  such that:

OH . OM   =   R2

To every primal edge between two primal faces correspond a dual edge connecting the two dual vertices so associated with those primal faces.

Modern Viewpoint :   Using the geometric concept of  inversion  introduced by Steiner in 1826, we may state that the vertices of the dual polyhedron and the projections of the center onto  [the planes of]  the faces of the primal polyhedron are inverses of each other.

What makes the above construction "work" is the following geometric fact:
Lemma :   In a plane, Let I be a point such that IH is perpendicular to OH and IH' is perpendicular to OH'.  If M and M' are points respectively located on OH and OH',  then I, M and M' are aligned if the following numerical relations hold:

OH . OM   =   OH' . OM'   =   OI2

Proof :   Consider the configuration of O,I,H and M  if IH is perpendicular to HM  (and/or to OH, since O, H and M are aligned).  IM is then a diameter of the circle containing IH and M.

The power of point O with respect to that circle is  OH.OM.  It's also equal to OI2 if and only if OI is tangent to the circle, which is to say, if and only if OI is perpendicular to the diameter IM.

The same argument holds for  O,I,H' and M'.  So, if the advertised numerical equalities hold, then IM and IM' are perpendicular to the same line (OI) which means that I, M and M' are aligned.

For a given edge of a polyhedron, we may apply the lemma to the orthogonal projection I onto that edge of the  (arbitrary)  center of inversion  O  in the plane orthogonal to the edge  (that plane contains the orthogonal projections H and H' of O on both of the faces adjacent to the featured edge).

The dual edge MM' thus belongs to the plane perpendicular to the primal edge  (that conclusion holds for every edge separately, whether or not the inversion radius is equal to OI, which we chose for convenience).

A so-called  canonical  polyhedron is endowed with a  midsphere  tangent to all its edges.  In that special case, the midsphere is the preferred inversion sphere  (or  invariant sphere)  for the above construction of the dual.  The dual of a canonical polyhedron is canonical and both share the same midsphere.  Every primal edge intersect its dual at right angle at the point where both are tangent to the midsphere.

The canonical case is important because any polyhedron is topologically equivalent to a canonical polyhedral shape  (uniquely defined, modulo a rotation and a scaling and/or translation).

(2000-11-28)   Enumeration of Polyhedra
How many polyhedra have a given number of faces and/or edges?

See the  (extendedtable on this site...

Polyhedra which are mirror images of each other are not counted as distinct. In the above, we counted  7  types of hexahedra.  That would be  8  if both chiralities of the tetragonal antiwedge were tallied.

There's only one tetrahedron:

There are two types of pentahedra: the triangular prism and the square pyramid.

There are 7 hexahedra (see previous article), 34 heptahedra, 257 octahedra, 2606 enneahedra, 32300 decahedra, 440564 hendecahedra, 6384634 dodecahedra, 96262938 tridecahedra, 1496225352 tetradecahedra, etc.

BenO (Ben Ocean, 2001-07-28; e-mail)   Platonic Solids
What are the x,y,z coordinates of the vertices of the 5 Platonic Solids?

Up to rotation and/or scaling,  there are only 5  convex  regular polyhedra.  These very special polyhedra are known as  Platonic solids.  (See below for a generalization to  n  dimensions.)

A B C D 2 2 -4 0 2Ö3 -2Ö3 0 0 -Ö2 -Ö2 -Ö2 3Ö2

Shown at right are the Cartesian coordinates of the vertices of a regular tetrahedron ABCD centered at the origin. These may be scaled and/or rotated. As given, this tetrahedron has:

• A side equal to 4Ö3.
• A height equal to 4Ö2.
• A circumscribed sphere of radius 3Ö2.
An alternate set of Cartesian coordinates for a smaller regular tetrahedron (of side 2Ö2, inscribed in a sphere of radius Ö3) would consist of every other vertex in a cube of side 2 (see below) centered at the origin. This highlights some geometrical relationships and dims others:

A = [+1,+1,+1],   B = [+1,-1,-1],   C = [-1,+1, -1],   D = [-1,-1,+1]

In such a regular tetrahedron, two vertices are seen from the center at an angle known as the tetrahedral angle (which is very familiar to chemists) whose cosine is -1/3 and whose value is 109.47122°... The dihedral angle between a pair of faces is supplementary to that angle; its cosine is 1/3 and its value is about 70.52878° (this may be called a cubic angle, for a reason which follows). Expressed in radians, three times this angle minus a flat angle (p) gives the value [in steradians, sr] of the solid angle at each corner of the tetrahedron, namely 0.55128559843... This is about 4.387 % of the solid angle of a whole sphere (4p). Astronomers may use the square degree as a unit of solid angle (a square degree equals p2/1802 sr); the solid angle at the corner of a regular tetrahedron is 1809.7638632... square degrees (or 6515150 square minutes).

We may choose  3  coordinates from the set  {-1,+1}  in  8  different ways.  Those correspond to the coordinates of the  8  vertices of a cube of side 2, centered at the origin (and inscribed in a sphere of radius Ö3).  Seen from the center of a cube, the angular separation between corners is either a flat angle (180° between diametrically opposed vertices), a tetrahedral angle of cosine -1/3 (about 109.47° between the opposite corners in a face), or a cubic angle whose cosine is 1/3 and which is supplementary to a tetrahedral angle (about 70.53° between adjacent corners).  The solid angle at each corner of a cube is clearly p/2, namely 1/8 of a whole sphere (4p).

There are  6  ways to choose  3  coordinates from the set {-1,0,+1} so that only one of them is nonzero.  These correspond to the coordinates of the  6  vertices of a  regular octahedron  of side  Ö2  centered at the origin (and inscribed in a sphere of radius 1).  Seen from the solid's center, two vertices are separated either by a  right angle  (90°) or by a  flat angle  (180°).

The volume of a regular dodecahedron is (15+7Ö5)/4 times the cube of its side.  The dihedral angle has a cosine of -1/Ö5 and a value of about 116.565°

Topical pages with other relevant data :   Paul Bourke, Ron Knott, VB Helper, etc.

(2013-01-30)   Symmetrical Polyhedra
Polyhedra having more than one isometric automorphism.

A polyhedron is  symmetric  if it's stable under at least one nontrivial isometric transformation  (i.e., a mirror reflection or a nonzero rotation).

As previously remarked, a  tetragonal antiwedge  can only have  one  such symmetry.  More complicated polyhedra are usually studied only if they have so many symmetries that there are only  very few  different types of nodes, edges or faces to consider...

Two commensurate components of a polyhedron  (e.g., two vertices, two edges or two faces)  are said to be  equivalent  if there's an isometry  (rotation or reflection)  which transforms one into the other.  When this results in only one  equivalence class  the following adjectives qualify the polyhedron:

• Isogonal  (or vertex-transitive)  when all vertices are equivalent.
• Isotoxal  (or edge-transitive)  when all edges are equivalent.
• Isohedral  (or face-transitive)  when all faces are equivalent.
• Isochoric  (or cell-transitive)  when all 3-cells are equivalent.

Since all polyhedra possess only  one  3-cell,  the term  isochoric  applies vacuously to all of them  (it's only interesting for  polychora  or other  polytopes  in more than three dimensions).  An isohedral polyhedron is called an  isohedron.  Convex isohedra are  intrinsically  fair dice.

The following terms are used for polyhedra which have at least two of the above symmetries  (besides  polychoric):

• Quasiregular  polyhedra are  isotoxal  and  isogonal.
• Noble  polyhedra are  isogonal  and  isohedral(Hess & Bruckner, 1900)
• Parahedral  polyhedra  (parahedra)  are  isohedral  and  isotoxal.

A polyhedron is said to be  regular  when it possesses  all  of the above symmetries  (in more than three dimensions, a polytope is said to be  regular  when all its flags are equivalent).

Uniform  polyhedra are  isogonal  and  equilateral  (i.e.,  all their edges have the same length but they're not necessarily equivalent).  Avoid the ambiguous term  "semi-regular";  it could mean either uniform or quasiregular!

The only regular  convex  polyhedra are the  five Platonic solids.  There are also four  nonconvex  regular polyhedra, dubbed  Kepler-Poinsot polyhedra  (two pairs, obtained by stellation of the regular convex dodecahedron and the regular convex icosahedron).

To respect the usual  inclusive meaning of mathematical terms,  all those regular polyhedra are  also  considered quasiregular, parahedral, noble and uniform.

Quasiregular  polyhedra are  uniform.  The converse isn't true:  Besides the cube and octahedron, uniform prisms and antiprisms  aren't isotoxal.  Besides platonic solids,  only two  convex  quasi-regular polyhedra exist: The  cuboctahedron  and  icosidodecahedron  (other Archimedean solids  have several types of edges).

The duals of these are the two rhombic polyhedra which are  isotoxal  and  isohedral  but not  isogonal,  namely the  rhombic dodecahedron  (at left) and the  rhombic triacontahedron  (at right).

Those were among the favorites of Bucky Fuller (1895-1983).

The only non-regular  convex  noble polyhedra are the  disphenoids.

### Inertial Symmetry :

For completeness, let's mention the type of symmetry that makes  moments  of the polyhedron with respect to something invariant under transformations of that thing.  The only example with any recognized practical importance pertains to  moments of inertia:  A solid is said to be  inertially symmetric  when it has the same moment of inertia with respect to any axis  (and/or any plane)  containing its center of gravity.  This happens when the three eigenvalues of its tensor of inertia are identical  (physicists call such a tensor  scalar ).  This characterization would make it possible to classify inertial symmetry among the equimetric properties discussed next.

Wikipedia :   Symmetrical polyhedra   |   Isogonal figure   |   Isotoxal figure   |   Isohedral figure (Isohedron)
Quasi-regular polyhedron   |   Noble polyhedron   |   Uniform polyhedron
The Kepler-Poinsot polyhedra  by  Tom Gettys (1995).
Louis Poinsot (1777-1859, X1794)

(2013-02-13)   Equimetric Polyhedra
Polyhedra in which some commensurate measures are constant.

In this context, the qualifier  commensurate  is simply used for things that can be compared to each other  (it doesn't make any sense to compare an edge to a face, for example).  A measure is a numerical function;  it's said to be  constant  when it's the same for all elements.

The prefix  equi-  indicates equality of some specific measures of such things, whereas the prefix  iso-  (which dominates the previous section)  indicates their complete equivalence, with respect to  any  possible measure or criterion,  on account of a global  symmetry.

Arguably, the term  "equifacial"  belongs to neither category.  It denotes a polyhedron whose faces are all congruent  (not necessarily equivalent).  The simplest example of an equifacial polyhedron which isn't isohedral is the  pseudo strombic icositetrahedron.  The lesser requirement of faces of equal area seems of little or no interest  (if needed, the qualifier "equiareal", listed below, could be used for that).

Simple examples of the measures that can be used to qualify polyhedra as  equimetric  include the length of all edges or the surface areas of all faces.

Other measures involve a prescribed point, which will naturally be called a  center  in case of  equimetry.  Examples of such central measurements, include radial distances  (to vertices, edges or faces)  angles subtended by edges,  solid angles subtended by faces,  etc.

In  symmetrical  cases the centers with respect to different measures will often coincide with some center of symmetry, but this needn't be so in general...  There may even be several centers with respect to which a given measure is the same for all relevant elements.  One example is the distance to [the planes of] faces in any tetrahedron:  There are  5  different centers that are equidistant to all  4  faces.

Among the lesser-investigated measures are various  moments  related to a central point.  That category includes the volume of so-called  radial pyramids  (apex at the center and one face as a base)  or the area of  radial triangles  (apex at the center and one edge as a base).

A traditional nomenclature exists for some equimetric concepts.  Others are best denoted by neologisms of recent origin:

• In  equiradial  polyhedra, all vertices are equidistant from the center.
• In  canonical  polyhedra, all edges are equidistant from the center.
• In  orthohedral  polyhedra, all faces are equidistant from the center.
• In  equilateral  polyhedra, all edges have the same length.
• In  equiareal  polyhedra, all faces have the same surface area.
• In  equicircular  polyhedra, all edges subtend the same angle.
• In  equispherical  polyhedra, all faces subtend the same solid angle.
• In  equitrigonal  polyhedra, all radial triangles have the same area.
• In  equipyramidal  polyhedra, all radial pyramids have the same volume.

Generally, the distance of a point to a set is the smallest distance from that point to a point of the set  (more correctly, the  greatest lower bound  of such distances).  However, in the context of polyhedra, the distance to a face or an edge is understood as the distance to the relevant linear support  (i.e.,  the plane of a face or the line containing an edge).  That's always obtained as the distance to the center's orthogonal projection.  Here are a few remarks:

• An  equiradial  polyhedron is  inscribed  in a sphere, its  circumscribed sphere  (whose radius is dubbed  circumradius ).  Its dual is  orthohedralSome authors call equiradial polyhedra  spherical,  although  spherical polyhedra  are normally understood to be figures drawn on the surface of a sphere, with edges represented by arcs of great circles instead of their chords  (the latter representation is degenerate or ambiguous for the monogonal or digonal faces of a spherical polyhedron).
• canonical  polyhedron is one that possesses a  midsphere  (i.e.,  a sphere to which all edges are tangent).  The distance from any edge to the solid's center is called  midradius. Isotoxal  polyhedra are canonical by symmetry.  So are  quasiregular  polyhedra, for which the tangency points are the middles of all the edges.
• An  orthohedral  polyhedron  (or  orthohedron)  is  circumscribed  to a sphere, called its  inscribed sphere  (whose radius is dubbed  inradius ).  Its dual is  equiradial.  In the special case of a  tetrahedron,  a distinction is made between the  inscribed  sphere inside the tetrahedron and four other  escribed spheres  located outside the tetrahedron and tangent to the four planes of its faces.
• The term  equicircular  refers to the  circular measure  of  planar angles  (note that  equiangular  denotes a different noncentral concept).
• Equispherical  refers to the  spherical measure  of  solid angles.

Clearly, symmetries imply many equimetries :

• Isotoxal polyhedra are canonical, equilateral, equicircular and equitrigonal.  The  quasiregular  ones are also equiradial.
• Isohedral polyhedra are orthohedral, equispherical, equiareal and equipyramidal.  The  noble  ones are also equiradial.

Circumsphere of equiradial polyhedra   |   Midsphere of canonical polyhedra   |   Insphere of orthohedra

(2001-07-28)   Uniform Polyhedra
uniform  polyhedron is both  isogonal  and  equilateral.

Two vertices of a polyhedron are called  equivalent  if one is the image of the other in an isometric transformation  (rotation or reflection)  of the polyhedron unto itself.  If all its vertices are equivalent, a polyhedron is  isogonal.  If an  isogonal  polyhedron is also  equilateral,  it's said to be  uniform.

All the vertices of a uniform polyhedron have the same arrangement of faces around them, but this condition isn't  sufficient  to ensure uniformity.  For example, the  elongated square gyrobicupola  (J37, pictured at left)  is  not  uniform  (it's known as the  pseudo-rhombicuboctahedron ).

The pseudo-rhombicuboctahedron has been rediscovered many times, by energetic amateurs or seasoned professionals, as a supposedly "forgotten" or "overlooked" 14-th Archimedean solid...  As late as 2012, Thomas C. Hales (1958-)  saw fit to lament about how "the pseudo rhombic cuboctahedron has been overlooked or illogically  [sic]  excluded from  [convex Archimedean polyhedra]".  He calls that  "one of the most persistent blunders in the history of mathematics"  (no less).

To preserve the distinction between actual symmetries and superficial clues, I beg to differ from the isolated opinion of Pr. Hales.  The duals of Archimedean solids are strict isohedra; the dual of J37 isn't.  As such, it's not guaranteed to be a fair die, in spite of the fact that all its faces are congruent:  Thoses faces are partitioned into two separate equivalence classes  (8 "polar" faces and 16 "equatorial" ones)  which don't play the same rôle as the die rolls on an horizontal surface.  This may result in a slight statistical bias about the nature of the face  (polar or equatorial)  that the die will eventually land on.

### 75  or  76  Nonprismatic Uniform Polyhedra :

Besides the infinite families of  prisms and antiprisms,  there is a grand total of  75  distinct  uniform  polyhedra.  18  of those are  convex  (the  5 Platonic solids  and the  13 Archimedean solids)  The other  57  are the  uniform star polyhedra  enumerated in 1954 by  J.C.P. Miller (1906-1981),  Donald Coxeter (1907-2003)  and  Michael S. Longuet-Higgins (1925-2016).

This includes  Miller's monster  (the great dirhombicosidodecahedron)  with 60 vertices, 240 edges and 124 faces  (its Euler characteristic is  c = -56).

On the edge-vertex skeleton of  Miller's monster,  one additional  (76-th)  uniform star polyhedron  with 204 faces can be constructed,  if  the definition of a polyhedron is relaxed to allow more than two faces to meet at one edge.  This last shape has an Euler characteristic of  24  (that's 60-240+204)  and is best known as  Skilling's figure.  It features 120 ordinary two-face edges and 120 four-face edges.  Alternately, the double edges can be considered to consist of 120 pairs of single edges that coincide in space  (for a grand total of 360 ordinary edges and, thus, a different Euler characteristic  c = -96).

With this  (optional)  addition, John Skilling (1945-)  proved, in 1970, that the previously known list of  75  nonprismatic uniform polyhedra was complete.  That result was formally published in 1975.

It's useful to observe that the  convex hull  of a uniform polyhedron is an  isogonal convex solid  having the same vertices.  This allows us to base a complete classification of all the uniform polyhedra on the much simpler classification of the convex ones...

### Convex Uniform Polyhedra :

The platonic solids are clearly  convex  and  uniform.
So are  equilateral  antiprisms  and uniform  prisms.

The only other  convex  uniform polyhedra are the  13  Archimedean solids  presented  below.

The Uniform Convex Solids   ( their duals are faur dice )
Platonic  Solids   F    E    V   Duals
VEF
Tetrahedron464Tetrahedron
Cube6128Octahedron
Octahedron8126Cube
Dodecahedron123020 Icosahedron
Icosahedron203012 Dodecahedron
Regular
Prisms  &  Antiprisms
FEV Duals
VEF
n-gonal Prism  (n ¹ 4)n+23n2n n-gonal Dipyramid
n-gonal Antiprism  (n > 3)2n+24n2n n-gonal Deltohedron
Archimedean  Solids FEV Duals
( Catalan Solids )
VEF
Truncated Tetrahedron81812 Triakis Tetrahedron
Cuboctahedron142412 Rhombic Dodecahedron
Truncated Cube143624 Triakis Octahedron
Truncated Octahedron143624 Tetrakis Cube
Rhombicuboctahedron264824 Deltoidal Icositetrahedron
(Strombic Icositetrahedron)
Great Rhombicuboctahedron
(= Truncated Cuboctahedron)
267248 Disdyakis Dodecahedron
Icosidodecahedron326030 Rhombic Triacontahedron
Truncated Icosahedron329060 Pentakis Dodecahedron
Truncated Dodecahedron329060 Triakis Icosahedron
Snub Cube386024 Pentagonal Icositetrahedron
Rhombicosidodecahedron6212060 Deltoidal Hexacontahedron
(Strombic Hexecontahedron)
Great Rhombicosidodecahedron
(= Truncated Icosidodecahedron)
62180120 Disdyakis Triacontahedron
Snub Dodecahedron92150 60Pentagonal Hexacontahedron
(or "hexecontahedron")
Yellow highlighting indicates chiral solids  (snubs and their duals).

A priori, the following pedantic nomenclature would be  perfectly acceptable,  but they are explicitly ruled out in the above table to avoid  duplication.

• cube  is a  "regular tetragonal prism".
• regular tetrahedron  is an  "equilateral digonal antiprism".
• regular octahedron  is an  "equilateral trigonal antiprism".

Note, however, that those are only  ad hoc  exclusions; it's occasionally quite useful to apply what's known for antiprisms to tetrahedra and octahedra!

(2001-07-28)   The  13  Archimedean Solids
Convex  uniform  solids,  besides  Platonic solids, prisms and antiprisms.

There are  13  Archimedean solids (two of which are chiral, the snub cube and snub dodecahedron).  Archimedes of Syracuse (c.287-212 BC) may have discovered them all, but only 12 of them were known during the Renaissance.  Kepler (1571-1630) added the  snub dodecahedron  when he reconstructed the whole set in 1619.

The truncated icosahedron (the shape of a traditional soccer ball) is now more commonly known as a buckyball ever since it was found to be the structure of a wonderful new molecule, now called fullerene (C60) in honor of the famous American architect R. Buckminster ("Bucky") Fuller (1895-1983), who created and advocated geodesic domes in the late 1940s.

The buckyball is one of 4 Archimedean solids without triangular faces.  The other three are the truncated octahedron (at left), the great rhombicosidodecahedron (at right) and the great rhombicuboctahedron.

The 4 Archimedean polyhedra illustrated so far are simplicial (i.e., only 3 edges meet at each vertex).  There are 3 others such simplicial polyhedra, illustrated next, which happen to be obtained
(like the buckyball and truncated octahedron above) by truncating a Platonic solid.
This leaves 4 nonchiral Archimedean solids with vertices of degree 4 :

Finally, 5 edges meet at every vertex of the two chiral Archimedean polyhedra:

Pappus  attributes the list to  Archimedes.  The  snub dodecahedron  (above right)  was lost until Johannes Kepler (1571-1630)  reconstructed the entire set, in 1619.

(2013-04-22)   Isogonal polyhedra generalize uniform ones.
Every uniform polyhedron typifies a d-dimensional isogonal family.

By definition,  in an  isogonal polyhedron  all vertices are  equivalent.  The edges around every vertex can be given  d  labels.  Let's show that the coordinates of each vertex are porportional to an affine function of the length associated to each label.

(2000-11-19)   Types of polyhedra named after a polygon :

Take a regular polygon (an hexagon, say) and construct a polyhedron by considering an identical copy of that hexagon in a parallel plane.  Join each vertex of the hexagon to the corresponding vertex in its copy and you obtain what's called an hexagonal prism Instead, you may twist the copy slightly and join each vertex to the two nearest vertices of the copy. What you obtain is an hexagonal antiprism. In such families, the polyhedron is named using the adjective corresponding to the name of the polygon it's built on (e.g., "hexagonal").

There are several other families besides  prisms  and  antiprisms  for which this pattern applies. For example, if you cut a prism with a plane containing some edge of either base polygon (but not intersecting the other), this "half" prism is called a  wedge  (it includes the base polygon and its featured edge).

Alternately, if the cutting plane contains only a single vertex, instead of a whole edge, the polyhedron we obtain by cutting a prism is an  hemiprism.

deltohedron  is what a regular n-gonal dipyramid becomes if we twist its upper pyramidal cone 1/2n of a turn with respect to the lower one: The intersection of the two cones becomes a solid whose faces are quadrilaterals [see figure at left].  Do not confuse this with the deltahedra defined below !

Some polyhedra based on an n-sided polygon
NameVerticesEdgesFacesRemarks
pyramidn+12nn+1One n-gon, n triangles
dipyramidn+23n2n2n triangles
deltohedron 2n+24n2n2n quadrilaterals (a cube is
a triangular deltohedron)
(a cube is a square prism)
antiprism2n4n2n+2Two n-gons, 2n triangles
cupola3n5n2n+2One 2n-gon, one n-gon,
orthobicupola
gyrobicupola
4n8n4n+2Two n-gons,
cupolapyramid
3n+17n4n+1One n-gon,
rotunda4n7n3n+2One 2n-gon, one n-gon,
n pentagons, 2n triangles
orthobirotunda
gyrobirotunda
6n12n6n+2Two n-gons,
2n pentagons, 4n triangles
rotundapyramid4n+19n5n+1One n-gon, n pentagons,
cupolarotunda
(ortho- / gyro-)
5n10n5n+2Two n-gons, n pentagons,
hemiprism2n-13n-1n+2Two n-gons (sharing one vertex),
wedge2n-23n-3n+1Two n-gons (sharing one edge),
antiwedge(s)2n-24n-62n-2Two n-gons, 2n-4 triangles
Only 4 vertices with 3 edges.

Topologically, we obtain what's called an  n-gonal antiwedge  by starting with an  n-gonal wedge  (as described above)  and splitting each of its n-3 lateral tetragonal faces into two triangles  (by introducing just one diagonal of each such quadrilateral as a new edge).  An equivalent  geometrical  construction starts with two  (non-coplanar)  n-gons  sharing an edge  (the so-called  hinge)  and the two triangular faces formed by an extremity of that hinge together with the adjacent vertices found on each n-gon.  We're then left with n-3 lateral tetragons (which are not, in general, planar quadrilaterals) from which we build 2n-6 additional triangular faces  (for a grand total of 2n-2 faces, including the two n-gons).  Note that there's only one way to split a  nonplanar  tetragon into two triangular faces to form a  convex  polyhedron.

A priori, the above constructions yield  2n-3  types of n-gonal antiwedges  which may differ either by their topology or their chirality.  However, some pairs of such configurations may be obtained from each other by a 180° rotation about an axis perpendicular to the  hinge.

Besides the trivial case of the "trigonal antiwedge"  (which is just a fancy name for an ordinary tetrahedron)  the simplest such polyhedron is the tetragonal antiwedge,  a remarkable hexahedron which stands out as the simplest example of a chiral polyhedron  (the two possible tetragonal antiwedges are mirror images of each other).

(2002-06-14)
Is there a systematic way to name polyhedra?

Only up to a point.  The most "generic" way is to use for polyhedra the same naming scheme as for polygons, by counting the number or their faces:  Thus, a tetrahedron has 4 faces, a pentahedron has 5, a dotriacontahedron (also called triacontakaidihedron) has 32 faces.

### The case of the Icosidodecahedron :

Counting faces is not nearly enough to describe a polyhedron, even from a topological standpoint.  In some cases, a nonstandard counting prefix is traditionally used for certain very specific polyhedra.  For example, the dotriacontahedron shown above is an Archimedean solid unambiguously known as an icosidodecahedron (literally, a polyhedron with 20+12 faces) because it includes 20 triangular faces and 12 pentagonal ones.  Because it's composed of two  pentagonal rotundas, the icosidodecahedron could also be called a pentagonal gyrobirotunda but that name would mask its much greater symmetry compared to the pentagonal orthobirotunda (J34) which is the other way to glue two such halves.  For the same reason, a special name has been given to the cuboctahedron  (at right)  which might otherwise be called a triangular gyrobicupola.

If the icosidodecahedron had not claimed the title, for the above reason, the name could have been given to another Archimedean solid with 32 faces, the so-called truncated dodecahedron (which has 20 triangular faces and 12 decagonal ones).  It wasn't...

The notoriety of the icosidodecahedron has made it tempting for some (knowledgeable) people to use the nonstandard  icosidodeca  prefix (instead of dotriaconta or triacontakaidi ) to name other unrelated things (like a 32-sided polygon).  Resist this temptation...

The general situation is similar to the naming of chemical compounds.  Certain families can be identified and a systematic naming can be introduced among such families.  The next article gives the most common such examples.

(2000-11-19)   Deltahedra
Don't confuse  deltahedra  with the aforementioned  deltohedra.

Deltahedra are simply polyhedra whose faces are all equilateral triangles (a polyhedron whose faces are triangles which are not all equilateral is best called an irregular deltahedron). A deltahedron [or an irregular deltahedron] has necessarily an even number of faces (2n faces, 3n edges, and n+2 vertices).

A noteworthy fact is that there are only 8 convex deltahedra (disallowing coplanar adjacent faces).  Namely:
 4 faces 6 faces 8 faces 10 faces

1. Regular tetrahedron (4 faces).
2. Triangular Dipyramid (6 faces).
3. Regular octahedron or "Square Dipyramid" (8 faces).
4. Pentagonal Dipyramid (10 faces).
5. Snub Disphenoid (12 faces), J84.
[The Snub Disphenoid was originally called "Siamese dodecahedron" by Freudenthal and van der Waerden, who first described it in 1947.]
6. Triaugmented Triangular Prism (14 faces), J51.
7. Gyroelongated Square Dipyramid (16 faces), J17.
8. Icosahedron or "Gyroelongated Pentagonal Dipyramid" (20 faces).
 12 faces 14 faces 16 faces 20 faces

All told, the convex deltahedra include

• 3 Platonic solids (tetrahedron, octahedron and icosahedron).
• 3 dipyramids (triangular, square and pentagonal).
• 3 Johnson solids (J17, J51, and J84).

This adds up to 8, instead of 9, because the regular octahedron happens to be counted twice (as a Platonic solid and a square dipyramid)...

Johnson Solids and Polyhedral Nomenclature

It's a challenge to enumerate all  convex  polyhedra whose faces are regular polygons.  Besides infinitely many prisms and antiprism, this includes only:

Norman W. Johnson (1930-)  gave the full classification in 1966, by adding the 92 polyhedra now collectively named after him.

Only 5 of the 92 Johnson solids are chiral, namely:

• J44 :   gyroelongated triangular bicupola.
• J45 :   gyroelongated square bicupola.
• J46 :   gyroelongated pentagonal bicupola.
• J47 :   gyroelongated pentagonal cupolarotunda.
• J48 :   gyroelongated pentagonal birotunda.

### Nomenclature :

To describe these and other common polyhedra, some systematic nomenclature is useful.  In particular, any polyhedron gives rise to many other types whose names include one or more of the following adjectives:

• Elongated: By replacing (one of) the largest m-sided polygon, with an m-gonal prism (that polygon may not be a face of the polyhedron, but an "internal" polygon with apparent edges).  This adds m vertices, 2m edges, and m faces.  The simplest example, shown at right, is the elongated tetrahedron (J7), which is an heptahedron.
• Gyroelongated: By replacing (one of) the largest m-sided polygon, with an m-gonal antiprism (that polygon is usually not a face of the polyhedron, but an "internal" polygon with apparent edges). This adds m vertices, 3m edges, and 2m faces. Gyroelongation can be performed in two different ways (often leading to different chiral versions of the same polyhedron).
• Snub: Snubbing is an interesting chiral process which, roughly speaking, amounts to loosening all faces of a polyhedron and rotating them all slightly in the same direction (clockwise or counterclockwise), creating 2 triangles for each edge and one m-sided polygon for each vertex of degree m. A polyhedron and its dual have the same snub(s)!  If a polyhedron has k edges, its snub has 5k edges, 2k vertices and 3k+2 faces.
• Truncated: By cutting off an m-gonal pyramid at one or more (usually all) of the vertices. This add (m-1) vertices, m edges and 1 face for each truncated vertex.
• Augmented: By replacing one or more of the m-sided faces with an m-gonal pyramid, cupola, or rotunda.
• etc.

Other terms are available to describe certain interesting special cases:

• Cingulum (Latin: girdle; cingere to gird).
• Fastigium (Latin: apex, height).
• Sphenoid (Greek: wedge).
• etc.

The rest of the nomenclature used in the context of Johnson solids, is best described in the words of Norman W. Johnson himself :

"  If we define a lune as a complex of two triangles attached to opposite sides of a square, the prefix spheno- refers to a wedgelike complex formed by two adjacent lunes. The prefix dispheno- denotes two such complexes, while hebespheno- indicates a blunter complex of two lunes separated by a third lune. The suffix -corona refers to a crownlike complex of eight triangles, and -megacorona, to a larger such complex of 12 triangles. The suffix -cingulum indicates a belt of 12 triangles. "

How do polyhedra generalize to 4 dimensions or more?

The equivalent of a polyhedron in dimension 4 is called a polychoron (plural polychora). Polychora are discussed extensively on beautifully illustrated pages proposed by George Olshevsky and Jonathan Bowers.

Although the introduction of the term polychoron is fairly recent, it seems now generally accepted, as there's no serious competition (the etymology of "polyhedroid" is poor and misleading). The term was coined in the 1990's by George Olshevsky, whose earlier proposal of "polychorema" (plural: "polychoremata") was unsuccessful. Olshevsky's new proposal had the early support of Norman W. Johnson, after whom the 92 "convex regular-faced solids" are named (Johnson solids).

polychoron is bounded by 3-dimensional faces, called cells. The four-dimensional equivalent of the Euler-Descartes formula is a topological relation which relates the number of vertices (V), edges (E), faces (F), and cells (C) in any polychoron enclosing a portion of hyperspace homeomorphic to a 4D open ball (provided edges, faces and cells are homeomorphic, respectively, to 1D, 2D and 3D open balls):

V - E + F - C   =   0

Such higher-dimensional generalizations of the Euler formula were first established in 1852 by  Ludwig Schläfli (1814-1895) but his results were only published in 1901  (by then, many others had rediscovered them).

In an unspecified number of dimensions, the counterpart of a 2D polygon, a 3D polyhedron, or a 4D polychoron is called a polytope,  a term coined by  Alicia Boole Scott (1860-1940) daughter of George Boole (1815-1865).

The boundary of an n-dimensional polytope consists of hyperfaces which are (n-1)-dimensional polytopes, joining at hyperedges, which are (n-2)-dimensional polytopes. (All the hyperfaces of an hyperface are thus hyperedges.) This vocabulary is consistent with the well-established term hyperplane to designate a vector space of codimension 1 (in a hyperspace with a finite number n of dimensions, a hyperplane is therefore a linear space of dimension n-1). We also suggest the term hyperline for a linear space of codimension 2 (and, lastly, hyperpoint to designate a space of codimension 3).

To denote the p-dimensional polytopes within a polytope of dimension n, the following terms may be used: vertex (p=0; plural "vertices"), edge (p=1), face (p=2), cell or triface (p=3), tetraface (p=4), pentaface (p=5), hexaface (p=6), ... hypervertex (p=n-3), hyperedge (p=n-2), hyperface (p=n-1), hypercell (p=n).

To establish and/or memorize the n-dimensional equivalent of the Euler-Descartes formula for "ordinary" polytopes in n dimensions, it's probably best to characterize each such polytope by the open region it encloses (boundary excluded), except in dimension zero (the 0-polytope is a single point). For the formula to apply, each such region should be homeomorphic [i.e., topologically equivalent] to the entire Euclidean space of the same dimension, or equivalently to an open ball of that dimension. An edge is an open segment, a face is an open disk, a cell is an open ball, etc. [for example, the ring between two concentric circles is not allowed as a face, and the inside of a torus is disallowed as a cell]. Then we notice that a number can be assigned to any polytope (and a number of other things) called its Euler characteristic (c), which is additive for disjoint sets, equal to 1 for a point and invariant in a topological homeomorphism (so that topologically equivalent things have the same c). For our purposes, this may be considered an axiomatic definition of c. It may be used to establish (by induction) that the c of n-dimensional Euclidean space is (-1)n, which is therefore equal to the c of our "ordinary" open polytopes (HINT: A hyperplane separates hyperspace into three parts; itself and 2 parts homeomorphic to the whole hyperspace). The c of all "ordinary" closed polytopes in n dimensions is the c of a closed n-dimensional ball and it may be obtained by inspecting the components of the boundary of any easy n-dimensional polytope like the hypercube or the simplex polytope discussed below. It turns out to be equal to 1 in any dimension n. If the hypercell itself (the polytope's interior) is excluded from the count, as it is in the traditional 3-dimensional Euler-Descartes formula, the RHS of the formula will therefore be 2 in an odd number of dimensions and zero in an even number of dimensions. For example, in 7 dimensions, if we denote by T the number of tetrafaces, by P the number of pentafaces (hyperedges), and by H the number of hexafaces (hyperfaces), we have:

V - E + F - C + T - P + H   =   2

We may focus on the n-dimensional equivalent of the Platonic solids, namely the regular convex polytopes, whose hyperfaces are regular convex polytopes of a lower dimension, given the fact that the concept reduces to that of a regular polygon [equiangular and equilateral] in dimension 2. In dimension 3, this gives the 5 regular polyhedra. In dimension 4, we have just 6 convex regular polychora, first described by Schläfli:

• The  regular pentachoron  (4-dimensional simplex).
• The  tesseract  (4-dimensional hypercube).
• The  hyper-diamond  (24 octahedral cells, 6 per vertex, 3 per edge).
• The  dodecaplex  (120 dodecahedral cells, 4 per vertex, 3 per edge).
• The  tetraplex  (600 tetrahahedral cells, 20 per vertex, 5 per edge).

In dimension 5 or more, only 3 regular polytopes exists which belong to one of the following three universal families (also existing in lower dimensions):

Familyn-Polytopedim. V
0
E
1
F
2
C
3
T
4
p-Faces
p
Simplex  n C(n+1,p+1)
Point 01
Segment 121
Triangle 2331
Tetrahedron 34641
Pentachoron 45101051
Cross-
Polytope
(orthoplex
or cocube)
n C(n,p+1) 2p+1
Point 01
Segment 121
Square 2441
Octahedron 361281
Hypercube  n C(n,p) 2n-p
Point 01
Segment 121
Square 2441
Cube 381261
Tesseract 416322481

The regular simplex polytope is obtained by considering n+1 vertices in dimension n, so that each one is at the same distance from any other (its hyperfaces are simplex polytopes of a lower dimension). Choosing as vertices all points whose Cartesian coordinates are from the set {-1,+1}, we obtain an n-dimensional hypercube (of side 2). The hyperfaces of an hypercube are hypercubes of a lower dimension. The dual of the above hypercube is the regular cross polytope whose vertices have a single nonzero coordinate, taken from the set {-1,+1}.

The  interactive hypercube  at right is from Kurt Brauchli (details here).  Click and drag with the mouse to turn the cube aound the chosen axes (H and V) indicated in the menu bar.  This 5D cube projects like a 3D cube if you rotate only around axes 0, 1 or 2.  The fourth and fifth dimensions appear with axes 3 and 4.

With the bottom cursor, you may choose a distant (left) or close-up perspective (right).
 Your browser is unable to run the Java "applet" corresponding to the interactive pictureat this location... Sorry.

The word  tesseract  was coined by  Charles Howard Hinton  (1853-1907)  the son-in-law of the logician George Boole (1815-1864).  C. Howard Hinton was obsessed with visualizations of the fourth dimension involving the tesseract, which he presented in his 1904 book entitled  The Fourth Dimension,  whose third edition (1912) has been made available online by rkumar, thanks to Banubula.

EnolaStraight (2002-05-07)
What's the radius of the circle touching 3 touching unit circles?
What's the radius of the sphere touching 4 touching unit spheres?
[In this (edited) question, "touching" means "externally tangent (to)".]

The generalization of this question to any number of dimensions is a classic demonstration that whatever geometrical intuition we may have developed in two or three dimensions may not be trusted in a space of more dimensions. The two-dimensional case [pictured at right] shows 3 congruent circles, centered on the vertices of an equilateral triangle, touching each other and a much smaller circle [pictured as a red disk] whose radius has to be determined. Based on this 2-D case [and, to a lesser extent, on the 3-D case] it would seem that such an inner ball [disk, sphere, or n-dimensional hypersphere] would always be small enough to fit inside the simplex [equilateral triangle, regular tetrahedron, n-dimensional regular simplex] formed by the centers of the congruent balls. This happens to be true for a dimension equal to 4 or less, but fails for a dimension of 5 or more. In a very large number of dimension, the (linear) size of the inner ball is about 41% the size of the outer ones. More precisely, Ö2-1 = 0.41421356... is the limit of that ratio when the number of dimensions tends to infinity. Read on...

Consider the center O of the n-dimensional simplex formed by the n+1 centers of the congruent balls [each of radius 1]. The critical quantity is the distance D(n) from the center O to any vertex; the radius of the inner ball is simply D(n)-1. Well, because O is the center of gravity of n+1 vertices, it is on the line that goes from a vertex to the center of gravity of the n others. It divides that line in a 1 to n ratio. The length of that line is therefore (1+1/n)D(n) and it is also one side of a right triangle whose other side is of length D(n-1) and whose hypotenuse is of length 2 (it's the side of the simplex, the distance between the centers of two balls). In other words, D(n) is given recursively by the relations:

D(1) = 1     and     D(n) = Ö[4-D(n-1)2] / (1+1/n)

This recursion can be used to prove, by induction, the following formula:

D(n)   =   Ö[ 2n / (n+1) ]

The simplicity of this result is a hint that there might be a more direct way to obtain it. D(2) = 2/Ö3 says that the radius of the inner circle in the above figure is 2/Ö3-1 [about 15.47 %] of the radius of any outer circle. Similarly, the corresponding ratio for spheres is ½Ö6-1 [about 22.4745 %]. The limit of D(n) is Ö2: In a space with a very large number of dimensions, the ratio of the radius of the inner hypersphere to the radius of any outer hyperspheres is thus slightly less than Ö2-1 [about 41.42 %].

In dimension n, the distance from the center O to any of the hyperplanes (of dimension n-1) of the "faces" is D(n)/n.  Therefore, if the radius D(n)-1 is greater than is D(n)/n., the inner hypersphere bulges outside of the n-dimensional regular simplex formed by the centers of the outer hyperspheres. This happens as soon as n2-5n+2 > 0, which is the case when n is at least equal to 5. This higher-dimensional configuration is contrary to the intuition we would form by looking only at the two-dimensional and/or three-dimensional cases...

Higher Dimensional Spheres  by  Kelsey Houston-Edwards  (PBS Infinite Series,  2016-11-17).

Dr. Murali V.R. (2004-02-25; e-mail)
What is the volume of a regular antiprism?

A regular antiprism is a polyhedron whose faces are two parallel n-gonal bases  [regular polygons with n sides]  and 2n equilateral triangles called lateral faces.

Look at the outline of such a solid from above, and what you see is a regular polygon with 2n sides (every other vertex is on the top base, and every other one is on the bottom).  The angle at each vertex of this outline is thus  q = p-p/n.

Now, each lateral face is seen as an isosceles triangle having an angle q at the top and featuring a base observed at its real size  a  (as the direction of observation is perpendicular to it).  The height of such an isosceles triangle is thus:

½ a cotan (q/2)   =   ½ a tan (p/2n)

This quantity is also equal to the length of a side of a right triangle whose hypotenuse is the true height of a lateral face (namely ½ aÖ3) and whose other side is the height  h  of the antiprism  (namely, the distance between its bases).  This gives the height  h  of the antiprism in terms of the length  a  of its edges:

 h   =   ½ a Ö 3 - tan2(p/2n)

Consider the circumscribed prism of height h  whose base is the 2n-gonal outline.  Each side of this outline is equal to  ½ a / cos(p/2n).  Its surface area is therefore:   (n a2/4) / sin(p/n)   and the volume of the prism is h times that.

The  antiprism  is obtained from this prism by removing 2n triangular pyramids of height h whose bases are all congruent to the above isosceles triangle, for a combined base area of  (n a2/2) tan(p/2n)   and a total volume h/3 times that.

The volume V of the antiprism is the difference between these two volumes:

 V   =   (n a3 / 24)  [ 3 / sin(p/n) - 2 tan (p/2n) ] Ö 3 - tan2(p/2n)

This can be rewritten in a much more palatable form, using   t = tan (p/2n)  :

V   =   n a3 ( 3 - t 2 ) 3/2  /  48t

 h   =   ½  a Ö 3 - tan2(p/2n) V   =   n h 3 / 6 tan(p/2n)

Two noteworthy special cases (for a = 1):

• V = 1 / Ö72   when n=2.  A regular tetrahedron! [A degenerate but valid case.]
• V = ½ Ö3   when n=3.  A regular octahedron...

(2006-01-18)   Szilassi Polyhedron  &  Császár Polyhedron
Two polyhedra of genus 1 [topology of a torus] dual of each other.

AuthorDateFacesEdgesVertices
Lajos Szilassi   (1942-)1977   7  hexagons  2114
Ákos Császár  (1924-)     1949   14  triangles217

In a polyhedron, a line between two nonadjacent vertices is called a diagonal.  When every pair of vertices is connected by an edge, the polyhedron has thus  no diagonals.  The tetrahedron is an example of a polyhedron without diagonals, so is  Császár's polyhedron.  By duality, this means that every face of the  Szilassi heptahedron  has an edge in common with each of the other 6 faces...

Geometrically, the Szilassi polyhedron has an axis of 180° symmetry: 3 pairs of congruent faces and a symmetrical face (darkest in the picture).  This symmetry allows one to build a full mental picture of the polyhedron from the image at right  (obtained from David Eppstein's Geometry Junkyard).

### Beyond Császár and Szilassi...

The Descartes-Euler formula for a polyhedral surface of  genus  G  is:

V - E + F   =   2 - 2G

In a polyhedron where all pairs of faces share an edge, E = (F-1)F/2.  Also, we have V = 2E/3, since any vertex must belong to only 3 faces.  Eliminating E and V using these two relations makes the above Euler formula boil down to:

G   =   (F-4)(F-3) / 12

As G is an integer, F must have definite values modulo 12, namely 0, 3, 4 or 7.  Beyond F=4 (the tetrahedron of genus 0) and F=7 (the Szilassi heptahedron of genus 1) the next possibility would thus be F=12, a dodecahedron of genus 6...

### Curved toroidal maps which can't be straightened :

The number of colors required to color a map (or a graph) so that no two adjacent patches (or nodes) are of the same color is called its  chromatic number.  For any given surface other than the  Klein bottle  (a surface of genus 1 on which any map can be colored with only 6 colors)  the maximal chromatic number of all maps drawn on it depends only on its geometric genus G, according to the following formula, proposed by P.J. Heawood in 1890  (A000934).

Maximal Chromatic Number   =   ë ½ ( 7  +  ( 48G + 1 ) ½ ) û

 Geometric Genus Heawood Number 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 4 7 8 9 10 11 12 12 13 13 14 15 15 16 16 16 17

For genus 0, this amounts to the celebrated  4-color theorem  for planar or spherical maps, as proved by Appel and Haken in 1977.  Otherwise, the formula was shown to be an upper bound by Heawood in 1890.  It was found to be  exact  (except for the Klein bottle)  in 1968, when Ringel and Youngs showed how a map with this many countries could always be drawn on a surface of genus G so that two countries always have a common boundary.

When  G  is  (F-4)(F-3) / 12,  the Heawood formula gives precisely a maximal chromatic number equal to F  [check it].  The challenge met by Szilassi for genus 1  (a topological torus)  was to draw a  Ringel-Youngs map with 7  flat countries.  Could the same feat be possible for the next cases,  starting with a genus-6 surface consisting of 12 planar faces?

Miro Dabrowski (of Bunbury, Australia. 2006-07-21)
What are the dihedral angles in a  buckyball ?
How do I taper 32 ready-to-glue wooden faces ?

First, let's consider how the overall size of the buckyball relates to the length of its edges  (a).  One easy way to do so is to consider the  equator  of the ball if the  polar axis  goes through the centers of two opposite pentagonal faces  (for example, let the polar axis be vertical in the above picture).  This equator is a  regular decagon  whose sides are of length  1.5 a  (namely, the distance between the middles of two nonadjacent edges in a  regular hexagon  of side  a).  The radius of the circle  circumscribed  to that decagon is the so-called  midradius  (r)  of the buckyball  (i.e., the distance from the ball's center to the middle of any  edge).  Thus,  3a/ 4r  is the  sine  of  p/10  (or 18°)  which is  1/2f.  This yields the first relation below.  The other equation gives the  circumradius  (R)  of the buckyball  (i.e., the distance  from the center to any  vertex )  obtained from the Pythagorean theorem  (r2 + a2/4  =  R).

 r / a   = 3/4  (1+Ö5)   =   3 f/2 =   2.42705098312484227230688025... R / a   = 1/4  Ö 58  +  18 Ö5 =   2.47801865906761553756640791...

solid  buckyball  (a truncated regular icosahedron)  consists of 32 straight regular pyramids  (12 pentagonal and 20 hexagonal ones)  with a common apex  at the center O.  All the lateral faces of those pyramids are congruent to an isosceles triangle of base  a  and sides  R.

A thick buckyball "shell" may be assembled from wooden faces (hexagonal or pentagonal) whose sides are tapered at an angle equal to the angle between the base and any lateral face of such a pyramid.  This is the only way to have the same taper for all hexagonal sides, in order to simplify tooling and final assembly.

The interior shape will not be a perfect buckyball itself if boards of the same thickness are used for both types of faces.  If needed, this flaw (which is normally hidden) could be eliminated by shaving about 5.1% off the boards used for the hexagonal blocks  (since the height of an hexagonal pyramid is about 94.9% of the height of a pentagonal one).
 Each n-gonal pyramid consists of n slices  (as pictured at left)  sharing an edge from O to the center  M  of the n-gon  (n is 5 or 6). a  is the length of every edge. R  is the  circumradius  (distance from the center O to any vertex). r  is the  midradius  (distance from O to [the middle of] any edge). bn  =  ½ a / tan(p/n)   is the inscribed radius of the n-gonal face.

The aforementioned taper angles  (q = 0  would be a straight cut)  are obtained from the formula   qn  =  arcsin ( bn / r ).  Numerically, this boils down to:

 q5   =   arcsin Ö (5+Ö5)  /  90 =   16.4722106927603966099...° q6   =   arcsin [ (Ö5 - 1)Ö3 / 6 ] =   20.9051574478892990329...°

Between an hexagon and an adjacent pentagon, the dihedral angle is  q5+q6.  The dihedral angle between two adjacent hexagons is 2q6.

Angular distances seen from the center of a buckyball :
Angle...SymbolExpressionValue in degrees
subtended
by an edge
a
 2 arcsin a 2R
23.28144627 °
from face center
to vertex
b5
 arcsin a 2R sin p/5
20.07675127 °
b6
 arcsin a 2R sin p/6
23.80018260 °
from face center
to [middle of] edge
q5
 arcsin a 2r tan p/5
16.47221069 °
q6
 arcsin a 2r tan p/6
20.90515745 °
between middles of edges, 1/3
of the way around an hexagon
p/5 36 °

A useful check is obtained by considering any journey along half a  great circle  (180°) which includes an edge between hexagons:

180°   =   p   =   a  +  2 (b5  +  q)  +  4 q6

A similar "trick" was used to obtain trivially the last entry of the above table  (actually, this was our starting point to obtain the midradius-to-edge ratio).

The  deficiency  at each of the 60 buckyball vertices is 12°, namely  4p/60 = p/15.  It may also be computed directly as  2p-2(2p/3)-3p/5.

The  deficiency  at a vertex of a polyhedron is defined to be what's missing from a full turn  (2p  or 360°)  when you add up all the angles of the faces which meet at that vertex.  The name is historically linked to Descartes'  deficiency theorem  which states that, in any polyhedron homeomorphic to a sphere, the sum of the deficiencies at all vertices is equal to an angle of 4p  (or 720°).

(2013-02-22)   Zonogons, Zonohedra, Zonotopes and Zonoids
zone  is a belt of polygons joined by  parallel edges.
In a  zonoid,  every 2n-gonal face belongs to n zones.

zonotope  is a convex polytope obtained as the Minkowski sum of line segments.  In three-dimensional space, a zonotope is called a  zonohedron.

zonotope  is said to be  nonsingular  when it's the Minkowski-sum of n line segments among which no 3 segments are coplanar  (some authors consider  only  this special kind of zonotopes).  The edges of such a  nonsingular zonotope  have  n  possible directions  (all the edges along one direction have the same length).  Each direction defines a single  zone  containing  2(n-1)  faces.  There are  n(n-1)  faces; every face is a parallelogram that belongs to  two  zones.

In a space of dimension d less than n, a nonsingular zonotope of order n can viewed as the  shadow  of an n-dimensional hypercube.  In three dimensions, a nonsingular zonohedron of order  n ≥ 3  has  n(n-1)  faces,  2n(n-1)  edges and  2+n(n-1)  vertices.

This doesn't apply to  singular  zonohedra whose faces  (zonogons)  can be dissected into parallelograms but need not be parallelograms themselves.

 My  favorite zonohedron  is the  scalene isogonal tetradecahedron  pictured at right  (with void rectangular faces, through which some  inner  faces are shown, with pencil marks on them).   It's rarely mentioned elsewhere, except in its more regular incarnation, commonly dubbed  truncated octahedron,  where the rectangular faces are square  (isosceles isogonal tetradecahedron).   The equilateral version  (uniform truncated octahedron) is an  Archimedean solid  which is space-filling  (see below)  and serves as the  node-skeleton  of  Kelvin's foam.

Zonohedra, zonohedrification  and  some zonohedra   by  George W. Hart
Zonohedra and Zonotopes  by  David Eppstein   (The Geometry Junkyard)
The Zonohedra Music Chart  by  Caspar Schwabe  (Polytopia Performance)
Zonohedra (MathWorld)   |   Zonohedra (Wikipedia)
Rhombo-hexagonal dodecahedron   =   Elongated dodecahedron   =   extended rhombic dodecahedron
Truncated rhombic dodecahedron   =   hexatruncated rhombic dodecahedron
Evgraf Stepanovich Fedorov (1853-1919)  Russian crystallographer who introduced zonotopes.

(2013-02-16)   Space-filling polyhedra  ( Plesiohedra )
There are  many  of those:

(2016-01-04)   Pyritohedron :   Space-filling pentagonal dodecahedron.
The only plesiohedron in a parametric family of isohedra.

The  pyrithedron  belongs to a one-parameter family of dodecahedral shapes which interpolate continuously between a  split-faced cube  and a  rhombic dodecahedron  (neither extreme case being included in the family).  They all feature  12  congruent  pentagonal faces,  30  edges and  20 vertices. just like the  regular dodecahedron,  which is one of them!

 This is illustrated by the morphing animation at right,  made in November 2009 by  Tom Ruen  (a computer programmer born in 1968).  His picture shows  8  fixed vertices   (±1, ±1, ±1)   and 12 that depend on  h  in the  interval  ]0,1[ :   (±(1-h2 ), ±(1+h), 0) (±(1+h), 0, ±(1-h2 ) )   and   (0, ±(1-h2 ), ±(1+h) )   The  6  edges parallel to a coordinate axis have length  2-2h2.  The other  24 edges have a length of Ö(1+h2+h4).

Both quantities are equal when  h  is  (Ö5-1)/2  (the inverse of the golden ratio).  In that special case, all edges have length  Ö5-1 while the horizontal diagonal has length 2, which makes the ratio of the diagonal to the size equal to the golden ratio, thus establishing that the faces are inddeed regular pentagons and, consequentely that the solid is a  a regular dodecahedron.

For the solid to be space-filling, a whole number of them must fit around each of the six special edges.  That number is 2 or 4 in the extreme cases where h is 0 or 1.  For a  proper  member of the family to be space-filling, the aforementioned number must be 3, which implies a dihedral angle of  120° and  h = 1/Ö3.  The corresponding shape is called a  pyritohedron.

h-parameter, edge lengths and volume of special related solids
Split-faced CubePyritohedronRegular DodecahedronRhombic Dodecahedron
h0
 Ö3 = 0.57725... 3
 Ö5-1 = 0.61803... 2
1
e624/3 = 1.333333...Ö5 - 1   =
1.236067977...
0
e241Ö13/3 = 1.20185...Ö3
V8 8+32/Ö27  =
14.158402871356+
10+2Ö5   =
14.472135955-
16

### Proofs of the above statements :

Let  1+h  be the altitude  (z)  of the horizontal top edge and  2d  be its length.  The extremities of that edge are  A' = (0,-d,1+h)  in the back and  A = (0,d,1+h)  in the front.

Let's now consider the pentagonal face whose highest point is the latter vertice .  (It's the pentagon with the lightest color in the animation).

In that pentagon,  A is connected to the fixed points  B=(1,1,1)  and  E=(-1,1,1).  Those two are respectively connected to two base points on the equator  (connected to each other by an horizontal edge).
Namely:  C = (d,1+h,0)  and  D = (-d,1+h,0).

First, we must establish the relation between  h  and  d  from the fact that A, B, C, D ane E  are coplanar.  One way to do so is to say that the determinant  of EA, EB and ED vanish  (EC will be in the same plane by symmetry):

 0   = 1 d-1 h =   2h2 + 2d - 2 2 0 0 1-d h -1

Therefore,   d  =  1 - h2   as advertised...

Each pentagonal face has a special side of length  2d = 2-2h2.  The other sides have a length whose square is  1+h2+(1-d)2  =  1+h2+h4.  The pentagon is equilateral when both lengths are equal:

(2-2h2 )2   =   1+h2+h4

This boils down to   3 ( h4 - 3 h2 + 1 )   =   0   a quadratic equation in  h2  whose only solution below 1 is  h2 = (3-Ö5)/2.  Therefore,  h = (Ö5-1)/2.

### Volume :

The polyhedron discussed so far can be dissected into  19  solids:

• A cube of side 2 and volume 8.
• Six prisms of height  2d,  based on a triangle of height h and base 2.
• Twelve pyramids of height h based on a rectangle of sides 2 and 1-d.

Thus, the total volume goes from  V = 8  (for h = 0)  to  V = 16  (for h = 1):

8  +  6 ( 2(1-h2 ) . h )  +  12 ( 2h2 . h/3 )   =   8 + 12h - 4h3   =   V(h)

 V   =   4 (2-h) (1+h)2

Space-filling polyhedra (MathWorld)   |   Space-filling polyhedra (Wikipedia)
Five Space-Filling Polyhedra  by  Guy Inchbald
The honeycomb theorem   |   Most economical foams
US Patent 5168677  by  Antonio C. Pronsato  &  Ernesto D. Gyurec   (1992)

### Belisha J. Price, polyhedron fan   (1938-)

Born in India, brought up in the London Blitz  and educated at the Duke of York's Royal Military SchoolBelisha  was a telephone technician in  London  before emigrating to  New Zealand  in 1961.  He taught Maths at Cambridge High School for 35 years.

Retired (2006)  |  Students  |  CyberSpace  by  Belisha Price  |  Belisha Price  by  Jim Hogan  |  Wikipedia