[画像: Symmetrical Tetragonal Antiwedge ] 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:

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

Joseph-Louis Lagrange (1736-1813) 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).

[画像: Come back later, we're still working on this one... ]

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.)

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

The duality of polyhedra is an involutive relationship (i.e., 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 Jakob Steiner (1796-1863) 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.

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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 respectively located on OH and OH', then I, M and M' are aligned when the following numerical relations hold:

OH . OM = OH' . OM' = OI²

The Crux of Polyhedral Duality

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

The power of point O with respect to that circle is defined to be OH.OM and it's 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 the configuration of O, I, H' and M'. So, if the advertised numerical equalities hold, then IM and IM' are both perpendicular to OI. Therefore, I, M and M' are aligned. QED

For a given edge of a polyhedron, we 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 onto 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 of the same chirality (uniquely defined, modulo a rotation, scaling and translation).

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

Counting Polyhedra See the (extended) table 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. [画像: Tetrahedron ]

There's only one tetrahedron:
[画像: Triangular Prism (pentahedron) ] [画像: Square Pyramid (pentahedron) ]

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.

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BenO (Ben Ocean, 2001年07月28日; 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.)

Regular Tetrahedron

ABCD

x = 2 2 -4 0

y = 2Ö3 -2Ö3 0 0

z = -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°).

[画像: Tetrahedron ] [画像: Cube ] [画像: Octahedron ] [画像: Dodecahedron ] [画像: Icosahedron ]

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°

[画像: Come back later, we're still working on this one... ]

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

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(2013年01月30日) 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!

[画像: Cuboctahedron ]
[画像: Icosidodecahedron ]

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).

[画像: Rhombic dodecahedron ] [画像: ] [画像: Rhombic triacontahedron ]

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)

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(2013年02月13日) 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 orthohedral. Some 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).
• A 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 :

• Isogonal polyhedra are equiradial.
• 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

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(2001年07月28日) Uniform Polyhedra
A 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's 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 we allow two pairs of faces to meet on the same edge. This last shape has an Euler characteristic of 24 (60-240+204) and is 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 Hexagonal antiprism Uniform Polyhedra :

The platonic solids are clearly convex and uniform.
So are equilateral antiprisms and uniform prisms. [画像: Hexagonal prism ]

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

The Uniform Convex Solids ( their duals are faur dice )

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

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

Those are only ad hoc exclusions; it's occasionally quite useful to apply what's known for antiprisms to tetrahedra and octahedra!

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(2001年07月28日) 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, Coat-of-Arms of Johannes Kepler (1571-1630) 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.

Buckyball 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.

Great Rhombicosidodecahedron

The buckyball is one of 4 Archimedean solids without triangular faces. [画像: Truncated Octahedron ] The other three are the truncated octahedron (at left), the great rhombicosidodecahedron (at right) and the great rhombicuboctahedron.
[画像: Great Rhombicuboctahedron ]

This leaves 4 nonchiral Archimedean solids with vertices of degree 4 :

[画像: Rhombicosidodecahedron ] [画像: Cuboctahedron ] [画像: Rhombicuboctahedron ] [画像: Icosidodecahedron ]

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

[画像: Snub Cube ] [画像: Snub Dodecahedron ]

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

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(2013年04月22日) 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.

[画像: Come back later, we're still working on this one... ]

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(2000年11月19日) Types of polyhedra named after a polygon :

[画像: Hexagonal Prism ] 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. [画像: Hexagonal Antiprism ] 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.

[画像: Pentagonal Deltohedron ]

A 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

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 2^n-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).

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(2002年06月14日)
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. [画像: Icosidodecahedron ]

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, [画像: Pentagonal rotunda (J6) ] the icosidodecahedron could also be called a pentagonal gyrobirotunda but that name would mask its much greater symmetry compared to the pentagonal orthobirotunda (J34) [画像: Cuboctahedron ] 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. [画像: Truncated dodecahedron ]

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...

[画像: Come back later, we're still working on this one... ]

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.

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(2000年11月19日) 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:

[画像: Tetrahedron ] [画像: Triangular Dipyramid ] [画像: Octahedron ] [画像: Pentagonal Dipyramid ]

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).

[画像: Snub Disphenoid ] [画像: Triaugmented Triangular Prism ] [画像: Gyroelongated Square Dipyramid ] [画像: Icosahedron ]

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)...

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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:

• 5 Platonic polyhedra
• 13 Archimedean solids
• 92 Johnson solids.

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. [画像:J7 heptahedron = elongated tetrahedron] 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. "

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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).

A 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 regular hexadecachoron (cross-polychoron).
• 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
0E
1F
2C
3T
4 p-Faces
p

Simplex n C(n+1,p+1)

Point 0 1

Segment 1 2 1

Triangle 2 3 3 1

Tetrahedron 3 4 6 4 1

Pentachoron 4 5 10 10 5 1

Cross-
Polytope
(orthoplex
or cocube) n C(n,p+1) 2^p+1

Point 0 1

Segment 1 2 1

Square 2 4 4 1

Octahedron 3 6 12 8 1

Hexadecachoron 4 8 24 32 16 1

Hypercube n C(n,p) 2^n-p

Point 0 1

Segment 1 2 1

Square 2 4 4 1

Cube 3 8 12 6 1

Tesseract 4 16 32 24 8 1

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 picture
at 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.

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EnolaStraight (2002年05月07日)
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)".]

[画像: Touching unit circles around a smaller circle ]

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...

Regular Polytopes in N Dimensions (26:18) by Carlo Séquin (Numberphile, 2016年03月23日).
Higher Dimensional Spheres by Kelsey Houston-Edwards (PBS Infinite Series, #1 2016年11月17日).

March 2016: Maryna Viazovska (1984-)

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[画像: Hexagonal Antiprism ] Dr. Murali V.R. (2004年02月25日; 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:

A lateral face seen from a direction perpendicular to both bases.

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:

Vinculum

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:

Vinculum

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

[画像: Regular Octahedron ] 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...

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

AuthorDateFacesEdgesVertices

Lajos Szilassi (1942-) 1977 7 hexagons 21 14

Ákos Császár (1924-) 1949 14 triangles 21 7

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...

Szilassi Heptahedron

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 an undirected 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 (1861-1955) in 1890 (A000934).

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

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?

A Six-Color Problem by Philip Franklin (April 1934).
Four, five, and six color theorems in Nature of Mathematics by William Tross (2010).

[画像: Truncated Icosahedron ]

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Miro Dabrowski (of Bunbury, Australia. 2006年07月21日)
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, [画像: The midradius is the radius of a circle circumscribed to a decagon of side 0.75 a. ] 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 = R² ).

r / a = ³/₄ (1+Ö5) = ^{3 f}/₂ = 2.42705098312484227230688025...

Vinculum

R / a = ¹/₄ Ö 58 + 18 Ö5 = 2.47801865906761553756640791...

A 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).

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

Vinculum

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 :

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 + q₅ ) + 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°).

Guy makes a big ball out of plywood (3:15) by Keith Williams (2018年01月27日).

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

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

A 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.

[画像: Hexagonal Prism ] [画像: Truncated Octahedron ] [画像: Great Rhombicuboctahedron ] [画像: Great Rhombicosidodecahedron ]
[画像: Cube ] [画像: Rhombic Dodecahedron ] [画像: Rhombic Triacontahedron ]

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.

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

[画像: Rhombic Dodecahedron ] [画像: Cuboctahedron ] [画像: Truncated Octahedron ]
[画像: Cube ] [画像: Triangular Prism ] [画像: Hexagonal Prism ]

Plesiohedron

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(2016年01月04日) 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!

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

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 = Vertical bar 1 d-1 h Vertical bar = 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.

[画像: Come back later, we're still working on this one... ]

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

[画像: Come back later, we're still working on this one... ]

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)

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Thumbs up, Teacher!

Belisha Price 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 School, Belisha was a telephone technician in London before emigrating to New Zealand in 1961. He taught Maths at Cambridge High School for 35 years.

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