Euler's polyhedral formula:   f + v = e + 2

All convex polyhedra verify this relation between the numbers of faces, of vertices and of edges. Actually, the Euler-Poincare characteristic  f+v-e  of a convex polyhedra is 2 (Descartes-Euler theorem).
Discussions between a professor and his students about the hypothesis of Euler's conjecture allows Imre Lakatos to give us an interesting analysis about the mathematical knowledge in Proofs and Refutations  (Cambridge University Press, 1976).
Other polyhedra satisfy the Euler formula: polyhedra without "holes", and more generally those which are topologically equivalent to a connected network (without intersections) drawn on a sphere. One can prove this relation with the help of Schlegel diagrams.
Convex polyhedra have many other numerical properties among which  e+6 ≤ 3f ≤ 2e  and  e+6 ≤ 3v ≤ 2e

Here are the characteristics of the semi-regular polyhedra of the first kind (for their duals we have just to exchange v and f):

   v   e   f    f3 f4 f5 f6 f8 f10 fn dual polyhedron (face)
truncated tetrahedron 12 18 8   4 - - 4 - - - triakis-tetrahedron (isosceles triangle)
truncated cube 24 36 14   8 - - - 6 - - triakis-octahedron (isosceles triangle)
truncated octahedron 24 36 14   - 6 - 8 - - - tetrakis-hexahedron (isosceles triangle)
cuboctahedron 12 24 14   8 6 - - - - - rhombic dodecahedron (rhombus)
small rhombicuboctahedron 24 48 26   8 18 - - - - - deltoidal icositetrahedron (kite)
great rhombicuboctahedron 48 72 26   - 12 - 8 6 - - disdyakis-dodecahedron (triangle)
snub cube 24 60 38   32 6 - - - - - pentagonal icositetrahedron (pentagon)
truncated dodecahedron 60 90 32   20 - - - - 12 - triakis-dodecahedron (isosceles triangle)
truncated icosahedron 60 90 32   - - 12 20 - - - pentakis-dodecahedron (isosceles triangle)
icosidodecahedron 30 60 32   20 - 12 - - - - rhombic triacontahedron (rhombus)
small rhombicosidodecahedron 60 120 62   20 30 12 - - - - deltoidal hexecontahedron (kite)
great rhombicosidodecahedron 120 180 62   - 30 - 20 - 12 - disdyakis-triacontahedron (triangle)
snub dodecahedron 60 150 92   80 - 12 - - - - pentagonal hexecontahedron (pentagon)
regular prisms (order n ≥ 3)  2n   3n  n+2   - n - - - - 2 regular diamonds (isosceles triangle)
regular antiprisms (order n ≥ 3) 2n 4n 2n+2   2n - - - - - 2 regular antidiamonds (kite)

In "rhombicuboctahedron" the prefix "rhombi" points out that 12 square faces belong to the faces of a rhombic dodecahedron (dual of the cuboctahedron).
Likewise the 30 square faces of the "rhombicosidodecahedra" belong to the faces of a rhombic triacontahedron (dual of the icosidodecahedron).
The great rhombicuboctahedron is also the truncated cuboctahedron; likewise the great rhombicosidodecahedron is the truncated icosidodecahedron.

Among the the Platonic and Archimedean solids the cuboctahedron is the only polyhedron inscribed in a sphere with radius equal to its edge (as the hexagon is the only regular polygon inscribed in a circle with radius equal to its side).
We notice the poverty of the tetrahedron family: the tetrahedron truncated using the midpoints of the edges is the octahedron, the snub tetrahedron is the icosahedron, but the truncation using the thirds of the edges gives the truncated tetrahedron.
The two snubs (and theirs duals) have no plane of symmetry; they exist in two mirror images forms.
Only five (semi-)regular polyhedra "fill the space": the cube, the triangular prism, the hexagonal prism, lord Kelvin's polyhedron (truncated octahedron) and the rhombic dodecahedron (dual of the cuboctahedron).

Generalization to non convex polyhedra:   f + v = e + 2 - 2t    where t is the number of tunnels of the polyhedron (its genus).

dodecahedron (wood) dodecahedron (copper wire) dodecahedron (origami)
Nice polyhedra can be built in many different ways: glued wood, welded copper wire, origami...

Descartes' defect theorem:   Σδj = 4π

The solid angle's defect δ of a polyhedron's vertex is the difference between 2π and the sum of face angles at that vertex:  δ=2π-Σai radians  (π radians = 180°).
The defect theorem (the sum of the defects at all vertices of a polyhedron is equal to 4π) is closely related to Euler's formula; it holds for any convex polyhedron (and more generally for any polyhedron homeomorphic to a sphere).
examples: cube:  8[2π-3(π/2)] = 4π
regular tetrahedron:  4[2π-3(π/3)] = 4π
regular dodecahedron:  20[360°-3x108°] = 720°      
cuboctahedron:  12[2π-2(π/2+π/3)] = 4π
regular pentagonal pyramid:  5[360°-(108°+2x60°)]+(360°-5x60°) = 720°
rhombic dodecahedron:  6[360°-4α]+8[360°-3(180°-α)] = 720°   α=35.265...°

Remark about the sum of all the face angles:   (Σαij)/2π = e-f = v-2
examples: cube:  8[3(π/2)]/2π = 6    12-6 = 8-2 = 6
regular tetrahedron:  4[3(π/3)]/2π = 2    6-4 = 4-2 = 2
regular dodecahedron: 20[3x108°]/360° = 18    30-12 = 20-2 = 18  
cuboctahedron:  12[2(π/2+π/3)]/2π = 10    24-14 = 12-2 = 10
regular pentagonal pyramid:  [5(108°+2x60°)+5x60°]/360° = 4    10-6 = 6-2 = 4
rhombic dodecahedron:  [6x4α+8x3(180°-α)]/360° = 12    24-12 = 14-2 = 12

other convex polyhedra ...

Convexity is a quite natural notion. Without giving an explicit definition it's easy to recognize a convex polyhedron: all its "real diagonals" (segments joining two vertices which don't belong to the same face) are inside.

Minkowski's condition:  Let  u1, u2, ... uk  be unit vectors that span the space, and  a1, a2, ... , ak  positive numbers. Then there exists a convex polyhedron (unique up to translation) having facet unit normals  u1, u2, ... uk  and corresponding facet areas  a1, a2, ... , ak  if and only if   a1 u1 + ... + ak uk = 0   (condition for convex polytopes)

Of course we can be interested in a specific category of convex polyhedra, for instance:
 •  the polyhedra with regular faces: we know those discovered by Plato and Archimedes, the prisms and antiprisms;
     the other are the 92 Johnson's solids (pyramids, diamonds, cupolas... assemblages of the previous),
 •  the different types of hexahedra, less easy to find as the tetrahedra (which have all four triangular faces) or the pentahedra (whose two types are the quadrilateral pyramids and the "roofs" formed by three quadrilaterals and two triangles),
 •  the deltahedra whose faces are all equilateral triangles,
 •  the pyramids which allow to produce jigsaws,
 •  the tectohedra which are obtained by successive cuttings of tetrahedra,
 •  the Waterman's polyhedra whose vertices are centers of spheres in a cubic close packing,
 •  the zonohedra whose faces have all a center of symmetry,
 •  the zonish polyhedra with "zones" of parallelograms,
 •  the rhombic polyhedra whose faces are all rhombi,
 •  the polyhedra with regular faces and golden rhombi,
 •  the polyhedra with regular faces and 1:1:1:2 isosceles trapezoids,
 •  the polyhedra with regular faces and specific rhombi (SR2 or golden rhombi),
 •  the spherical polyhedra (domes),
 •  the Goldberg's polyhedra,
 •  the 144 polyhedra which we get by truncations of edges of the cube,
 •  the symmetrohedra: convex polyhedra with "not much" types of faces, among them "many" regular ones (pdf document),
 •  simplest examples of canonical polyhedra with one of the 17 types of symmetry,
 •  . . .
Many other examples have interesting properties: the alveolus of a honeycomb, the crystals, the fullerenes (molecules of carbon), the cut gems . . .

gems

Polyhedra with only pentagonal and hexagonal faces have always twelve pentagonal faces.
The fullerenes have such structures, for example the C60 (truncated icosahedron with 20 hexagonal faces) and the C80 (truncated rhombic triacontahedron with 30 non regular hexagonal faces).

Fedorov's parallelohedra verify three conditions: the faces are two by two opposite and parallel, each edge belongs to a set of parallel edges (4 or 6 edges), and they fill the space using only translations. There are only five, to within affine transformations: the truncated octahedron, the elongated rhombic dodecahedron, the rhombic dodecahedron, the hexagonal prism (not necessarily right) and the cube (thus also all the parallelepipeds); starting with the 6 set of 6 parallel edges of the truncated octahedron, we suppress, by steps, sets of parallel edges to end up at the 3 set of 4 edges of the cube.

The convex hull (or convex envelope) of a polyhedron P is the unique smallest convex polyhedron which contains P.

three funny exercises  (one should of course SEARCH before looking at the solutions)

 •  create polyhedra (last problem, all levels, of the Australian Mathematics Competition - 2000)
Five boxes contain respectively two squares and eight triangles, three squares and two triangles, three squares and four triangles, four squares and three triangles, five squares and four triangles. The lengths of all the sides of all the squares and all the triangles are equal. We try to build polyhedra by assembling all the pieces of a given box. With how many boxes can the construction been successfully achieved?
 •  shades of polyhedra
Find a maximum of polyhedra whose shade can be a square, respectively a regular hexagon.
 •  pentagonal cross section of a cube (Cédric Villani)
Everyone knows that the cross section of a cube by a plane may be a square, an equilateral triangle or a regular hexagon. How to proceed to get a cross section which is a regular pentagon?

references:    •  Twenty Proofs of Euler's Formula  on "The Geometry Junkyard"
•  If you wish to handle, modify the appearance, colour according to your taste, and print the templates of 147 convex polyhedra, then POLY is the program you need!
•  With STELLA you may access an infinity of polyhedra (convex or not) and create new ones. A "must" for those who want to go deeper into the polyhedra world!
•  A nice set of 30 convex wooden polyhedra.
•  Some results about convex polyhedra by Gérard P. Michon.


  summary   April 1999
updated 17-04-2016