60 mirrorsymmetric hexagons 
80 hexagons, among them 20 regular 
120 hexagons, among them 60 irregular 
If we insert three rings of four squares along the edges of a regular octahedron (12 squares and the 6 square intersections) we get the small rhombicuboctahedron. 

We can insert a ring of six squares along each of the four "hexagonal equators" of the cuboctahedron (faces in dark blue).
With one ring we get the Johnson's solid J36. With the four "rings" we have 24 squares and 12 rhombi (faces of a rhombic dodecahedron) at the intersections, that is (6+8)+24+12=50 faces. 

Likewise the icosidodecahedron (faces in dark blue) has six "decagonal equators".
With one ring of ten squares we get another Johnson's solid: J43. Three "rings" can be arranged in two different ways to form two nice polyhedra, one flat and the other stretched. With the six "rings" we have 60 squares and 30 rhombi (faces of a rhombic triacontahedron) at the intersections, that is (20+12)+60+30=122 faces. 

These two polyhedra stem from the small rhombicosidodecahedron in which the 30 squares have been replaced with golden rectangles. There are two ways to arrange the rectangles; with rectangles of same size we thus get two different polyhedra (on the two solids the 20 triangles and the 12 pentagons have different sizes).
The presence of triangles along two opposite sides of each rectangle creates an optical illusion; the 2×30 rectangles are identical. 


A noble polyhedron is one which is isohedral (all faces are identical) and isogonal (all vertices are identical). The dual of a noble polyhedron is also noble. A disphenoid is an equifacial tetrahedron (four identical faces); the faces of a tetragonal disphenoid are isosceles and arranged in such a way that they create a double wedge (sphenoid comes from the Greek word for wedge). The faces of this tetragonal disphenoid have side's lengths a√3, a√3 and 2a. Its net can be obtained by simply folding a format A sheet (2a × 2a√2). It fills the space. 

This curious pentahedron is a tectohedron of order 4 (tetrahedron with a vertex cut off).
To get the mirror image it suffices to turn the net over and to fold it upside down. 

Non trivial polyhedra with heptagonal faces are seldom seen; this one, constructed by Marcel Tünnissen, has full tetrahedral symmetry.
Its faces are all equilateral (edges of same length): 12 heptagons and 16 triangles. 

With 8 pentagons and 12 triangles, this icosahedron has outstanding properties:
• the coordinates of its vertices are "small" integers, • the lengths of its edges are integers, and curiously its volume too, • it is autodual and its edges are tangent to a sphere (midsphere), ... (details and more surprising properties in A polyhedron full of surprises by Hans L. Fetter) 



More examples on an other page.
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