a dual pair, icosahedron and dodecahedron . From the icosahedron, the platonic (3,5) polyhedron, with 20 triangular faces and [hence] 12 vertices and 30 edges (let us call it Ic), a (5,6,6) Archimedean polyhedron, with 60 vertices, 90 edges and 32 faces [20 triangular and 12 pentagonal] can be constructed by slicing off a pentagonal neighborhood [of radius 1/3 of the edge-length of Ic] at any vertex of Ic. The result is accordingly called the truncated icosahedron. In Europe the leather version is called a football. Carbon fullerene In chemistry its skeleton is known as the model of the fullerene C60 carbon molecule, depicted on the stamp to the right.
. The cuboctahedron, a quasiregular (3,4,3,4) polyhedron with 12 vertices, 24 lines and 14 faces [6 squares and 8 triangles], can be obtained from either the cube, or its dual the octahedron, by a similar, but more drastic, truncating process. In the case of the cube, one cuts off a triangular neighborhood of radius 1/2 of the edge-length [at each vertex]; in the case of the octahedron, halfsized square pyramids have to be removed, to get the same result [as a balancing exercise in the pub you might try to create a paper model by using 6 square beer mats (leaving the triangular faces open)] A second drastic truncation operation then leads to a non-uniform polyhedron with 24 vertices, 48 edges [of size a or b = asqrt(2)], and 26 faces [8 equilateral triangles of side a, 6 squares of size bxb and 12 axb rectangles]. Unfold it to get a better understanding. Then shrink the edges of length b to length a to finally obtain the so-called small rhombicuboctahedron, a (3,4,4,4) Archimedean polyhedron [with 24 vertices 48 edges and 26 faces, namely, 8 triangles and 18 squares]. As above in accordance with Euler's formula: v - e + f = 2 , which together with an icosahedron appears on a stamp on my Algebra page. Let us see what Maple has to offer.

restart: with(geom3d):
icosahedron(Ic,point(o,0,0,0),1): draw(Ic);   Archimedean(F1,_t([3,5]),point(o,0,0,0),1): draw(F1);

Here _t stands for truncation, which transforms e.g. the green central triangle of (3,5) [= Ic] into the lighter central hexagon of (5,6,6) [= F1]. The circum-radius is set to 1. The star on the Berlin Xmas stamp below looks like a starred version of F1, but it is not, unfortunately, for the 6 pyramids surrounding the central hexagon are clearly congruent
icosahedronfootball Berlin Xmas the icosahedron tensegrity on our campus,
constructed by seven students,
one early morning in April(!) 1974
Kenneth Snelson , the inventor of tensegrities
Catalogue of symmetric tensegrities
George W. Hart, mathematical sculptor, computer scientist

The very instructive "examples,stellate" worksheets in Maple also provide the means to get pictures and numerical data of polyhedra. We may e.g. ask for the edge-length s, the mid-radius m [i.e. the radius of the mid-sphere, touching all the edges] and the name of the polyhedra at hand. For reasons of space, the output will be presented column-wise. The plottools package has its own powerful stellate facility.

s:=sides(F1);     evalf(s); m:=MidRadius(F1): evalf(m); form(F1);     simplify(s/m); simplify(sides(Ic)/MidRadius(Ic)); origami
s evalf(s)
evalf(m)

t(3,5) 		s/m

s(i)/m(i)

The origami polyhedron hovering over the city of Kyoto, the seat of the International Congress of Mathematicians in 1990, looks like a stella octangula, a compound of a tetrahedron and its dual [obtained by a point reflection in its center]. The 12 face diagonals of a common cube accordingly form the edges of a stella octangula. It is available in Maple as a stellation of the octahedron through [octahedron(oc,point(o,0,0,0),1): stellate(stoc,oc,1): draw(stoc);] The star on the German Xmas stamp below looks like a starred version of a (3,4,4,4) with no pyramids on the 8 triangular faces, of which only 2 are visible on the stamp.

Archimedean(co,[[3],[4]],point(o,0,0,0),1): draw(co);     Archimedean(F2,_r([[3],[4]]),point(o,0,0,0),1): draw(F2);

cuboctahedron small rhombicuboctahedron German Xmas

sco:=sides(co); s2:=sides(F2);     form(co); form(F2);     mco:=MidRadius(co): m2:=MidRadius(F2): evala(sco/mco); evala(s2/m2);
sco

s2 		cuboctahedron

small rhombicuboctahedron 		sco/mco
s2/m2

a cardboard cube 3-compound model made by Frits Göbel Maple can construct stellations [starformed extensions of a specific kind] of 7 polyhedra, the 5 platonic and the 2 quasiregular ones, among them the cuboctahedron co [also available through cuboctahedron(co,point(o,0,0,0),1): draw(co); ]. It has five stellations parameterized by n, where n = 0, 1, ..., 4, including the trivial one with n = 0, identical to co. The pictures are too nice to be missed. Within Maple the pictures can be rotated by moving your mouse over them [or by choosing different orientations].
stellate(G0,co,0): draw(G0,style=patch,orientation=[66,100],lightmodel=light2);
stellate(G1,co,1): draw(G1,style=patch,orientation=[66,100],lightmodel=light2);
stellate(G2,co,2): draw(G2,style=patch,orientation=[66,100],lightmodel=light2);
stellate(G3,co,3): draw(G3,style=patch,orientation=[66,100],lightmodel=light2);
stellate(G4,co,4): draw(G4,style=patch,orientation=[66,100],lightmodel=light2);
G0 G1 G2 G3 G4

Reference: L. Fejes Tóth, Regular Figures, MacMillan, New York (1964) [with nice stereograms of various polytopes, to be viewed with red-green glasses]

Finally the small stellated dodecahedron {5/2, 5} found by Kepler [and rediscovered by Poinsot ] and its dual the great dodecahedron {5, 5/2} found by Poinsot. The cardboard model to the left is again a gift from Frits Göbel. It luckily has survived cats, children and other curious creatures. The huge steel model to the right, made in 1997 by the technical staff of the physics department after a design by M.C. Escher from 1952, stands before the Mesa+ clean room on our campus nearby where we live.
small stellated dodecahedron small stellated dodecahedron
caption1 		great dodecahedron
caption2		 before the clean room
Homage to the Hexagon
 Bridges
[Mathematical Connections in Art, Music, and Science]