Truncation of Platonic solids (regular
polyhedra ) means that the corners and/or the edges are cut off to make Archimedean
solids (semi-regular polyhedra).
The original description of the thirteen Archimedean
solids is lost, but Pappi Alexandrini has in Collection ,
Book V, described all the polyhedra.
However, truncation is described in Collection only for no.1-4. Pierro della
Francesca truncated no. 1-5 and 8. Luca
Pacioli /Leonardo da Vinci truncated no. 1,2,4,5,8,9,10. Also Dürer,
Barbaro, Stevin, and Jamnitzer truncated some of them.
Truncation is described in Th. Heath:
History of Greek Mathematics, Vol.II, p.98-101.However, there is no
complete calculation. A. Badoureau gives a complete calculation in J.
Ecole Polytechnique, Tome XXX, 1881, p. 65; also of no.10 and 13 ,
which by other authors have been considered impossible to construct by
truncation. The ratios of edges of Platonic and Archimedian solids (EP/EA) are given in H.M.
Cundy & A.P Rollett: Mathematical Models, 1997,
but there is no detailed description of truncation.
Johannes Kepler (1619) reconstructed all the Platonic and
Archimedean polyhedra in Harmonices Mundi, Book 2, but not by
truncation; instead he fitted faces together round a vertex; a method he
clearly borrowed from Plato´s Thimaios. Solids, which required
distortion to make their faces regular, acquired the prefix “ rhombi” (No. 5,
6, and 11). Kepler also gave the geometric formulas: the
in-circle and circum-circle-radius, surface, volume etc.
Number and
type of polygons: e.g. 4/3 means four triangles.
EP is the edge in Platonic solids. EA is the edge in Archimedean solids .
Data of
polyhedra are quoted from R. Williams: The Geometrical Foundation of
Natural Structures, 1979.
First, the distance from the polyhedra centre to the centre of the faces
in Figure 3 are used to calculate the ratio EP/EA. Second, the circum-circle and
in-circle radius of polygons are taken from Table 2-1- in order to calculate
the position of the Archimedean polygons (red) on the Platonic polygons.
New
polygons of Archimedean polyhedra applied on faces of Platonic polyhedra are
red.
The parts
of Platonic polyhedra to be truncated are yellow. The polygons formed by
truncation are bluegreen.
1. Truncated
tetrahedron 4/3 4/6 from tetrahedron.
2a. Cuboctahedron 8/3
6/4 from cube
3a. Truncated octahedron from cube
4a.Truncated cube 8/3
6/8 from cube
5a. Rhombicuboctahedron
8/3 18/4 from
cube
6. Truncated
cuboctahedron 12/4 8/6 6/8 a. from cube and b. from octahedron
7a. Icosadodecahedron 20/3
12/5 from icosahedron
8. Truncated icosahedron
12/5 20/6 a. from
icosahedron and b. from dodecahedron
9. Truncated
dodecahedron 20/3
12/10 a. from dodecahedron and
b. from icosahedron
10. Snub cube 32/3
6/4 a. from cube and b. from octahedron
11.
Rhombicosidodecahedron (Small Rhombicosidodecahedron) 20/3
30/4 12/5 a.
from
icosahedron and b. from dodecahedron
12. Truncated icosadodecahedron (Great Rhombicosidodecahedron)
30/4 20/6 12/10
a. from
icosahedron and b. from dodecahedron
13. Snub dodecahedron 80/3 12/5 a. from
dodecahedron and b. from
icosahedron