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