MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

Mathematical Recreations 117
(a)
Figure 4.21: Icosahedron: (a) Icosahedron Net. (b) Five-Banded Icosahedron.
[130]
Recreation 18 (Plaited Polyhedra [130]). Traditionally [66], paper models
of the five Platonic solids are constructed from “nets” like that shown in Figure
4.21(a) for the icosahedron. The net is cut out along the solid line, folded
along the dotted lines, and the adjacent faces are then taped together. In 1973,
Jean J. Pedersen of Santa Clara University discovered a method of weaving or
braiding (“plaiting”) the Platonic solids from n congruent straight strips. Each
strip is of a different color and each model has the properties that every edge
is crossed at least once by a strip, i.e. no edge is an open slot, and every color
has an equal area exposed on the model’s surface. (An equal number of faces
will be the same color on all Platonic solids except the dodecahedron, which has
bicolored faces when braided by this technique.) She has proved that if these
two properties are satisfied then the number of necessary and sufficient bands
for the tetrahedron, cube, octahedron, icosahedron and dodecahedron are two,
three, four, five and six, respectively [130].
With reference to Figure 4.21(b), the icosahedron is woven with five valleycreased
strips. A visually appealing model can be constructed with each color
on two pairs of adjacent faces, the pairs diametrically opposite each other.
All five colors go in one direction around one corner and in the opposite direction,
in the same order, around the diametrically opposite corner. Each
band circles an “equator” of the icosahedron, its two end triangles closing the
band by overlapping. In making the model, when the five overlapping pairs of
ends surround a corner, all except the last pair can be held with paper clips,
which are later removed. The last overlapping end then slides into the proper
slot. Experts may dispense with the paper clips [130]. Previous techniques of
polyhedral plaiting involved nets of serpentine shape [322].
Recreation 19 (Pool-Ball Triangles [133]). Colonel George Sicherman of
Buffalo asked while watching a game of pool: Is it possible to form a “difference
triangle” in arranging the fifteen balls in the usual equilateral triangular
configuration at the beginning of a game?
(b)