2.1 Uniform tessellations
From sets of regular polygons one can construct uniform tessellations, in which
the same arrangement of polygons occurs at each vertex. The example shown is
unselfreflexible or cheiral, that is to say it differs from its mirror image. Uniform tessellations can also be constructed from 12-sided
coins.
2.2 Matt’s Bricks
In three dimensions the commonest building blocks are cubical or rectangular,
but other types are more attractive. A toy, “Matt’s Bricks”
(designed and made by the author in 1952 for Matthew Hinton, as a present on
his sixth birthday), consists of coloured tetrahedra and octahedra in similarly
shaped boxes. The blocks can easily be made from painted cardboard.
2.3 Stellations of regular polyhedra 2.5 Wire models, and their construction 2.6 Flexible models: degrees of freedom 2.7 A model of DNA
Other examples of other polyhedra constructed in this way include the five regular
polyhedra and their stellations, some of which are cheiral.
2.4 Compound polyhedra
Symmetric combinations of the regular polyhedra include the “2-dodecahedra”
and “2 icosahedra” (which have cubic symmetry); “3-cubes,”
“3-octahedra” and “6-tetrahedra;” “4-cubes,”
“4-octahedra” and “8-tetrahedra,” and the well-known
“5-cubes,” “5-octahedra” and “10-tetrahedra.”
There is another method of construction which is essentially simpler. The only
material used is garden wire, and the only tool is a pair of pliers. A cuboctahedron,
for example, is constructed from 4 hexagonal rings, which hold together through
their springiness. Such models have been made of all the uniform polyhedra.
Examples include the “snub” uniform polyhedra, some of which are
unselfreflexible.
An interesting and delightful toy consists of a set of wooden rods and flexible
(plastic) joints. With this one can construct polyhedra which are either “just
stiff,” or which have a certain number of degrees of freedom. After removing
6 of the edges of a stiff icosahedron in a cubically symmetric way, the structure
can be made to expand into a cuboctahedron or to collapse into an octahedron
or even further into a flat triangle. Further, one can construct a flexible
ring of 6 or more tetrahedra which can be used to demonstrate a generalization
of Euler’s theorem on the number of faces, edges and vertices of any convex
polyhedron.
From the toy “Playsticks” one can construct a double helix with
cross-links, which illustrates the structure of DNA, the molecule that carries
most of the genetic information in living cells.