MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

112 Mathematical Recreations
of placing n equal, nonoverlapping circles on a sphere so that the radius of
each circle is maximized [124]. The solution for the cases n = 3, 4, 12 appear
in Figure 4.14 and all involve equilateral triangles. The solution for cases
2 ≤ n ≤ 12 and n = 24 are known, otherwise the solution is unknown [124].
Figure 4.15: Rep-4 Pentagon: The Sphinx [146]
Recreation 15 (Replicating Figures: Rep-tiles [125]). In 1964, S. W.
Golomb gave the name “rep-tile” to a replicating figure that can be used to assemble
a larger copy of itself or, alternatively, that can be dissected into smaller
replicas of itself [146]. If four copies are required then this is abbreviated rep-4.
Figure 4.15 contains a rep-4 pentagon, known as the Sphinx, which may be
regarded as composed of six equilateral triangles or two-thirds of an equilateral
triangle [146]. The Sphinx is the only known 5-sided rep-tile [296, p. 134].
Figure 4.16 contains three examples of rep-4 nonpolygonal figures composed
of equilateral triangles: the Snail, the Lamp and the Carpenter’s Plane [125].
Each of these figures, shown at the left, is formed by adding to an equilateral
triangle an endless sequence of smaller triangles, each one one-fourth the size
of its predecessor. In each case, four of these figures will fit together to make a
larger replica, as shown on the right. (There is a gap in each replica because the
original figure cannot be drawn with an infinitely long sequence of triangles.)
Recreation 16 (Hexiamonds [126]). Hexiamonds were invented by S. W.
Golomb in 1954 and officially named by T. H. O’Beirne in 1961. Each hexiamond
is composed of six equilateral triangles joined along their edges. Treating
mirror images as identical, there are exactly 12 of them (Figure 4.17) [126].
Much is known about the mathematical properties of hexiamonds. For
example, the six-pointed star of Figure 4.18(a) is known to have the unique
eight-piece solution of Figure 4.18(b) [126]. Sets of plastic hexiamonds were
marketed in the late 1960’s, under various trade names, in England, Japan,
and West Germany.