MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

Mathematical Properties 73
Property 80 (Partitions of an Equilateral Triangle). Let T denote a
closed unit equilateral triangle. For a fixed integer n, let dn denote the infimum
of all those x for which it is possible to partition T into n subsets, each subset
having a diameter not exceeding x. Recall that the diameter of a plane set
A is given by d(A) = sup a,b∈A ρ(a, b) where ρ(a, b) is the Euclidean distance
between a and b. R. L. Graham [149] has determined dn for 1 ≤ n ≤ 15.
Figure 2.65 gives an elegant partition of T into 15 sets each having diameter
d15 = 1/(1 + 2 √ 3).
Property 81 (Dissecting a Polygon into Nearly-Equilateral Triangles).
Every polygon can be dissected into acute triangles. On the other hand,
a polygon P can be dissected into equilateral triangles with (interior) angles
arbitrarily close to π/3 radians if and only if all of the angles of P are multiples
of π/3. For every other polygon, there is a limit to how close it can come
to being dissected into equilateral triangles [64, pp. 89-90].
Figure 2.66: Regular Simplex [60]
Property 82 (Regular Simplex). A regular simplex is a generalization of
the equilateral triangle to Euclidean spaces of arbitrary dimension [60]. Given
a set of n + 1 points in R n which are pairwise equidistant (distance = d), an
n-simplex is their convex hull.
A 2-simplex is an equilateral triangle, a 3-simplex is a regular tetrahedron
(shown in Figure 2.66) , a 4-simplex is a regular pentatope and, in general, an
n-simplex is a regular polytope [60]. The convex hull of any nonempty proper
subset of the given n+1 mutually equidistant points is itself a regular simplex
of lower dimension called an m-face. The n + 1 0-faces are called vertices, the
n(n+1)
1-faces are called edges, and the n + 1 (n − 1)-faces are called facets.
2
� �
n + 1
In general, the number of m-faces is equal to and so may be found
m + 1