MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

Mathematical Properties 55
Figure 2.35: Closed Light Paths [57]
Figure 2.36: Erdös-Moser Configuration [235]
Property 55 (Erdös-Moser Configuration). An equilateral triangle of
side-length one is called a unit triangle. A set of points S is said to span
a unit triangle T if the vertices of T belong to S. n points in the plane are said
to be in strictly convex position if they form the vertex set of a convex polygon
for which each of the points is a corner. Pach and Pinchasi [235] have proved
that any set of n points in strictly convex position in the plane has at most
⌊2(n−1)/3⌋ triples that span unit triangles. Moreover, this bound is sharp for
each n > 0.
This maximum is attained by the Erdös-Moser configuration of Figure 2.36.
This configuration contains ⌊(n − 1)/3⌋ congruent copies of a rhombus with
side-length one and obtuse angle 2π/3, rotated by small angles around one of
its vertices belonging to such an angle [235].
Property 56 (Reuleaux Triangle). With reference to Figure 2.37(a), the
Reuleaux triangle is obtained by replacing each side of an equilateral triangle by
a circular arc with center at the opposite vertex and radius equal to the length
of the side [125].