MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

Mathematical Recreations 111
Recreation 13 (Erdös’ Sphere Coloring Problem [131]). Paul Erdös
proposed the following unsolved problem in graph theory. What is the minimum
number of colors required to paint all of the points on the surface of a unit
sphere so that, no matter how we inscribe an equilateral triangle of side √ 3
(the largest such triangle that can be so inscribed), the triangle will have each
corner on a different color (Figure 4.13(a)) [131]?
E. G. Straus has shown that five colors suffice. In his five-coloring (shown in
Figure 4.13(b)), the north polar region is open with boundary circle of diameter
√ 3. The rest of the sphere is divided into four identical regions, each closed
along its northern and eastern borders, as indicated by the heavy black line on
the dark shaded region. One color is given to the cap and to the south pole.
The remaining four colors are assigned to four quadrant regions. G. J. Simmons
has shown that three colors are not sufficient so that at least four colors are
necessary. It is unknown whether four or five colors are both necessary and
sufficient. The analogous problem for the plane, i.e. the minimum number of
colors which ensures that every equilateral triangle of unit side will have its
corners on different colors, is also open. Indeed, it is equivalent to asking for a
minimal coloring of the plane so that every unit line segment has its endpoints
on different colors, a problem which was discussed at the end of the previous
Recreation. This problem may be recast in terms of the chromatic number of
planar graphs [131].
(a) (b) (c)
Figure 4.14: Optimal Spacing of Lunar Bases: (a) n=3. (b) n=4. (c) n=12.
[124]
Recreation 14 (Optimal Spacing of Lunar Bases [124]). Assume that
the moon is a perfect sphere and that we want to establish n lunar bases as far
apart from one another as possible.
I.e., how can n points be arranged on a sphere so that the smallest distance
between any pair of points is maximized? This problem is equivalent to that