MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

Mathematical Properties 61
Property 63 (Equilateral Lattice). Let c denote the minimum distance
between two points in a unit lattice, i.e. one�constructed from an arbitrary
2
parallelogram of unit area [172]. Then, c ≤ √3 , and this upper bound is
achieved by the lattice generated by a parallelogram that is composed of two
equilateral triangles (i.e. by a “regular rhombus”).
Moreover, for a given value of c, this lattice has the smallest possible generating
parallelogram. Asymptotically, of all lattices with a given c, the lattice
composed of equilateral triangles has the greatest number of points in a given
large region. Finally, the lattice of equilateral triangles gives rise to the densest
≈ .907.
packing of circles of radius c
2
(Figure 2.44) with density D = π
2 √ 3
Property 64 (No Equilateral Triangles on a Chess Board). There is
no equilateral triangle whose vertices are plane lattice points [96].
This was one of 24 theorems proposed in a survey on The Most Beautiful
Theorems in Mathematics [321]. It came in at Number 19. In general,
a triangle is embeddable in Z 2 if and only if all of its angles have rational
tangents [17]. Of course, the equilateral triangle is embeddable in Z n (n ≥ 3):
{(1, 0, 0, . . . ), (0, 1, 0, . . . ), (0, 0, 1, . . . )}. M. J. Beeson has provided a complete
characterization of the triangles embeddable in Z n for each n [17].
Figure 2.45: Equilateral Triangular Mosaic [199]
Property 65 (Regular Tessellations of the Plane). The only regular tessellations
of the plane by polygons of the same kind meeting only at a vertex
are provided by equilateral triangles, squares and regular hexagons. If a vertex
of one polygon is allowed to lie on the side of another then the only such tessellations
are afforded by equilateral triangles (Figure 2.45) and squares [199,
pp. 199-202].