MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

Mathematical Properties 75
(a)
(b) (c)
Figure 2.68: The Hyperbolic Plane: (a) Embedded Patch [302]. (b) Poincaré
Disk. (c) Thurston Model [20]. [138]
Property 84 (The Hyperbolic Plane). The crochet model of Figure 2.68(a)
displays a patch of H 2 embedded in R 3 [302].
As shown in Figure 2.68(b), it may also be modeled by the Poincaré disk
whose geodesics are either diameters or circular arcs orthogonal to the boundary
[33]. In this figure, the disk has been tiled by equilateral hyperbolic triangles
meeting 7 at a vertex. This tiling ultimately led to the Thurston model of
the hyperbolic plane shown in Figure 2.68(c) [20]. In this model, 7 Euclidean
equilateral triangles are taped together at each vertex so as to provide novices
with an intuitive feeling for hyperbolic space [318]. However, it is important
to note that the Thurston model can be misleading if it is not kept in mind
that it is but a qualitative approximation to H 2 [20].
Property 85 (The Minkowski Plane). In 1975, L. M. Kelly proved the
conjecture of M. M. Day to the effect that a Minkowski plane with a regular
dodecagon as unit circle satisfies the norm identity [192]:
||x|| = ||y|| = ||x − y|| = 1 ⇒ ||x + y|| = √ 3.
Stated more geometrically, the medians of an equilateral triangle of side
length s are of length √ 3 · s just as they are in the Euclidean plane. Midpoint
2
in this context is interpreted vectorially rather than metrically.
Property 86 (Mappings Preserving Equilateral Triangles). Sikorska
and Szostok [281] have shown that if E is a finite-dimensional Euclidean space
with dim E ≥ 2 then f : E → E is measurable and preserves equilateral
triangles implies that it is a similarity transformation (an isometry multiplied
by a positive constant).
Since such a similarity transformation preserves every shape, this may be
paraphrased to say that if a measurable function preserves a single shape, i.e.
that of the equilateral triangle, then it preserves all shapes. In [282], they
extend this result to normed linear spaces.