Unilateral and equitransitive tilings by equilateral triangles

Tilings of Convex Sets by Mutually Incongruent Equilateral Triangles Contain Arbitrarily Small Tiles

“…This name is adopted from perfect tilings by squares [2]. Figure 1 illustrates a family of tilings of the full plane R 2 by equilateral triangles depending on a parameter α ∈ 0, 1 2 . These tilings are not perfect since there are only three (if α < 1 2 ) or two (if α = 1 2 ) different sizes of tiles.…”

“…Figure 1 illustrates a family of tilings of the full plane R 2 by equilateral triangles depending on a parameter α ∈ 0, 1 2 . These tilings are not perfect since there are only three (if α < 1 2 ) or two (if α = 1 2 ) different sizes of tiles. The limit case α ↓ 0 (with triangles of size 0 being ignored) would give the hexagonal tiling by triangles of unit size.…”

“…Let T be a tiling of a convex subset of the plane by at least two equilateral triangles with a positive lower bound on the side lengths of all tiles. Then T contains two triangles that share a side or T is similar to a tiling of the type depicted in Figure 1 with suitable α ∈ 0, 1 2 . Theorem 4 improves on a recent result on tilings of the complete plane R 2 by Aduddell, Ascanio, Deaton and Mann [1, Theorem 1] in two ways.…”

“…That is, [(i, j), (k, l)] ∈ E ⇐⇒ (k, l) ∈ {(i − 1, j − 1), (i + 1, j − 1), (i, j + 2)} (see Figure 8). We consider the random walk on Γ with transition probability p((i, j), (k, l)) = 1 3 for [(i, j), (k, l)] ∈ E (and in turn p((i, j), (k, l)) = 0 for [(i, j), (k, l)] / ∈ E). This is described by the Markov chain Z 2 even , P with state space Z 2 even and transition matrix P = (p((i, j), (k, l))) (i,j),(k,l)∈Z 2 even (cf.…”

Tilings of convex polygons by equilateral triangles of many different sizes

“…Finally, let us point out that the study of dissections into incongruent equilateral triangles is a fruitful field of ongoing research, see e.g. [4, Section C11], [6, Exercise 2.4.10], [9,Problem 4] and [1,8,11,12,13,14,15].…”