Kolam indiens, dessins sur le sable aux îles Vanuatu, courbe de Sierpinski et morphismes de monoïde

Periodicity of Certain Piecewise Affine Planar Maps

“…The problem currently seems hopeless for cases (4) and (5). However, a nice observation on rational values of λ with prime-power denominator p n is exhibited in [9].…”

“…By subtracting 1 from each term of the sequence of exponents (the run-lengths of 0's and 1's) we obtain the sequence 0 10 01110010 01001110011100111001001001110010 · · · which is easily shown to be the fixed point of the substitution 0 → 010, 1 → 01110 (see [5]), which is equal to σ in the cases λ = −1/γ, λ = 1/γ, λ = −γ, and to σ 1 in the case λ = √ 3. In case λ = γ, we have that σ ∞ (1) is the image of this word by the morphism 0 → 10, 1 → 110 since σ(10) = (10)(110)(10) and σ(110) = (10)(110)(110)(110)(10).…”

Von Koch and Thue-Morse Revisited

“…This approach can be generalized to situations where the representation does not necessarily lie on a straight line: see in particular the papers of Golomb [22] and Giles [19,20,21] where fixed points of morphisms and automatic sequences are not mentioned but can be found, and the papers of Dekking [9,10,11]. In the same direction see the papers of Dekking, Dekking and Mendès France, Deshouillers, Mendès France, and Mendès France and Shallit [12,13,15,29,30], but also the paper of Salon with an appendix by Shallit [32], where two-dimensional automatic sequences enter the picture, without forgetting the article by Prusinkiewicz and Hammel [31], and the article by Allouche, Allouche and Shallit [2]. We would also like to quote two papers by von Haeseler, Peitgen, Skordev [23,24] and the references therein, and a paper by Barbé and von Haeseler [5].…”

A Topological Approach to Creating Any <I>Pulli Kolam</I>, an Artform from South India

“…Many previous pioneering works exist that have provided mathematical insights into the form of a kolam over the past four decades. These include converting kolams into numbers and linear diagrams [1], using graph, picture, and array grammers [2][3][4][5][6][7][8][9][10], extended pasting schemes [11], morphism of monoids [12], L-and P-systems [13,14], gestural lexicons [15], knot theory [16], and mirror curves [17]. Of these, the work of Yanagisawa & Nagata [1], has similarities to this work.…”