Yoshikazu Yamagishi

Triangular spiral tilings

The topology of spiral tilings is intimately related to phyllotaxis theory and continued fractions. A quadrilateral spiral tiling is determined by a suitable chosen triple (ζ, m, n), where , and m and n are relatively prime integers. We give a simple characterization when (ζ, m, n) produce a triangular spiral tiling. When m and n are fixed, the admissible generators ζ form a curve in the unit disk. The family of triangular spiral tilings with opposed parastichy pairs (m, n) is parameterized by the divergence angle arg (ζ), while triangular spiral tilings with non-opposed parastichy pairs are parameterized by the plastochrone ratio 1/|ζ|. The generators for triangular spiral tilings with opposed parastichy pairs are not dense in the complex parameter space, while those with non-opposed parastichy pairs are dense. The proofs will be given in a general setting of spiral multiple tilings. We present paper-folding (origami) sheets that build spiral towers whose top-down views are triangular tilings.

Cantor bouquet of holomorphic stable manifolds for a periodic indeterminate point

In the rational dynamics of the complex projective plane, we construct a full 2-shift family of holomorphic stable manifolds of a periodic indeterminate point with two periodic orbits.

Stripes on Penrose tilings

We consider a class of cut-and-project sets Λ = ΛF × Z in the plane. Let L = Λ + wR, w ∈ R, be a countable union of parallel lines. Then either (1) L is a discrete family of lines, (2) L is a dense subset of R, or (3) each connected component of the closure of L is homeomorphic to [0, 1] × R.We study the existence of one-dimensional quasicrystal structures on the vertex setP of a Penrose tiling, in an arbitrary direction w ∈ C.I fwR ∩ Z(ζ) � 0, ζ = e 2πi/5 ,thenP +wRisadiscretefamilyoflinesthathasaone-dimensional quasicrystal structure. Conversely, if w � 0 and wR ∩ Z(ζ) = 0, � P + wR is a dense subset of C. We also have a weak analog of Kroneckers approximation theorem.

Geometrical study of phyllotactic patterns by Bernoulli spiral lattices

Geometrical studies of phyllotactic patterns deal with the centric or cylindrical models produced by ideal lattices. van Iterson (Mathematische und mikroskopisch – anatomische Studien über Blattstellungen nebst Betrachtungen über den Schalenbau der Miliolinen, Verlag von Gustav Fischer, Jena, 1907) suggested a centric model representing ideal phyllotactic patterns as disk packings of Bernoulli spiral lattices and presented a phase diagram now called Van Itersons diagram explaining the bifurcation processes of their combinatorial structures. Geometrical properties on disk packings were shown by Rothen & Koch (J. Phys France, 50(13), 1603‐1621, 1989). In contrast, as another centric model, we organized a mathematical framework of Voronoi tilings of Bernoulli spiral lattices and showed mathematically that the phase diagram of a Voronoi tiling is graph‐theoretically dual to Van Itersons diagram. This paper gives a review of two centric models for disk packings and Voronoi tilings of Bernoulli spiral lattices.

On holomorphic flows on Stein surfaces: Transversality, dicriticalness and stability

We study the classification of the pairs

On the local convergence of Newton's method to a multiple root

(N, \,X)

Spiral disk packings

where

Transversality of complex linear distributions with spheres, contact forms and Morse type foliations

N

Design methods of origami tessellations for triangular spiral multiple tilings

is a Stein surface and

Shape Limit in Triangular Spiral Tilings

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