Can you see the bubbles in a foam?

Kurusa, Árpád

doi:10.14232/actasm-015-299-1

Cited by 6 publications

(5 citation statements)

References 18 publications

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“…A multicurve (see also [11]) is a finite family of simple curves, called the members of the multicurve, which are parameterized on non-degenerate closed finite intervals, and any point of the plane belongs to at most one member, or it is the endpoint of exactly two members. If F and G are multicurves, F = G, and every member of F is the union of some members of G, we say that G is a partition of F. Definition 2.11.…”

Section: Definition 210mentioning

confidence: 99%

Tiling a Circular Disc with Congruent Pieces

2020

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In this note, we prove that any monohedral tiling of the closed circular unit disc with $$k \le 3$$ k ≤ 3 topological discs as tiles has a k-fold rotational symmetry. This result yields the first nontrivial estimate about the minimum number of tiles in a monohedral tiling of the circular disc in which not all tiles contain the center, and the first step towards answering a question of Stein appearing in the problem book of Croft, Falconer, and Guy in 1994.

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Section: Definition 210mentioning

confidence: 99%

Tiling a Circular Disc with Congruent Pieces

2020

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“…Definition 2. A multicurve (see also [12]) is a finite family of simple curves, called the members of the multicurve, which are parameterized on nondegenerate closed finite intervals, and any point of the plane belongs to at most one member, or it is the endpoint of exactly two members. If F and G are multicurves, F = G, and every member of F is the union of some members of G, we say that G is a partition of F. We finish with a remark and a definition.…”

Section: Preliminariesmentioning

confidence: 99%

On monohedral tilings of a regular polygon

Basit¹,

Lángi²

2021

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A tiling of a topological disc by topological discs is called monohedral if all tiles are congruent. Maltby (J. Combin. Theory Ser. A 66: 40-52, 1994) characterized the monohedral tilings of a square by three topological discs. Kurusa, Lángi and Vígh (Mediterr. J. Math. 17: article number 156, 2020) characterized the monohedral tilings of a circular disc by three topological discs. The aim of this note is to connect these two results by characterizing the monohedral tilings of any regular n-gon with at most three tiles for any n ≥ 5.

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“…Following [9], the masking number 1 M T (P) of the trace M T = Tr r J of a regular multicurve r J is M T (P) = (1/2) S 1 #(T ∩ (P, w)) dw, where (P, w) is the straight line through the point P ∈ R 2 with direction w ∈ S 1 and # is the counting measure. If T is a closed convex curve, then the masking number M T (P) is twice of the point projection (see [2]) and the shadow picture (see [6]).…”

Section: Notations and Preliminariesmentioning

confidence: 99%