Related papers: The Pattern Complexity of the Sierpi\'{n}ski Trian…
We give an exact formula for the number of distinct square patterns of a given size that occur in the Squiral tiling.
We present an exact formula for the number of distinct crease patterns in a square shaped region of a given size that appear in the 2 dimensional paperfolding structure.
We find the exact formula for the number of distinct $n \times n$ square patterns which appear in a Robinson tiling made of one infinite order supertile.
In this note we show that the system of equations t_{x}+t_{y}=t_{p},\quad t_{y}+t_{z}=t_{q},\quad t_{x}+t_{z}=t_{r}, where $t_{x}=x(x+1)/2$ is a triangular number, has infinitely many solutions in integers. Moreover we show that this system…
The triangulation complexity of a compact 3-manifold is the minimal number of tetrahedra in any triangulation of the 3-manifold. We compute the triangulation complexity of all elliptic 3-manifolds and all sol 3-manifolds, to within a…
We present a, hopefully, elementary mathematical treatment of the computational aspects of congruent numbers, such that an amateur could understand the problem and perform their own calculations.
Motivated by a question of Erd\"{o}s and inquiries by Beeson and Laczkovich, we explore the possible $N$ for which a triangle $T$ can tile into $N$ congruent copies of a triangle $R$. The \emph{reptile} cases (where $T$ is similar to $R$)…
In this paper the problem of finding a normal form of triangles and plane quadrilaterals up to similarity is considered. Several normal forms for triangles and a normal form for quadrilaterals of special case are described. Normal forms of…
How complex must two finite 2-complexes be to admit a common, but not finite common, covering? We obtain an almost answer: the minimum possible number of triangles in a pseudo-simplicial triangulation of each complex is 3, 4, or 5.
By examining arithmetic operations between decimal numbers in a given base m we uncover fractal structures that generalize the Sierpinski triangle
There are four characteristic circles for each triangle on a plane. All for are tangential to the three straight lines containing the triangles' three sides. Three are exterior circles, the fourth is the in-circle. When the triangle is…
We show upper and lower bounds for angles in iterations of trisections of certain triangulations.
A couple of complex projective plane curves are said to make a Zariski pair if they have the same degree and the same type of singularities, but their embeddings in the projective plane are topologically different. In this paper, we present…
Except for crystalline or random structures, an agreed definition of complexity for intermediate and hence interesting cases does not exist. We fill this gap with a notion of complexity that characterises shapes formed by any finite number…
We give some new advances in the research of the maximum number of triangles that we may obtain in a simple arrangements of n lines or pseudo-lines.
We give the complete list of possible torsion subgroups of elliptic curves with complex multiplication over number fields of degree 1-13. Additionally we describe the algorithm used to compute these torsion subgroups and its implementation.
A formula for the irregularity of a cyclic multiple plane associated to a branch curve that has arbitrary singularities and is transverse to the line at infinity is established. The irregularity is expressed as a sum of superabundances of…
A multicomplex, also known as a twisted chain complex, has an associated spectral sequence via a filtration of its total complex. We give explicit formulas for all the differentials in this spectral sequence.
These notes represent an extended version of a talk I gave for the participants of the IMO 2009 and other interested people. We introduce diophantine equations and show evidence that it can be hard to solve them. Then we demonstrate how one…
The article contains some important classes of multisets. Combinatorial proofs of problems on the number of m-submultisets and m-permutations of multiset elements are considered and effective algorithms for their calculation are given. In…