Related papers: Apollonian Equilateral Triangles

Let X(1,3,a) be a crepant resolution of the quotient singularity C^3/G, where G is a diagonal cyclic subgroup of SL(3,\C) acting on C^3 with weights (1,3,a). For each such X(1,3,a), we construct a (Q,W)-configuration of spherical objects in…

We show, using acylindrical hyperbolicity, that a finitely generated group splitting over $\Z$ cannot be simple. We also obtain SQ-universality in most cases, for instance a balanced group (one where if two powers of an infinite order…

We describe various kaleidoscopic and self-similar aspects of the integral Apollonian gaskets - fractals consisting of close packing of circles with integer curvatures. Self-similar recursive structure of the whole gasket is shown to be…

In groups with involution a nonassociative product of elements is defined, which leads to the definition of a certain type of quasigroups. These quasigroups are represented by square tables of complex numbers, with inverses, which differ…

We investigate the construction and properties of Clifford algebras by a similar manner as our previous construction of the octonions, namely as a twisting of group algebras of Z_2^n by a cocycle. Our approach is more general than the usual…

In this paper we first construct an analytic realization of the $C_\lambda$-extended oscillator algebra with the help of difference-differential operators. Secondly, we study families of $d$-orthogonal polynomials which are extensions of…

We introduce a new class of division algebras, the hyperpolyadic algebras, which correspond to the binary division algebras $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$, $\mathbb{O}$ without considering new elements. First, we use the matrix…

A perfect Euler cuboid is a rectangular parallelepiped with integer edges, with integer face diagonals, and with integer space diagonal as well. Finding such parallelepipeds or proving their non-existence is an old unsolved mathematical…

A generalization of the Macdonald polynomials depending upon both commuting and anticommuting variables has been introduced recently. The construction relies on certain orthogonality and triangularity relations. Although many…

Multiplicities of periodic orbit lengths for non-arithmetic Hecke triangle groups are discussed. It is demonstrated both numerically and analytically that at least for certain groups the mean multiplicity of periodic orbits with exactly the…

This paper concentrates on the homogeneous (conformal) model of Euclidean space (Horosphere) with subspaces that intuitively correspond to Euclidean geometric objects in three dimensions. Mathematical details of the construction and…

The spaces of triangulations of a given manifold have been widely studied. The celebrated theorem of Pachner~\cite{Pachner} says that any two triangulations of a given manifold can be connected by a sequence of bistellar moves, or Pachner…

Many questions about triangles and quadrilaterals with rational sides, diagonals and areas can be reduced to solving certain Diophantine equations. We look at a number of such questions including the question of approximating arbitrary…

There is a natural action of the braid group on the symmetric matrices with units on the diagonal, appearing in various fields as Singularity Theory, Frobenius Manifolds or Isomonodromic deformations of certain classes of linear…

The Babylonian graph B has the positive integers as vertices and connects two if they define a Pythagorean triple. Triangular subgraphs correspond to Euler bricks. What are the properties of this graph? Are there tetrahedral subgraphs…

We propose a construction of the quantum-corrected Coulomb branch of a general 3d gauge theory with $\mathcal{N}=4$ supersymmetry, in terms of local coordinates associated with an abelianized theory. In a fixed complex structure, the…

We consider $48$ parastrophically uncancellable quadratic functional equations with four object variables and two quasigroup operations in two classes: balanced non--Belousov (consists of 16 equations) and non--balanced non--gemini…

A latin bitrade is a pair of partial latin squares which are disjoint, occupy the same set of non-empty cells, and whose corresponding rows and columns contain the same set of entries. Dr\'apal (\cite{Dr9}) showed that a latin bitrade is…

We say that a group is {\em almost abelian} if every commutator is central and squares to the identity. Now let $G$ be the Galois group of the algebraic closure of the field $\QQ$ of rational numbers in the field of complex numbers. Let…

Let $f(x)=x^4+ax^3+d\in {\mathbb Z}[x]$, where $ad\ne 0$. Let $C_n$ denote the cyclic group of order $n$, $D_4$ the dihedral group of order 8, and $A_4$ the alternating group of order 12. Assuming that $f(x)$ is monogenic, we give necessary…