Searcharxiv⌕ Search

arXiv subjects

Jerzy Kocik

Publications and source records attributed to Jerzy Kocik.

At least 19 recordsLinked to original sources
arXiv2021

Integral spinors, Apollonian disk packings, and Descartes groups

We show that every irreducible integral Apollonian packing can be set in the Euclidean space so that all of its tangency spinors and all reduced coordinates and co-curvatures are integral. As a byproduct, we prove that in any integral Descartes configuration, the sum of the curvatures of two adjacent disks can be written as a sum of two squares. Descartes groups are defined, and an interesting occurrence of the Fibonacci sequence is found.

math.GR↗
arXiv2020

Apollonian depth, spinors, and the super-Dedekind tessellation

The configuration space of tricycles (triples of disks in contact) is shown to coincide with the complex plane resulting as a projective space costructed from the tangency and Pauli spinors. Remarkably, the fractal of the depth functions assumes a particularly simple and elegant form. Moreover, the factor space due to a certain symmetry group provides a parametrization of the Apollonian disk packings.

math.MG↗
arXiv2020

Induced orders in free monoids of words

A family of partial orders in the free monoid of words, induced from a partial order in alphabet, is presented. The induced orders generalize the chronological posets that have been defined for the two-letter alphabet only, and the morphological order. We show that the induced orders are natural with respect to alphabet homomorphisms.

math.CO↗
arXiv2020

Traces, symmetric functions, and a raising operator

The polynomial relationship between elementary symmetric functions (Cauchy enumeration formula) is formulated via a ``raising operator" and Fock space construction. A simple graphical proof of this relation is proposed. The new operator extends the Heisenberg algebra so that the number operator becomes a Lie product. This study is motivated by natural appearance of these polynomials in the theory of invariants for Lax equations and in classical and topological field theories.

math-ph↗
arXiv2020

Fibonacci numbers and Ford circles

An amusing connection between Ford circles, Fibonacci numbers, and golden ratio is shown. Namely, certain tangency points of Ford circles are concyclic and involve Fibonacci numbers. They form four circles that cut the x-axis at points related to the golden ratio.

math.NT↗
arXiv2020

Apollonian depth and the accidental fractal

The depth function of three numbers representing curvatures of three mutually tangent circles is introduced. Its 2D plot leads to a partition of the moduli space of the triples of mutually tangent circles/disks that is unexpectedly a beautiful fractal, the general form of which resembles that of an Apollonian disk packing, except that it consists of ellipses instead of circles.

math.MG↗
arXiv2020

Spinors, lattices, and classification of integral Apollonian disk packings

A parametrization of integral Descartes configurations (and effectively Apollonian disk packings) by pairs of two-dimensional integral vectors is presented. The vectors, called here tangency spinors defined for pairs of tangent disks, are spinors associated to the Clifford algebra for three-dimensional Minkowski space. A version with Pauli spinors is given. The construction provides a novel interpretation to the known Diophantine equation parametrizing integral Apollonian packings.

math.MG↗
arXiv2019

On the Dedekind tessellation

The Dedekind tessellation -- the regular tessellation of the upper half-plane by the Mobius action of the modular group -- is usually viewed as a system of ideal triangles. We change the focus from triangles to circles and give their complete algebraic characterization with the help of a representation of the modular group acting by Lorentz transformations on Minkowski space. This interesting example of the interplay of geometry, group theory and number theory leads also to convenient algorithms for computer drawing of the Dedekind tessellation.

math.HO↗
arXiv2019

The loop of relativistic velocities as a deformation of the menhir loop

The algebra of the relativistic composition of velocities is shown to be isomorphic to an algebraic loop defined on division algebras. This makes calculations in special relativity effortless and straightforward, unlike the standard formulation, which consists of a rather convoluted algebraic equation. The elegant appearance of the new formula brings about an additional value.

math-ph↗
arXiv2019

Tessellations and Descartes disk configurations

An intriguing correspondence between certain finite planar tessellations and the Descartes circle arrangements is presented. This correspondence may be viewed as a visualization of the spinor structure underlying Descartes circles.

math.MG↗
arXiv2019

A note on unbounded Apollonian disk packings

A construction and algebraic characterization of two unbounded Apollonian Disk packings in the plane and the half-plane are presented. Both turn out to involve the golden ratio.

math.MG↗
arXiv2019

Making sense of relativistic composition of velocities

What does it mean to ``add'' velocities relativistically -- clarification of the conceptual problems, new derivations of the related formulas, and identification of the source of the non-associativity of the standard vector version of the addition formula are addressed.

gr-qc↗
arXiv2019

Apollonian coronas and a new Zeta function

We find a formula for the area of disks tangent to a given disk in an Apollonian disk packing (corona) in terms of a certain novel arithmetic Zeta function. The idea is based on "tangency spinors" defined for pairs of tangent disks.

math.MG↗
arXiv2019

Spinors and Descartes configurations of disks

We define spinors for pairs of tangent disks in the Euclidean plane and prove a number of theorems, one of which may be interpreted as a "square root of Descartes Theorem". In any Apollonian disk packing, spinors form a network. In the Apollonian Window, a special case of Apollonian disk packing, all spinors are integral.

math.MG↗
arXiv2018

On geometry of phenomenological thermodynamics

We present the formalism of phenomenological thermodynamics in terms of the even-dimensional symplectic geometry, and argue that it catches its geometric essence in a more profound and clearer way than the popular odd-dimensional contact structure description. Among the advantages are a number of conceptual clarifications: the geometric role of internal energy (not made as an independent variable), the lattice of potentials, and the gauge interpretation of the theory.

math-ph↗