Jerzy Kocik | Southern Illinois University at Carbondale (original) (raw)
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Papers by Jerzy Kocik
Series on knots and everything, Sep 30, 2023
arXiv (Cornell University), Aug 7, 2014
We present a geometric theorem on a porism about cyclic quadrilaterals, namely the existence of a... more We present a geometric theorem on a porism about cyclic quadrilaterals, namely the existence of an infinite number of cyclic quadrilaterals through four fixed collinear points once one exists. Also, a technique of proving such properties with the use of pseudounitary traceless matrices is presented. A similar property holds for general quadrics as well as the circle. 1
arXiv (Cornell University), Feb 16, 2007
Krawtchouk's polynomials occur classically as orthogonal polynomials with respect to the binomial... more Krawtchouk's polynomials occur classically as orthogonal polynomials with respect to the binomial distribution. They may be also expressed in the form of matrices, that emerge as arrays of the values that the polynomials take. The algebraic properties of these matrices provide a very interesting and accessible example in the approach to probability theory known as quantum probability. First it is noted how the Krawtchouk matrices are connected to the classical symmetric Bernoulli random walk. And we show how to derive Krawtchouk matrices in the quantum probability context via tensor powers of the elementary Hadamard matrix. Then connections with the classical situation are shown by calculating expectation values in the quantum case.
arXiv: Number Theory, Oct 17, 2007
A family of sequences produced by a non-homogeneous linear recurrence formula derived from the ge... more A family of sequences produced by a non-homogeneous linear recurrence formula derived from the geometry of circles inscribed in lenses is introduced and studied. Mysterious "underground" sequences underlying them are discovered in this paper. 1. Introduction 2. Recurrence formula from geometry 3. More on lens geometry 4. Basic algebraic properties of lens sequences 5.
A remarkably simple Diophantine quadratic equation is known to generate all Apollonian integral g... more A remarkably simple Diophantine quadratic equation is known to generate all Apollonian integral gaskets (disk packings). A new derivation of this formula is presented here based on inversive geometry. Also occurrences of Pythagorean triples in such gaskets is discussed.
An interpretation of Krawtchouk matrices in terms of discrete version of the Feynman path integra... more An interpretation of Krawtchouk matrices in terms of discrete version of the Feynman path integral is given. Also, an algebraic characterization in terms of the algebra of split quaternions is provided. The resulting properties include an easy inference of the spectral decomposition. It is also an occasion for an expository clarification of the role of Krawtchouk matrices in different areas, including quantum information.
A geometric diagram that allows one to visualize the Poincar\'e formula for relativistic addi... more A geometric diagram that allows one to visualize the Poincar\'e formula for relativistic addition of velocities in one dimension is presented. An analogous diagram representing the angle sum formula for trigonometric tangent is given.
We present the formalism of phenomenological thermodynamics in terms of the even-dimensional symp... more 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.
We show that the space of Euclid's parameters for Pythagorean triples is endowed with a natur... more We show that the space of Euclid's parameters for Pythagorean triples is endowed with a natural symplectic structure and that it emerges as a spinor space of the Clifford algebra R_2,1, whose minimal version may be conceptualized as a 4-dimensional real algebra of "kwaternions." We observe that this makes Euclid's parameterization the earliest appearance of the concept of spinors. We present an analogue of the "magic correspondence" for the spinor representation of Minkowski space and show how the Hall matrices fit into the scheme. The latter obtain an interesting and perhaps unexpected geometric meaning as certain symmetries of an Apollonian gasket. An extension to more variables is proposed and explicit formulae for generating all Pythagorean quadruples, hexads, and decuples are provided.
We show that the relation between the Schrödinger equation and diffusion processes has an algebra... more We show that the relation between the Schrödinger equation and diffusion processes has an algebraic nature and can be revealed via the structure of "duplex numbers." This helps one to clarify that quantum mechanics cannot be reduced to diffusion theory. Also, a generalized version of quantum mechanics where is replaced by a normed algebra with a unit is proposed.
Journal of the Optical Society of America B, 2019
Contemporary Mathematics, 2016
An interpretation of Krawtchouk matrices in terms of discrete version of the Feynman path integra... more An interpretation of Krawtchouk matrices in terms of discrete version of the Feynman path integral is given. Also, an algebraic characterization in terms of the algebra of split quaternions is provided. The resulting properties include an easy inference of the spectral decomposition. It is also an occasion for an expository clarification of the role of Krawtchouk matrices in different areas, including quantum information.
American Mathematical Monthly, 2009
Abstract. We show that the space of Euclid’s parameters for Pythagorean triples is endowed with a... more Abstract. We show that the space of Euclid’s parameters for Pythagorean triples is endowed with a natural symplectic structure and that it emerges as a spinor space of the Clifford algebra R21, whose minimal version may be con-ceptualized as a 4-dimensional real algebra of “kwaternions. ” We observe that this makes Euclid’s parameterization the earliest appearance of the concept of spinors. We present an analogue of the “magic correspondence ” for the spinor representation of Minkowski space and show how the Hall matrices fit into the scheme. The latter obtain an interesting and perhaps unexpected geomet-ric meaning as certain symmetries of an Apollonian gasket. An extension to more variables is proposed and explicit formulae for generating all Pythagorean quadruples, hexads, and decuples are provided.
Physical Review D
Using scaling arguments and the property of self-similarity we derive the Casimir energies of Sie... more Using scaling arguments and the property of self-similarity we derive the Casimir energies of Sierpinski triangles and Sierpinski rectangles. The Hausdorff-Besicovitch dimension (fractal dimension) of the Casimir energy is introduced and the Berry-Weyl conjecture is discussed for these geometries. We propose that for a class of fractals, comprising of compartmentalized cavities, it is possible to establish a finite value to the Casimir energy even while the Casimir energy of the individual cavities consists of divergent terms.
arXiv: Metric Geometry, 2019
We define spinors for pairs of tangent disks in the Euclidean plane and prove a number of theorem... more 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.
arXiv: General Mathematics, 2021
A simple matrix formulation of the Fibonacci, Lucas, Chebyshev, and Dixon polynomials polynomials... more A simple matrix formulation of the Fibonacci, Lucas, Chebyshev, and Dixon polynomials polynomials is presented. It utilizes the powers and the symmetric tensor powers of a certain matrix.
A family of sequences produced by a non-homogeneous linear recurrence formula derived from the ge... more A family of sequences produced by a non-homogeneous linear recurrence formula derived from the geometry of circles inscribed in lenses is introduced and studied. Mysterious “underground” sequences underlying them are discovered in this paper.
Series on knots and everything, Sep 30, 2023
arXiv (Cornell University), Aug 7, 2014
We present a geometric theorem on a porism about cyclic quadrilaterals, namely the existence of a... more We present a geometric theorem on a porism about cyclic quadrilaterals, namely the existence of an infinite number of cyclic quadrilaterals through four fixed collinear points once one exists. Also, a technique of proving such properties with the use of pseudounitary traceless matrices is presented. A similar property holds for general quadrics as well as the circle. 1
arXiv (Cornell University), Feb 16, 2007
Krawtchouk's polynomials occur classically as orthogonal polynomials with respect to the binomial... more Krawtchouk's polynomials occur classically as orthogonal polynomials with respect to the binomial distribution. They may be also expressed in the form of matrices, that emerge as arrays of the values that the polynomials take. The algebraic properties of these matrices provide a very interesting and accessible example in the approach to probability theory known as quantum probability. First it is noted how the Krawtchouk matrices are connected to the classical symmetric Bernoulli random walk. And we show how to derive Krawtchouk matrices in the quantum probability context via tensor powers of the elementary Hadamard matrix. Then connections with the classical situation are shown by calculating expectation values in the quantum case.
arXiv: Number Theory, Oct 17, 2007
A family of sequences produced by a non-homogeneous linear recurrence formula derived from the ge... more A family of sequences produced by a non-homogeneous linear recurrence formula derived from the geometry of circles inscribed in lenses is introduced and studied. Mysterious "underground" sequences underlying them are discovered in this paper. 1. Introduction 2. Recurrence formula from geometry 3. More on lens geometry 4. Basic algebraic properties of lens sequences 5.
A remarkably simple Diophantine quadratic equation is known to generate all Apollonian integral g... more A remarkably simple Diophantine quadratic equation is known to generate all Apollonian integral gaskets (disk packings). A new derivation of this formula is presented here based on inversive geometry. Also occurrences of Pythagorean triples in such gaskets is discussed.
An interpretation of Krawtchouk matrices in terms of discrete version of the Feynman path integra... more An interpretation of Krawtchouk matrices in terms of discrete version of the Feynman path integral is given. Also, an algebraic characterization in terms of the algebra of split quaternions is provided. The resulting properties include an easy inference of the spectral decomposition. It is also an occasion for an expository clarification of the role of Krawtchouk matrices in different areas, including quantum information.
A geometric diagram that allows one to visualize the Poincar\'e formula for relativistic addi... more A geometric diagram that allows one to visualize the Poincar\'e formula for relativistic addition of velocities in one dimension is presented. An analogous diagram representing the angle sum formula for trigonometric tangent is given.
We present the formalism of phenomenological thermodynamics in terms of the even-dimensional symp... more 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.
We show that the space of Euclid's parameters for Pythagorean triples is endowed with a natur... more We show that the space of Euclid's parameters for Pythagorean triples is endowed with a natural symplectic structure and that it emerges as a spinor space of the Clifford algebra R_2,1, whose minimal version may be conceptualized as a 4-dimensional real algebra of "kwaternions." We observe that this makes Euclid's parameterization the earliest appearance of the concept of spinors. We present an analogue of the "magic correspondence" for the spinor representation of Minkowski space and show how the Hall matrices fit into the scheme. The latter obtain an interesting and perhaps unexpected geometric meaning as certain symmetries of an Apollonian gasket. An extension to more variables is proposed and explicit formulae for generating all Pythagorean quadruples, hexads, and decuples are provided.
We show that the relation between the Schrödinger equation and diffusion processes has an algebra... more We show that the relation between the Schrödinger equation and diffusion processes has an algebraic nature and can be revealed via the structure of "duplex numbers." This helps one to clarify that quantum mechanics cannot be reduced to diffusion theory. Also, a generalized version of quantum mechanics where is replaced by a normed algebra with a unit is proposed.
Journal of the Optical Society of America B, 2019
Contemporary Mathematics, 2016
An interpretation of Krawtchouk matrices in terms of discrete version of the Feynman path integra... more An interpretation of Krawtchouk matrices in terms of discrete version of the Feynman path integral is given. Also, an algebraic characterization in terms of the algebra of split quaternions is provided. The resulting properties include an easy inference of the spectral decomposition. It is also an occasion for an expository clarification of the role of Krawtchouk matrices in different areas, including quantum information.
American Mathematical Monthly, 2009
Abstract. We show that the space of Euclid’s parameters for Pythagorean triples is endowed with a... more Abstract. We show that the space of Euclid’s parameters for Pythagorean triples is endowed with a natural symplectic structure and that it emerges as a spinor space of the Clifford algebra R21, whose minimal version may be con-ceptualized as a 4-dimensional real algebra of “kwaternions. ” We observe that this makes Euclid’s parameterization the earliest appearance of the concept of spinors. We present an analogue of the “magic correspondence ” for the spinor representation of Minkowski space and show how the Hall matrices fit into the scheme. The latter obtain an interesting and perhaps unexpected geomet-ric meaning as certain symmetries of an Apollonian gasket. An extension to more variables is proposed and explicit formulae for generating all Pythagorean quadruples, hexads, and decuples are provided.
Physical Review D
Using scaling arguments and the property of self-similarity we derive the Casimir energies of Sie... more Using scaling arguments and the property of self-similarity we derive the Casimir energies of Sierpinski triangles and Sierpinski rectangles. The Hausdorff-Besicovitch dimension (fractal dimension) of the Casimir energy is introduced and the Berry-Weyl conjecture is discussed for these geometries. We propose that for a class of fractals, comprising of compartmentalized cavities, it is possible to establish a finite value to the Casimir energy even while the Casimir energy of the individual cavities consists of divergent terms.
arXiv: Metric Geometry, 2019
We define spinors for pairs of tangent disks in the Euclidean plane and prove a number of theorem... more 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.
arXiv: General Mathematics, 2021
A simple matrix formulation of the Fibonacci, Lucas, Chebyshev, and Dixon polynomials polynomials... more A simple matrix formulation of the Fibonacci, Lucas, Chebyshev, and Dixon polynomials polynomials is presented. It utilizes the powers and the symmetric tensor powers of a certain matrix.
A family of sequences produced by a non-homogeneous linear recurrence formula derived from the ge... more A family of sequences produced by a non-homogeneous linear recurrence formula derived from the geometry of circles inscribed in lenses is introduced and studied. Mysterious “underground” sequences underlying them are discovered in this paper.
arXiv:1604.05698, 2016
We present a novel representation of the Lorentz group, the geometric version of which uses "reve... more We present a novel representation of the Lorentz group, the geometric version of which uses "reversions" of a sphere while the algebraic version uses pseudo-unitary 2 × 2 matrices over complex numbers and quaternions, and Clifford algebras in general. A remarkably simple formula for relativistic composition of velocities and an accompanying geometric construction follow. The method is derived from the diffeomorphisms of the celestial sphere induced by Lorentz boost.
arXiv:2002.04135, 2020
The depth function of three numbers representing curvatures of three mutually tangent circles is ... more 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.
arXiv:2001.05866, 2020
A parametrization of integral Descartes configurations (and effectively Apollonian disk packings)... more 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 3-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.
arXiv:1912.05768, 2019
The Dedekind tessellation-the regular tessellation of the upper half-plane by the Möbius action o... more The Dedekind tessellation-the regular tessellation of the upper half-plane by the Möbius 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.
arXiv:1910.09174, 2019
In his talk "Integral Apollonian disk Packings" Peter Sarnak asked if there is a "proof from the ... more In his talk "Integral Apollonian disk Packings" Peter Sarnak asked if there is a "proof from the Book" of the Descartes theorem on circles [Sar]. A candidate for such a proof is presented in this note. Figure 1: Four disks in a Descartes configuration-special cases Then the outer circle in (b) represents the boundary of an unbounded disk outside of circle D, for which we assume a negative radius and curvature. Each pair of disks are tangent externally. Theorem 1 (Descartes formula, 1643 [Ped]). The curvatures a, b, c and d of four pair-wise externally tangent disks satisfy (a + b + c + d) 2 = 2 (a 2 + b 2 + c 2 + d 2). A system of such four pair-wise tangent disks is said to form Descartes' configuration.
arXiv:1910.06785, 2019
What does it mean to "add" velocities relativistically-clarification of the conceptual problems, ... more 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.
arXiv:1910.06556, 2019
The algebra of the relativistic composition of velocities is shown to be isomorphic to an algebra... more 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 the standard formulation, which consists of a rather convoluted algebraic equation. The elegant appearance of the new formula brings about an additional value.
arXiv:1912.05768, 2019
An intriguing correspondence between certain finite planar tessellations and the Descartes circle... more 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.
arXiv:1909.09941, 2019
We find a formula for the area of disks tangent to a given disk in an Apollonian disk packing (co... more 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.
arXiv:1909.06994, 2019
We define spinors for pairs of tangent disks in the Euclidean plane and prove a number of theorem... more 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.