© Hindawi Publishing Corp. THE COMBINATORIAL STRUCTURE OF TRIGONOMETRY

The native mathematical language of trigonometry is combinatorial. Two interrelated combinatorial symmetric functions underlie trigonometry. We use their characteristics to derive identities for the trigonometric functions of multiple distinct angles. When applied to the sum of an infinite number of infinitesimal angles, these identities lead to the power series expansions of the trigonometric functions. When applied to the interior angles of a polygon, they lead to two general constraints satisfied by the corresponding tangents. In the case of multiple equal angles, they reduce to the Bernoulli identities. For the case of two distinct angles, they reduce to the Ptolemy identity. They can also be used to derive the De Moivre-Cotes identity. The above results combined provide an appropriate mathematical combinatorial language and formalism for trigonometry and more generally polygonometry. This latter is the structural language of molecular organization, and is omnipresent in the natural phenomena of molecular physics, chemistry, and biology. Polygonometry is as important in the study of moderately complex structures, as trigonometry has historically been in the study of simple structures

The combinatorial structure of trigonometry

The native mathematical language of trigonometry is combinatorial. Two interrelated combinatorial symmetric functions underlie trigonometry. We use their characteristics to derive identities for the trigonometric functions of multiple distinct angles. When applied to the sum of an infinite number of infinitesimal angles, these identities lead to the power series expansions of the trigonometric functions. When applied to the interior angles of a polygon, they lead to two general constraints satisfied by the corresponding tangents. In the case of multiple equal angles, they reduce to the Bernoulli identities. For the case of two distinct angles, they reduce to the Ptolemy identity. They can also be used to derive the De Moivre-Cotes identity. The above results combined provide an appropriate mathematical combinatorial language and formalism for trigonometry and more generally polygonometry. This latter is the structural language of molecular organization, and is omnipresent in the natural phenomena of molecular physics, chemistry, and biology. Polygonometry is as important in the study of moderately complex structures, as trigonometry has historically been in the study of simple structures

Discrete Harmonic Oscillator: A Short Compendium of Formulas

peer reviewedSince 2002, we have, in seven papers, studied the problem of the discrete harmonic oscillator, analytically, numerically, and graphically, and made progress on a number of fronts. We identified the natural frequency of oscillation of the dual incursive oscillators, studied the system bifurcation generated by incursive discretization, and the frequency dependent correlation between the resulting incursive oscillators. We also studied the synchronization of the discretizing time interval with the frequency of oscillation. From this analysis there emerged a nonlinear (or more precisely bilinear) formalism that is perfectly stable at all time scales, and fully conserves the total energy of the system. The formalism is applicable to the discrete harmonic oscillator specifically, and to discrete Hamiltonian systems in general. In retrospect the important themes and crucial steps can more easily be identified, and this is the purpose of this short note. In it we give, in a unified notation, a short compendium of the key formulas. This summary can serve as a guide for navigating through the seven papers, as well as a practical concise manual for using the formalism

Discrete Harmonic Oscillator: Evolution of Notation and Cumulative Erratum

peer reviewedAs the project on the discrete harmonic oscillator advanced, the notation evolved in parallel with our understanding of the problem, and it became progressively clear that a change of notation concerning the natural angular frequency of oscillation of the system, and related parameters, was required in order to conform to the spirit of the physics being developed. Specifically the notation needed to reflect the fact that the discrete harmonic oscillator is the more general entity, with the continuous harmonic oscillator being a special (and to some extent unrealistic) limiting case corresponding to dt-->0, where dt is the discretizing time interval. Furthermore, as the discrete nature of time (and eventually space), incursion (anticipation), system bifurcation, hyperincursion, frequency dependent correlation, and synchronization, where progressively integrated into the fabric of the discretizing algorithm, a new discrete, perfectly stable and energy conserving, formalism emerged, and with it evolved, in parallel, the expression for the total energy. In this short note we will focus on the evolution, throughout the project, of the notation for frequency, and the expression for the discrete total energy. We also provide a cumulative erratum for the seven papers in the project so far, as well as a table summarizing the evolution of the notation for the angular frequency and related parameters

Time-Symmetric Discretization of the Harmonic Oscillator

peer reviewedWe explicitly and analytically demonstrate that simple time-symmetric discretization of the harmonic oscillator (used as a simple model of a discrete dynamical system), leads to discrete equations of motion whose solutions are perfectly stable at all time scales, and whose energy is exactly conserved. This result is important for both fundamental discrete physics, as well as for numerical analysis and simulation