Two-dimensional hypercomplex numbers and related trigonometries and geometries

Catoni, F.; Cannata, Roberto; Catoni, Vincenzo; Zampetti, Paolo

doi:10.1007/s00006-004-0008-2

Cited by 28 publications

(23 citation statements)

References 8 publications

(30 reference statements)

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“…Thanks to this correspondence and by pointing out the analogies and the differences with these two number systems, the space-time trigonometry has been formalized with the same rigour as Euclidean one [10], [12]. In fact the theorems of Euclidean trigonometry are usually obtained through elementary geometry observations.…”

Section: Basic Definitionsmentioning

confidence: 99%

“…This mathematical tool is based on the use of hyperbolic numbers, introduced by S. Lie in the late XIX century [11]. In analogy with the procedures applied in the case of complex numbers, it is possible to formalize, also for the hyperbolic numbers, a space-time geometry and a trigonometry following the same Euclidean axiomatic-deductive method [10,12]. In this paper, we first summarize the introduction of the hyperbolic space-time trigonometry and then apply it to formalize the twin paradox for inertial motions as well as for accelerated ones.…”

Section: Introductionmentioning

confidence: 99%

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Space-Time Trigonometry and Formalization of the “Twin Paradox” for Uniform and Accelerated Motions

Boccaletti

Catoni

2006

AACA

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-The formal structure of the early Einstein's Special Relativity follows the axiomatic deductive method of Euclidean geometry. In this paper we show the deep-rooted relation between Euclidean and space-time geometries that are both linked to a two-dimensional number system: the complex and hyperbolic numbers, respectively. By studying the properties of these numbers together, pseudo-Euclidean trigonometry has been formalized with an axiomatic deductive method and this allows us to give a complete quantitative formalization of the twin paradox in a familiar "Euclidean" way for uniform motions as well as for accelerated ones.

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Section: Basic Definitionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Space-Time Trigonometry and Formalization of the “Twin Paradox” for Uniform and Accelerated Motions

Boccaletti

Catoni

2006

AACA

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“…Generalizations of Euclidean geometry (Pytaghoras theorem, Hero formula in the spirit of Ref. [14]). …”

Section: Discussionmentioning

confidence: 99%

Complex algebras on n-order polynomials and generalizations of trigonometry, oscillator model and Hamilton dynamics

Yamaleev

2005

AACA

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A generator of the complex algebra within the framework of general formulation obeys the quadratic equation of the type e 2 = a1e − a0. In this paper we construct the general complex algebras of the n-th order where the generators obey n-order polynomial equation of the type e n = an−1e n−1 − an−2e n−2 + . . . + (−) n+1 a0, with real coefficients a k , k = 0, 1, ..., n−1. This algebra induces a generalized trigonometry ((n + 1)-gonometry), subyacent to the n-th order oscillator model and to the n-th order Hamilton equations.

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“…The three main classes of bidimensional H-numbers are distinguished according to the value of ∆ [10,11]. The first class is that of the elliptic H-numbers, for which ∆ < 0; in this case, Eq.…”

Section: Bidimensional Hypercomplex Numbers and Complex Numbersmentioning

confidence: 99%

Commutative hypercomplex numbers and functions of hypercomplex variable: a matrix study

Catoni

Cannata

Nichelatti

et al. 2005

Adv. Appl. Clifford Algebras

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Systems of hypercomplex numbers, which had been studied and developed at the end of the 19 th century, are nowadays quite unknown to the scientific community. It is believed that study of their applications ended just before one of the fundamental discoveries of the 20 th century, Einstein's equivalence between space and time. Owing to this equivalence, not-defined quadratic forms have got concrete physical meaning and have been recently recognized to be in strong relationship with a system of bidimensional hypercomplex numbers. These numbers (called hyperbolic) can be considered as the most suitable mathematic language for describing the bidimensional space-time, in spite of some unfamiliar algebraic properties common to all the commutative hypercomplex systems with more than two dimensions: they are decomposable systems and there are non-zero numbers whose product is zero. With respect to the famous Hamilton quaternions, one can introduce the differential calculus for the hyperbolic numbers and for all the commutative hypercomplex systems; moreover, one can even introduce functions of hypercomplex variable.The aim of this work is to study the systems of commutative hypercomplex numbers and the functions of hypercomplex variable by describing them in terms of a familiar mathematical tool, i.e. matrix algebra.

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Two-dimensional hypercomplex numbers and related trigonometries and geometries

Cited by 28 publications

References 8 publications

Space-Time Trigonometry and Formalization of the “Twin Paradox” for Uniform and Accelerated Motions

Space-Time Trigonometry and Formalization of the “Twin Paradox” for Uniform and Accelerated Motions

Complex algebras on n-order polynomials and generalizations of trigonometry, oscillator model and Hamilton dynamics

Commutative hypercomplex numbers and functions of hypercomplex variable: a matrix study

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