An axiomatic foundation of Cayley-Klein geometries

Struve, Rolf

doi:10.1007/s00022-015-0293-z

Cited by 13 publications

(6 citation statements)

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“…Observers are interpreted to be labels for inertial coordinate systems. Quantities are used to specify coordinates, lengths and related quantities, and we assume that 1 Metric geometries corresponding to these two structures also appear among Cayley-Klein geometries; see, e.g., [Str16] and [PSS17,§6].…”

Section: Frameworkmentioning

confidence: 99%

Groups of Worldview Transformations Implied by Isotropy of Space

Madarász¹,

Stannett²,

Székely³

2020

Preprint

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Given any Euclidean ordered field, Q, and any 'reasonable' group, G, of (1+3)-dimensional spacetime symmetries, we show how to construct a model M G of kinematics for which the set W of worldview transformations between inertial observers satisfies W = G. This holds in particular for all relevant subgroups of Gal cPoi, and cEucl (the groups of Galilean, Poincaré and Euclidean transformations, respectively, where c ∈ Q is a model-specific parameter corresponding to the speed of light in the case of Poincaré transformations).In doing so, by an elementary geometrical proof, we demonstrate our main contribution: spatial isotropy is enough to entail that the set W of worldview transformations satisfies either W ⊆ Gal, W ⊆ cPoi, or W ⊆ cEucl for some c > 0. So assuming spatial isotropy is enough to prove that there are only 3 possible cases: either the world is classical (the worldview transformations between inertial observers are Galilean transformations); the world is relativistic (the worldview transformations are Poincaré transformations); or the world is Euclidean (which gives a nonstandard kinematical interpretation to Euclidean geometry). This result considerably extends previous results in this field, which assume a priori the (strictly stronger) special principle of relativity, while also restricting the choice of Q to the field R of reals.As part of this work, we also prove the rather surprising result that, for any G containing translations and rotations fixing the time-axis t, the requirement that G be a subgroup of one of the groups Gal, cPoi or cEucl is logically equivalent to the somewhat simpler requirement that, for all g ∈ G: g[t] is a line, and if g[t] = t then g is a trivial transformation (i.e. g is a linear transformation that preserves Euclidean length and fixes the time-axis setwise).2010 Mathematics Subject Classification. Primary 51P05 (83A05, 70B99); 20A15; 46B20.7 This can be expressed in our formal language without any assumption about the structure of quantities as: R(4 , here R( p ) 2 , R( p ) 3 , and R( p ) 4 denotes the second, third and fourth component of R( p ) ∈ Q 4 , i.e. if R( p ) = (t, x, y, z), then R( p ) 2 = x, R( p ) 3 = y, and R( p ) 4 = z.

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Section: Frameworkmentioning

confidence: 99%

Groups of Worldview Transformations Implied by Isotropy of Space

Madarász¹,

Stannett²,

Székely³

2020

Preprint

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show abstract

“…Over the reals, by choosing whether distances are measured along the groups SO(2) if Q(L) = −1, SO(1, 1) if Q(L) = 1 and R if Q(L) = 0, and analoguously for angles and P , the 3 × 3 possibilities represent the 9 Cayley-Klein geometries, as seen in [15]:…”

Section: Cayley-klein Geometriesmentioning

confidence: 99%

“…Another way of approaching the similarities is using projective geometry and comparing how distances and angles are measured. As seen in [15], we have three types of measurement for both distances and angles: spherical, Euclidean and hyperbolic. This gives us 9 possible plane geometries.…”

Section: Introductionmentioning

confidence: 99%

“…This way distances and angles can be calculated for any field, without the need to use classical trigonometrical functions. The 9 plane geometries in [15] are revealed in a natural way. Furthermore, similar enumerations may be done for all fields, presented here for the field of complex numbers and all finite fields of odd characteristic, resulting in a strikingly similar pattern of 9 plane geometries for finite fields.…”

Section: Introductionmentioning

confidence: 99%

“…The Poincaré model permits the construction of Möbius planes, which over arbitrary fields become the Miquelian Möbius planes ( [3]). Other attempts at describing conformal geometries over arbritrary fields include [15], [17] and [16]. This article hopes to extend and incorporate all these constructions and shed some insight into these relationships through linear algebra.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A universal linear algebraic model for conformal geometries

Juhász

2018

J. Geom.

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This article describes an entirely algebraic construction for developing conformal geometries, which provide models for, among others, the Euclidean, spherical and hyperbolic geometries. On one hand, their relationship is usually shown analytically, through a framework comparing the measurement of distances and angles in Cayley-Klein geometries, including Lorentzian geometries, as done by F. Bachmann and later R. Struve. On the other hand, such a relationship may also be expressed in a purely linear algebraic manner, as explained by D. Hestens, H. Li and A. Rockwood. The model described in this article unifies these approaches via a generalization of Lie sphere geometry. Like the work of N. Wildberger, it is a purely algebraic construction, and as such it works over any field of odd characteristic. It is shown that measurement of distances and angles is an inherent property of the model that is easy to identify, and the possible models are classified over the real, complex and finite fields, and partially in characteristic 2, revealing a striking analogy between the real and finite geometries.

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The Axiomatic Destiny of the Theorems of Pappus and Desargues

Pambuccian

Schacht

2019

Geometry in History

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An axiomatic foundation of Cayley-Klein geometries

Cited by 13 publications

References 13 publications

Groups of Worldview Transformations Implied by Isotropy of Space

Groups of Worldview Transformations Implied by Isotropy of Space

A universal linear algebraic model for conformal geometries

The Axiomatic Destiny of the Theorems of Pappus and Desargues

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