Inversive Geometry

Wilker, J. B.

doi:10.1007/978-1-4612-5648-9_27

Cited by 31 publications

(22 citation statements)

References 49 publications

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“…The main results in this paper are theorems in inversive geometry, as described in Wilker [33], also Alexander [2] and Schwerdtfeger [26]. Inversive geometry is the geometry that preserves spheres and their incidences, which consists of the study of geometric properties preserved by the group Möb(n) of conformal transformations of the spaceR n = R n ∪ {∞} ≈ S n .…”

Section: Resultsmentioning

confidence: 99%

Beyond the Descartes Circle Theorem

Lagarias¹,

Mallows²,

Wilks³

2002

The American Mathematical Monthly

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The Descartes circle theorem states that if four circles are mutually tangent in the plane, with disjoint interiors, then their curvatures (or "bends"). We show that similar relations hold involving the centers of the four circles in such a configuration, coordinatized as complex numbers, yielding a complex Descartes Theorem. These relations have elegant matrix generalizations to the n-dimensional case, in each of Euclidean, spherical, and hyperbolic geometries. These include analogues of the Descartes circle theorem for spherical and hyperbolic space.

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Section: Resultsmentioning

confidence: 99%

Beyond the Descartes Circle Theorem

Lagarias¹,

Mallows²,

Wilks³

2002

The American Mathematical Monthly

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“…The parameter space property (P1) generalizes to all dimensions, as was shown by three of the authors in [23]. Results in this direction were established earlier by Wilker [35], who used the term cluster for an ordered, but not oriented, Descartes configuration. There is a notion of augmented curvature-center coordinates (ACC-coordinates) for an ordered, oriented n-dimensional Descartes configuration, and a parameter space M n D of all such configurations, specified by a matrix condition that intertwines two quadratic forms in n + 2 variables under conjugacy, as was shown in [23].…”

Section: Resultsmentioning

confidence: 53%

“…One checks that all the above matrices are actually in Aut(Q W,n ) ↑ , so that the map so far defines a homomorphism of Möb(n) into Aut(Q W,n ) ↑ ≃ O(n + 1, 1) ↑ , identified with the isochronous Lorentz group. The group Möb(n) acts simply transitively on ordered Descartes configurations, as observed by Wilker [35,Theorem 3,p. 394], and the group Aut(Q W,n ) acts simply transitively on ordered, oriented Descartes configurations, as implied by Theorem 3.1.…”

Section: Appendix Möbius Group Actionmentioning

confidence: 84%

“…B. Wilker [35], who introduced in spherical geometry a coordinate system analogous to augmented curvature-center coordinates, see [35, §2 p. 388-390 and §9]. However he did not formulate any result explicitly exhibiting a quadratic form like Q W,n ; see the remark on [23, p. 349 ].…”

Section: Descartes Configurations and Acc-coordinatesmentioning

confidence: 99%

“…Wilker [35,Theorem 3]. Thus it suffices to prove the result for a single such configuration, and we consider the configuration of n + 1 mutually touching spheres of equal radii whose centers are at the vertices of a regular n-simplex, and tangency points of the spheres are the midpoints of its edges.…”

Section: N-dimensional Duality Operatormentioning

confidence: 99%

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Apollonian Circle Packings: Geometry and Group Theory III. Higher Dimensions

Graham

Lagarias

Mallows

et al. 2005

Discrete Comput Geom

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This paper gives n-dimensional analogues of the Apollonian circle packings in parts I and II. Those papers considered circle packings described in terms of their Descartes configurations, which are sets of four mutually touching circles. They studied packings that had integrality properties in terms of the curvatures and centers of the circles. Here we consider collections of n-dimensional Descartes configurations, which consist of n + 2 mutually touching spheres.We work in the space M n D of all n-dimensional oriented Descartes configurations parametrized in a coordinate system, ACC-coordinates, as those (n + 2) × (n + 2) real matrices W with, and Q D,n and Q W,n are their corresponding symmetric matrices. On the parameter space M n D of ACC-matrices the group Aut(Q D,n ) acts on the left, and Aut(Q W,n ) acts on the right. Both these groups are isomorphic to the (n + 2)-dimensional Lorentz group O(n + 1, 1), and give two different "geometric" actions. The right action of Aut(Q W,n ) (essentially) corresponds to Möbius transformations acting on the underlying Euclidean space R n while the left action of Aut(Q D,n ) is defined only on the parameter space M n D .1 Partially supported by NSF grants DMS-0070574, DMS-0245526 and a Sloan Fellowship. This author is also affiliated with Dalian University of Technology, China.We introduce n-dimensional analogues of the Apollonian group, the dual Apollonian group and the super-Apollonian group. These are finitely generated groups in Aut(Q D,n ), with the following integrality properties: the dual Apollonian group consists of integral matrices in all dimensions, while the other two consist of rational matrices, with denominators having prime divisors drawn from a finite set S depending on the dimension. We show that the Apollonian group and the dual Apollonian group are finitely presented, and are Coxeter groups. We define an Apollonian cluster ensemble to be any orbit under the Apollonian group, with similar notions for the other two groups. We determine in which dimensions there exist rational Apollonian cluster ensembles (all curvatures rational) and strongly rational Apollonian sphere ensembles (all ACC-coordinates rational).

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On Coxeter’s Loxodromic Sequences of Tangent Spheres

Weiss

1981

The Geometric Vein

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Inversive Geometry

Cited by 31 publications

References 49 publications

Beyond the Descartes Circle Theorem

Beyond the Descartes Circle Theorem

Apollonian Circle Packings: Geometry and Group Theory III. Higher Dimensions

On Coxeter’s Loxodromic Sequences of Tangent Spheres

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