The Bowl of Integers and the Hexlet

Soddy, Frederick

doi:10.1038/139077a0

Cited by 42 publications

(32 citation statements)

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“…We show in §5 that for all n ≥ 3 the n-dimensional Apollonian group is a hyperbolic Coxeter group (Theorems 5.1 and 5.2). For n = 3 the group has an extra relation, which explains the existence of the structures "The Bowl of Integers" and "The Hexlet" studied by Soddy [31], [32] [33] and Gosset [17]. It may be true that in all dimensions the dual Apollonian group and super-Apollonian group are also hyperbolic Coxeter groups, but we leave these as open questions.…”

Section: Resultsmentioning

confidence: 99%

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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Section: Resultsmentioning

confidence: 99%

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

Graham

Lagarias

Mallows

et al. 2005

Discrete Comput Geom

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“…Using the transformed configuration he deduced that the ring of C's closes with the sixth ball. This observation, which Soddy had made earlier by using his bowl of integers [9], sparked an interesting correspondence in Nature ([4]- [10]) concerning the possible sizes of the six balls which occur when the ring is in general position.…”

Section: The Hexletmentioning

confidence: 90%

Inversive geometry and the hexlet

Wilker

1981

Geom Dedicata

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“…For example, one might be given a large circle containing five smaller circles of varying sizes and asked to find the value of one of these smaller circles given the diameter of another. These problems ranged in difficulty, and their subject matter included results similar to the Malfatti circles (see [2]), Casey's theorem [3], and Soddy's hexlet theorem [22] which appeared on sangaku prior to being known in Europe [17]. These problems were elaborately decorated using pigment paints.…”

Section: The Sangaku Traditionmentioning

confidence: 99%

Solving Sangaku: A Traditional Solution to a Nineteenth Century Japanese Temple Problem

Hosking¹

2017

Journal for History of Mathematics

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This paper demonstrates how a nineteenth century Japanese votive temple problem known as sangaku from Okayama prefecture can be solved using traditional mathematical methods of the Japanese Edo (1603-1868 CE). We compare a modern solution to a sangaku problem from Sacred Geometry: Japanese Temple Problems of Tony Rothman and Hidetoshi Fukagawa with a traditional solution of Ōhara Toshiaki (?-1828).Our investigation into the solution of Ōhara provides an example of traditional Edo period mathematics using the tenzan jutsu symbolic manipulation method, as well as producing new insights regarding the contextual nature of the rules of this technique.

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The Bowl of Integers and the Hexlet

Cited by 42 publications

References 0 publications

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

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

Inversive geometry and the hexlet

Solving Sangaku: A Traditional Solution to a Nineteenth Century Japanese Temple Problem

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