On a Theorem in Geometry

Pedoe, D.

doi:10.2307/2314247

Cited by 20 publications

(13 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Stigler [31]. Proofs of the n-dimensional theorem appear in Pedoe [23] and Coxeter [9]. Pedoe observes that this result is actually a theorem 3 of real algebraic geometry, rather than of complex algebraic geometry, in dimensions 3 and above.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Beyond the Descartes Circle Theorem

Lagarias¹,

Mallows²,

Wilks³

2002

The American Mathematical Monthly

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 99%

“…Another independent rediscovery with a complete proof was given in 1842 by H. Beecroft [4], and is described in Coxeter [8]. Many other proofs have been discovered (and rediscovered), some of which appear in Pedoe [23].…”

Section: Introductionmentioning

confidence: 99%

Beyond the Descartes Circle Theorem

Lagarias¹,

Mallows²,

Wilks³

2002

The American Mathematical Monthly

View full text Add to dashboard Cite

show abstract

“…11. The algorithm based on the Descartes theorem allows calculating the pore size distribution from the bead size distribution (see [20,33]). The calculated pore size distributions are shown by the two left curves in Fig.…”

Section: Determining the Jamming Ratio Dependency Of The Correlation mentioning

confidence: 99%

Improved population balance model for straining-dominant deep bed filtration using network calculations

Yuan

You

Shapiro

et al. 2013

Chemical Engineering Journal

View full text Add to dashboard Cite

“…This describes the relationship between the curvatures of four mutually tangent circles in the plane. A selection of proofs of this are given in [9]. …”

Section: Self-similar Half-plane Packingsmentioning

confidence: 99%

Apollonian circle packings of the half-plane

Ching

Doyle

2012

Journal of Combinatorics

View full text Add to dashboard Cite

Abstract. We consider Apollonian circle packings of a half Euclidean plane. We give necessary and sufficient conditions for two such packings to be related by a Euclidean similarity (that is, by translations, reflections, rotations and dilations) and describe explicitly the group of self-similarities of a given packing. We observe that packings with a non-trivial self-similarity correspond to positive real numbers that are the roots of quadratic polynomials with rational coefficients. This is reflected in a close connection between Apollonian circle packings and continued fractions which allows us to completely classify such packings up to similarity.

show abstract

On a Theorem in Geometry

Cited by 20 publications

References 0 publications

Beyond the Descartes Circle Theorem

Beyond the Descartes Circle Theorem

Improved population balance model for straining-dominant deep bed filtration using network calculations

Apollonian circle packings of the half-plane

Contact Info

Product

Resources

About