On a Theorem in Geometry

Pedoe, D.

doi:10.1080/00029890.1967.12000012

Cited by 24 publications

(11 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The Theorem was rediscovered by Philip Beecroft almost 200 years later and then by Frederick Soddy almost 100 years later. This makes the quadricentennial history of Geometrical Predictions look like Tercentennial history of the Law of Large Numbers that started with publication of Bernoulli's Theorem and had its Centenary and Bicentenary anniversaries commemorated by Laplace's publication of his Treatise on Probability, translations of Ars Conjectandi and publications on the history of the LLN for the Bicentenary celebrations The Tricentenary that is also the 250th anniversary of the first public presentation of Thomas Bayes's work was commemorated in Paris in at a colloquium entitled L'art de conjecturer des Bernoulli [10][11][12][20][21].…”

Section: Resultsmentioning

confidence: 99%

Multidimensional Time Model for Probability Cumulative Function Applied to Geometrical Predictions

Fundator¹

2018

ACM

View full text Add to dashboard Cite

Multidimensional Time Model for Probability Cumulative Function can be reduced to finite-dimensional time model, which can be characterized by Boolean algebra for operations over events and their probabilities and index set for reduction of infinite dimensional time model to finite number of dimensions of time model through application of Boolean prime ideal theorem and Stone duality and can be indexed by an index set considering also the fractal-dimensional time arising from alike supersymmetrical properties of probability through consideration of extension of the classical Stone duality to the category of Boolean spaces, locally compact Hausdorff spaces. The introduction of probabilistical prediction philosophically based on Erdős-Rényi LLN for the prediction through Descartes' cycles, Gauss methods of trigonometric interpolation and least squares to reduce error in determination of the orbits of planetary bodies, and Farey series continued by sampling on the Sierpinski gasket.

show abstract

Section: Resultsmentioning

confidence: 99%

Multidimensional Time Model for Probability Cumulative Function Applied to Geometrical Predictions

Fundator¹

2018

ACM

View full text Add to dashboard Cite

show abstract

“…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]. Theorem 3.8 (Descartes' Circle Theorem).…”

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

“…The Monthly has published several papers concerning this fascinating topic: [1,3,4,5,6]. It was only in 2001 [3] that is was noticed, and proved, that the "curvature-centers" (curvature times center where the center is considered a complex number) satisfy the same relation.…”

mentioning

confidence: 99%