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Two Remarkable Spherical Arrangements of Circles

  • Hellmuth StachelEmail author
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 809)

Abstract

Inspired by recent publications of K. Myrianthis and of A.W. Akopyan and A.I. Bobenko, two different arrangements of circles on the sphere are studied. The first one originates from Phyllotaxis, a topic in plant morphogenesis, and gives rise to a polyhedron with hexagonal faces and a covering of the sphere with circles in a spiral arrangement. The second is related to a Poncelet grid on the sphere. The extended n sides of a closed spherical billiard within a conic form a net with a finite number of quadrilaterals with incircles. Orthogonality transforms it into a configuration of n concurrent lines where each pair is ‘concircular’ with \(n-2\) other pairs, i.e., the four lines are inscribed into a cone of revolution.

Keywords

Spiral grid Hyperbolic screw motion Incircular net Ivory’s theorem 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Vienna University of TechnologyViennaAustria

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