From line-systems to sphere-systems - Schläfli's double six, Lie's line-sphere transformation, and Grace's theorem

Authors:
Hiroshi Maehara;Norihide Tokushige
Affiliations:
College of Education, Ryukyu University, Nishihara, Okinawa, 903-0213, Japan;College of Education, Ryukyu University, Nishihara, Okinawa, 903-0213, Japan
Venue:
European Journal of Combinatorics
Year:
2009

Citing 3
Cited 0

Helly-type theorems for spheres

Discrete & Computational Geometry
Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics)

Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics)
Ball-Polyhedra

Discrete & Computational Geometry

Quantified Score

Hi-index	0.00

Visualization

Abstract

If each four spheres in a set of five unit spheres in R^3 have nonempty intersection, then all five spheres have nonempty intersection. This result is proved using Grace's theorem: the circumsphere of a tetrahedron encloses none of its escribed spheres. This paper provides self-contained proofs of these results; including Schlafli's double six theorem and modified version of Lie's line-sphere transformation. Some related problems are also posed.