An interstice relationship for flowers with four petals

摘要

Given three mutually tangent circles with bends (related to the reciprocal of the radius) a, b and c respectively, an important quantity associated with the triple is the value ({langle a,b,c rangle:=ab+ac+bc}) . In this note we show in the case when a central circle with bend b 0 is “surrounded” by four circles, i.e., a flower with four petals, with bends b 1, b 2, b 3,b 4 that either $$sqrt{langle b_{0},b_{1},b_{2} rangle}+sqrt{langle b_{0},b_{3},b_{4} rangle}=sqrt{langle b_{0},b_{2},b_{3} rangle}+sqrt{langle b_{0},b_{4},b_{1} rangle}$$or $$sqrt{langle b_{0},b_{1},b_{2} rangle}=sqrt{langle b_{0},b_{2},b_{3} rangle}+sqrt{langle b_{0},b_{3},b_{4} rangle}+sqrt{langle b_{0},b_{4},b_{1} rangle}$$ (where ({langle b_{0},b_{1},b_{2} rangle}) is chosen to be maximal). As an application we give a sufficient condition for the alternating sum of the ({sqrt{langle a,b,crangle}}) of a packing in standard position to be 0. (A packing is in standard position when we have two circles with bend 0, i.e., parallel lines, and the remaining circles are packed in between.) Mathematics Subject Classification (2010) 52C26 Keywords Interstice apollonian petals flowers Page %P Close Plain text Look Inside Reference tools Export citation EndNote (.ENW) JabRef (.BIB) Mendeley (.BIB) Papers (.RIS) Zotero (.RIS) BibTeX (.BIB) Add to Papers Other actions Register for Journal Updates About This Journal Reprints and Permissions Share Share this content on Facebook Share this content on Twitter Share this content on LinkedIn Related Content Supplementary Material (0) References (10) ReferencesBrinkmann, G., McKay, B.: Guide to using. plantri. http://​cs.​anu.​edu.​au/​~bdm/​plantri/​ (2001)Butler S., Graham R., Guettler G., Mallows C.: Irreducible apollonian configurations and packings. Discret. Comput. Geom. 44, 487–507 (2010)MathSciNetCrossRefMATHCollins C., Stephenson K.: A circle packing algorithm. Comput. Geom. 25, 233–256 (2003)MathSciNetCrossRefMATHGraham R., Lagarias J., Mallows C., Wilks A., Yan C.: Apollonian circle packings: number theory. J. Number Theory 100, 1–45 (2003)MathSciNetCrossRefMATHGraham R., Lagarias J., Mallows C., Wilks A., Yan C.: Apollonian circle packings: geometry and group theory I. The Apollonian group. Discret. Comput. Geom. 34, 547–585 (2005)MathSciNetCrossRefMATHGraham R., Lagarias J., Mallows C., Wilks A., Yan C.: Apollonian circle packings: geometry and group theory II. Super-Apollonian group and integral packings. Discret. Comput. Geom. 35, 1–36 (2006)MathSciNetCrossRefGraham R., Lagarias J., Mallows C., Wilks A., Yan C.: Apollonian circle packings: geometry and group theory III. Higher dimensions. Discret. Comput. Geom. 35, 37–72 (2006)MathSciNetCrossRefMATHHidetoshi F., Rothman T.: Sacred Mathematics: Japanese Temple Geometry. Princeton University Press, Princeton (2008)Stephenson K.: Introduction to Circle Packing. Cambridge University Press, Cambridge (2005)MATHStephenson K.: Circle packing: a mathematical tale. Notices Am. Math. Soc. 50, 1376–1388 (2003)MathSciNetMATH About this Article Title An interstice relationship for flowers with four petals Journal Journal of Geometry Volume 104, Issue 3 , pp 421-438 Cover Date2013-12 DOI 10.1007/s00022-013-0173-3 Print ISSN 0047-2468 Online ISSN 1420-8997 Publisher Springer Basel Additional Links Register for Journal Updates Editorial Board About This Journal Manuscript Submission Topics Geometry Keywords 52C26 Interstice apollonian petals flowers Industry Sectors Finance, Business & Banking Authors Steve Butler (1) Ron Graham (2) Gerhard Guettler (3) Colin Mallows (4) Author Affiliations 1. Iowa State University, Ames, IA, 50011, USA 2. UC San Diego, La Jolla, CA, 92093, USA 3. University of Applied Sciences Giessen Friedberg, 35390, Giessen, Germany 4. Avaya Labs, Basking Ridge, NJ, 07920, USA Continue reading... To view the rest of this content please follow the download PDF link above.