Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-29T15:24:38.575Z Has data issue: false hasContentIssue false
  • English
  • Français

ON A FABRIC OF KISSING CIRCLES

Part of: Sequences and sets Discrete geometry

Published online by Cambridge University Press:  08 January 2021

VIERA ČERŇANOVÁ Open the ORCID record for VIERA ČERŇANOVÁ [Opens in a new window]
VIERA ČERŇANOVÁ*
Affiliation:
Department of Mathematics and Computer Science, Faculty of Education, Trnava University, Priemyselná 4, P.O. Box 9, 918 43Trnava, Slovakia e-mail: [email protected]
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Applying circle inversion on a square grid filled with circles, we obtain a configuration that we call a fabric of kissing circles. We focus on the curvature inside the individual components of the fabric, which are two orthogonal frames and two orthogonal families of chains. We show that the curvatures of the frame circles form a doubly infinite arithmetic sequence (bi-sequence), whereas the curvatures in each chain are arranged in a quadratic bi-sequence. We also prove a sufficient condition for the fabric to be integral.

Keywords

curvaturefabric of kissing circlesframeinversion in a circlePL chain

MSC classification

Primary: 52C26: Circle packings and discrete conformal geometry
Secondary: 11B37: Recurrences
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 104 , Issue 2 , October 2021 , pp. 320 - 329
Check for updates
Copyright
© 2020 Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Čerňanová, V., ‘On a configuration resulting from circle inversions’, Proc. 19th Conf. Applied Mathematics APLIMAT 2020 (Curran Associates, Red Hook, NY, 2020), 238–243.Google Scholar
Descartes, R., ‘Correspondance avec Elisabeth’ (Egmond du Hoef, November 1643). https://web.archive.org/web/20050306191834/http://www.ac-nice.fr:80/philo/textes/Descartes-Elisabeth/Descartes-Elisabeth.htm.Google Scholar
Fukagawa, H. and Pedoe, D., Japanese Temple Geometry Problems. San Gaku (The Charles Babbage Research Centre, Winnipeg, Canada, 1989).Google Scholar
Fukagawa, H. and Rothman, T., Sacred Mathematics: Japanese Temple Geometry (Princeton University Press, Princeton, NJ, 2008).Google Scholar
Lagarias, J. C., Mallows, C. L. and Wilks, A. R., ‘Beyond the Descartes circle theorem’, Amer. Math. Monthly 109(4) (2002), 338–361.CrossRefGoogle Scholar
Soddy, F., ‘The kiss precise’, Nature 137(3477) (1936), 1021.CrossRefGoogle Scholar
Stephenson, K., ‘Circle packing: a mathematical tale’, Notices Amer. Math. Soc. 50(11) (2003), 1376–1388.Google Scholar