Tangent Circles on the Plane Using Variable Compass
Publication: Journal of Architectural Engineering
Volume 23, Issue 2
Abstract
The present study analyzes how to trace a circle packing using a variable compass. Several of these circle packings have been used by artisans and designers to make ornamental patterns in architecture and arts. The present paper begins by analyzing tangent circles between secant straight lines based on the Archimedes theorem and then studies other ways to trace tangent circles between secant straight lines, showing how craftsmen applied this knowledge to their artistic or architectural creations. Packing circles with different diameters has led to analyzing various geometric constructions used to design Gothic rosettes and is how craftsmen were forced to face new geometric and mathematical problems, such as the Apollonius problem. Geometric constructions of rosettes led to the knowledge and design of different spiral patterns, which involved a new relationship with mathematicians. Similar to a previous study, this paper is limited to geometric constructions developed on the plane, leaving three-dimensional analysis for further study.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
The author especially thanks Antonio Casas for support in multiple issues on geometry and mathematics and Manuela Arzola for the revision of the translation of this manuscript.
References
Abbey Church. (2016). “The rose window in Abbey Church of Saint Alban, United Kingdom.” 〈https://www.flickr.com/photos/4colourprogress/4315268617〉 (Oct. 13, 2016).
Billings, R. W. (1842). Illustrations of geometric tracery from the paneling belonging to Carlisle Cathedral, Thomas and William Boone, London 〈https://archive.org/stream/illustrationsofg00billuoft#page/n97/mode/2up〉 (Oct. 13, 2016).
Billings, R. W. (1851). The power of form applied to geometric tracery, William Blackwood and Sons, London 〈https://archive.org/details/powerofformappli00bill_0〉 (Oct. 13, 2016).
Boyer, C. B. (1996). Cap. IX: Apolonio de perga. Historia de la matemática. Trans. Mariano Martínez Pérez, 5th Ed., Alianza Editorial, Madrid, Spain, 189–208 (in Spanish).
Calter, P. (2000). “How to construct a logarithmic rosette (without even knowing it).” Nexus Network J., 2(1), 25–32.
Chartres Cathedral. (2016). “The rose window in Chartres Cathedral.” 〈http://viajerosblog.com/wp-content/uploads/2011/09/Catedral-Chartres-Vidrieras.jpg〉 (Oct. 13, 2016).
Chateau d’Anne. (2016). “Le Chateau d’Anne, France.” 〈http://www.brynmawr.edu/cities/Cities/wld/04610/04610j.jpg〉 (Oct. 13, 2016).
Courant, R., and Robbins, H. (1996). What is mathematics? An elementary approach to ideas and methods, Oxford University Press, New York.
Fernández-Cabo, M. C. (2013). “Analysis of different hypotheses about the geometric pattern of the Pantheon’s coffered dome.” Nexus Network J., 15(3), 527– 547.
Gauss, C. F. (1965). Disquisitiones aritmethmeticae. Trans. A. A. Clarke, Yale University Press, Princeton, NJ.
Knorr, W. R. (1986). Ancient tradition of geometric problems, Birkhauser, Boston.
Kostovskii, A. N. (1961). Geometrical constructions using compasses only, Blaisdell, New York.
León Cathedral. (2016). “West façade rose window from León Cathedral, Spain.” 〈https://mw2.google.com/mw-panoramio/photos/medium/124558587.jpg〉 (Oct. 13, 2016).
Lleida. (2016). “Seu Vella Cloister, Lleida, Spain.” 〈http://lleidacb.turismedelleida.com/els-nostres-serveis-es/seu-vella〉 (Oct. 13, 2016).
Lyon Vieux. (2016). “St. Jean Cathedral in Lyon Vieux.” 〈http://www.therosewindow.com/TheRoseWindow2/lyon-cath-Ww.htm〉 (Oct. 13, 2016).
Maple 2016 academic edition [Computer software]. Maplesoft, Cybernet Systems Co., Ltd., Tokyo.
Meaux Cathedral. (2016). “St. Ethiene Cathedral, Meaux, France.” 〈https://www.flickr.com/photos/112709184@N02/18706659260〉 (Oct. 13, 2016).
O'Connor, J. J., and Robertson, E. F. (1980). “Menelaus of Alexandria.” History of Mathematics Archive, MacTutor, Univ. of St. Andrews, St. Andrews, U.K.
Panofsky, E. (2005). The life and art of Albrecht Du¨rer, Princeton University Press, Princeton, NJ.
Roman Mosaic. (2016). “Roman mosaic in Museum of Zaragoza (Spain).” 〈http://c7.alamy.com/comp/C4XE38/roman-mosaic-in-museum-zaragoza-aragon-spain-C4XE38.jpg〉 (Oct. 13, 2016).
Roman Mosaic of Medusa. (2016). 〈http://cdn.shopify.com/s/files/1/0130/8502/products/RMN-04_4.jpg?v=1466449314〉 (Oct. 13, 2016).
Rosin, P. L. (2001). “Rosettes and other arrangements of circles.” Nexus Network J., 3(2), 113–126.
Solomon Cup. (590–628). “Khosrow II Parviz Cup, King of Persia, 590–628, known as Solomon Cup, National Library of France, Room of Medals, Paris, France.” 〈http://www.segre.com/es/noticias/cultura/2016/11/06/la_seu_vella_escogida_para_participar_un_concurso_25_catedrales_para_escoger_mejor_espana_5456_1112.html〉 (Oct. 13, 2016).
Sutton, A. (2009). Ruler and compass, Walker, New York.
Triana Bridge. (2016). “Triana bridge, Seville, Spain.” 〈https://farm8.staticflickr.com/7175/13558522673_b7a645b3e0_o.jpg〉 (Oct. 13, 2016).
Ver Eecke, P. (1933). Pappus d’Alexandrie: La collection mathématique avec une introduction et des notes, Fondation Univ. de Belgique, Albert Blanchard, Paris.
Vernet, J., and Catalá, A. (1968). “Arquímedes árabe: El tratado de los círculos tangentes.” Al-Andalus, 33:1 (in Spanish).
Wantzel, P. L. (1837). “Recherches sur les moyens de reconnai^tre si un problème de Géométrie peut se résoudre avec la règle et le compas.” J. Mathématiques Pures Appliquées, 1(2), 366–372.
Williams, K. (1998). Italian pavements: Patterns in space, Anchorage, Houston.
Williams, K. (1999). “Spirals and rosettes in architectural ornament.” Nexus Network J., 1(1), 129–138.
Information & Authors
Information
Published In
Copyright
© 2017 American Society of Civil Engineers.
History
Received: Feb 10, 2016
Accepted: Sep 12, 2016
Published online: Jan 19, 2017
Published in print: Jun 1, 2017
Discussion open until: Jun 19, 2017
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.