Geometric and Aesthetic Concepts Based on Pentagonal Structures

  • Cornelie LeopoldEmail author
Living reference work entry

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The relationship between geometry and art will be examined using the example of pentagonal structures. The work of contemporary Dutch artist Gerard Caris is based on those pentagonal structures. He calls his art work Pentagonism and questions how art creations and design processes can rely on strong, geometric, structural thinking. Pentagonal structures in plane as well as in space will be analyzed from a geometrical point of view and compared to corresponding art approaches. A review of geometric research on tessellations will be followed by a discussion on previous attempts to tile the Pentagrid with regular pentagons. The fundamental role of the Pentagrid and derivable Kite-Dart-Grid in Caris’ art design processes will also be explained. A step into the three-dimensional space leads to the dodecahedron and derived rhombohedra configurations for tessellations, or packings, in space. The geometric background refers to fundamental works by Plato, Euclid, Dürer, and Kepler as well as recent research results. The investigation will end with a discussion of the aesthetic categories of redundancy and innovation, their application to art evaluation and the differentiation of geometry and art. The example of Caris’ art, which concentrates on the regular pentagon and the spatial counterpart dodecahedron, points out the possibilities of aesthetic expressions on the basis of geometric structures. Art enables the exploration of those structures in a playful and self-explanatory way and often precedes scientific research.


Geometry Tessellation Pentagon Pentagrid Kite-dart-grid Dodecahedron Packing Aesthetics Gerard Caris 



Many thanks to the artist Gerard Caris for the opportunity to visit him in his atelier, showing and explaining his work to me, and allowing me to get an inside view of his creation processes. The images of his works of art in the figures here are used with his kind permission, and they are managed and supported by VG Bild-Kunst, Bonn. I am grateful to Margriet Caris for helping me with all of my questions and requests.

Thank you to Zvi Hecker, who agreed to allow the use of his drawings and photos of Ramot Polin housing project to explain his design background.

Finally, many thanks to Vera Viana for her discussions on the relationship of recent topological interlocking research and Gerard Caris’ respective artworks, as well as for creating the drawings/renderings in Fig. 24 for this paper. Thank you also to Jasmine Segarra for proofreading this chapter.


  1. Bense M (1965) Aesthetica. Einführung in die neue Ästhetik. Agis, Baden-Baden, 2nd expanded edn 1982Google Scholar
  2. Bill M (1949) Die mathematische Denkweise in der Kunst unserer Zeit. Werk 36, 3, Winterthur. English version: The mathematical way of thinking in the visual art of our time. In: Emmer M (ed) (1993) The visual mind: art and mathematics. MIT Press, Cambridge, pp 5–9Google Scholar
  3. Bill M (1965) Structure as art? Art as structure? In: Kepes G (ed) Structure in art and in science. Braziller, New York, pp 150–151Google Scholar
  4. Birkhoff GD (1933) Aesthetic measure. Harvard University Press, CambridgeCrossRefGoogle Scholar
  5. Bourbaki N (1948) Éléments de mathématique. Paris 1939 ff, L’Architecture des MathématiquesGoogle Scholar
  6. Caris G (2018) Pentagonism. Accessed 10 Oct 2018
  7. Conway JH, Lagarias JC (1990) Tiling with polyominoes and combinatorial group theory. J Combin Theory Ser A 53:183–208. Figures available Accessed 10 Oct 2018MathSciNetCrossRefGoogle Scholar
  8. Coxeter HSM (1951) Extreme forms. Can J Math 3:391–441. Scholar
  9. Dürer A (1525) Underweysung der Messung, mit dem Zirckel und Richtscheyt, in Linien, Ebenen und gantzen corporen. Nürnberg, p 66–69. Online Edition: Accessed 10 Oct 2018Google Scholar
  10. Dyskin A, Estrin Y, Kanel-Belov A, Pasternak E (2003) Topological interlocking of platonic solids: a way to new materials and structures. Philos Mag Lett 83(3):197–203CrossRefGoogle Scholar
  11. El-Said I, Parman A (1976) Geometric concepts in Islamic art. World of Islam Festival Publishing Company Ltd, London, p 82ffGoogle Scholar
  12. Estrin Y, Dyskin A, Pasternak E (2011) Topological interlocking as a material design concept. Mater Sci Eng C 31(6):1189–1194CrossRefGoogle Scholar
  13. Euclid (300 BC) Elements Book XIII. English Version by David E. Joyce, 1996. Accessed 11 Oct 2018
  14. Ghyka MC (1977) The geometry of art and life, 2nd edn. Dover, New YorkzbMATHGoogle Scholar
  15. Gianetti C (2004) Cybernetic aesthetics and communication. Media Art Net. Accessed 14 Nov 2018
  16. Groß D (2007) Planet “Goldener Diamant”. In: Leopold C (ed) Geometrische Strukturen. Technische Universität Kaiserslautern, Kaiserslautern, pp 28–33Google Scholar
  17. Grünbaum B, Shephard GC (1987) Tilings and patterns. W. H. Freeman, New York, pp 537–547zbMATHGoogle Scholar
  18. Hecker Z (2018.) and corrections, sent by email. Accessed 27 Oct 2018
  19. Hegel GWF (1835) Vorlesungen über die Aesthetik. In: Hotho HG (ed) Duncker & Humblot, Berlin, p CXVIIIGoogle Scholar
  20. Jansen G, Weibel P (eds) (2007) Gerard Caris. Pentagonismus/Pentagonism. Walther König, KölnGoogle Scholar
  21. Kanel-Belov A, Dyskin A, Estrin Y, Pasternak E, Ivanov-Pogodaev I (2008) Interlocking of convex polyhedra: towards a geometric theory of fragmented solids. Mosc Math J 10(2):337–342. (ArXiv08125089 Math)MathSciNetCrossRefGoogle Scholar
  22. Kant I (1783) Prolegomena zu einer jeden künftigen Metaphysik, die als Wissenschaft wird auftreten können. Johann Friedrich Hartknoch, Riga.!start.htm
  23. Kepler J (1619) Harmonices Mundi. Lincii Austriae, Linz. Online Edition Accessed 10 Oct 2018Google Scholar
  24. Kuperberg G, Kuperberg W (1990) Double-lattice packings of convex bodies in the plane. J Discrete Comput Geom 5:389–397. Scholar
  25. Leopold C (2011) Prolegomena zu einer geometrischen Ästhetik. In: Kürpig F (ed) Ästhetische Geometrie – Geometrische Ästhetik. Shaker, Aachen, pp 61–65Google Scholar
  26. Leopold C (2012) Strukturelles Denken als Methode. In: Warmburg J, Leopold C (eds) Strukturelle Architektur. Zur Aktualität eines Denkens zwischen Technik und Ästhetik. Transcript, Bielefeld, pp 9–29Google Scholar
  27. Leopold C (2016) Geometry and aesthetics of pentagonal structures in the art of Gerard Caris. In: Torrence E et al (eds) Proceedings bridges Finland. Tessellations Publishing, Phoenix, pp 187–194Google Scholar
  28. Leopold C (2018) Pentagonal structures as impulse for art. In: Emmer M, Abate M (eds) Imagine Math 6. Between culture and mathematics. Springer International Publishing, Cham. Scholar
  29. MacMahon MPA (1921) New mathematical pastimes. University Press, Cambridge, p 101zbMATHGoogle Scholar
  30. Miyazaki K (1986) An adventure in multidimensional space: the art and geometry of polygons, polyhedra, and polytopes. Wiley, New YorkzbMATHGoogle Scholar
  31. Moles AA (1966) Information theory and esthetic perception. Urbana, University of Illinois Press. French original 1958Google Scholar
  32. Nake F (2012) Information aesthetics: an heroic experiment. J Math Arts 6(2–3):65–75. Scholar
  33. Plato (360 BC) Timaeus. Translated by Jowett B. Online Edition Accessed 12 Oct 2018
  34. Pöppe C (2015) Unordentliche Fünfeckspflasterungen. Spektrum der Wissenschaft 11/2015, pp 62–67. Accessed 12 Oct 2018
  35. Rao M (2017) Exhaustive search of convex pentagons which tile the plane. Manuscript: 16, Bibcode: 2017arXiv170800274R. Accessed 10 Oct 2018
  36. Shannon CE (1948) A mathematical theory of communications. Bell Tech J 27:379–423; 623–656MathSciNetCrossRefGoogle Scholar
  37. Valéry P (1895) Introduction à la méthode de Léonard de Vinci. La Nouvelle Revue Française, ParisGoogle Scholar
  38. van de Craats J (2007) Rhombohedra in the work of Gerard Caris. In: Jansen G, Weibel P (eds) Gerard Caris. Pentagonismus/Pentagonism. Walther König, Köln, pp 44–48Google Scholar
  39. Viana V (2018a) From solid to plane tessellations, and back. Nexus Netw J. 20:741–768
  40. Viana V (2018b) Topological interlocking of convex regular Polyhedra. In: Leopold C, Robeller C, Weber U (eds) RCA 2018. Research culture in architecture x international conference on cross-disciplinary collaboration. Conference book. Fatuk – Faculty of Architecture, Technische Universität Kaiserslautern, 2018, pp 254–257Google Scholar
  41. Walther E (2004) Philosoph in technischer Zeit – Stuttgarter Engagement. Interview mit Elisabeth Walther, Teil 2. In: Büscher B, von Herrmann H-G, Hoffmann C (eds) Ästhetik als Programm. Max Bense/Daten und Streuungen. Diaphanes, Berlin, pp 62–73, translated by Cornelie LeopoldGoogle Scholar
  42. Weisstein EW “Dual Tessellation”. From MathWorld – a Wolfram web resource. According Williams R (1979) The geometrical foundation of natural structure: a source book of design. Dover, New York, p 37. Accessed 10 Oct 2018

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.FATUK – Faculty of ArchitectureTUK KaiserslauternKaiserslauternGermany

Section editors and affiliations

  • Bharath Sriraman
    • 1
  • Kyeong-Hwa Lee
    • 2
  1. 1.Department of Mathematical SciencesThe University of MontanaMissoulaUSA
  2. 2.Department of Mathematics Education, College of EducationSeoul National UniversitySeoulSouth Korea

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