# Geometric and Aesthetic Concepts Based on Pentagonal Structures

## Abstract

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.

## Keywords

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

to present the idea for the immediate intuition in a sensual form and not in the form of thinking and pure spirituality in general. (Hegel 1835)

Aesthetics, as Bense brings it into play, is the principle of order par excellence. Aesthetics is order, and order on the other hand is describable by mathematics. Therefore, aesthetics is important as structuring the world for techniques as well as architecture, literature, etc., for all what will be created. Whenever we take something out of the chaos of existing and assemble it new, we need an aesthetic foundation. (Walther 2004)

Such principles of order as an aesthetic foundation of art can be found in the geometry of tessellations, patterns, and its spatial variants. This corresponds to the statement of Paul Valéry (1895) that patterns and ornaments are fundamental to art. He compared the role of the ornamental drawing in art with the role of mathematics in the sciences. This fundamental role can be also confirmed by the appearance of geometric patterns in various traditions of ethnic handicrafts found in transcultural studies.

By studying the art of Gerard Caris (Leopold 2016, 2018) in its developments, it is possible to follow the process from an early stage of geometrically ordered structures to the later stage of aesthetic expressions. His concentration on the regular pentagon and its spatial counterpart, the dodecahedron, allows him to explore manifold pentagonal structures in plane as well as in space. Caris calls his art “Pentagonism” (Jansen and Weibel 2007). The suffix “-ism” denotes a condition or system; thus, the word describes his art as a system based on the pentagon. The following sections discuss this pentagonal system, its geometric foundations, and the received art creations stimulated by those structural considerations.

## Tessellations and Their Dualizations

There are three possible tessellations that can be formed with regular polygons: triangle, square, and hexagon. These three regular tessellations, or Platonian tessellations, are followed by eight semiregular or Archimedean tessellations. Archimedean tessellations are characterized by two or more convex regular polygons for which the same polygons in the same order surround each polygonal vertex.

The result of the dualization of a semiregular tessellation is a dual tessellation with one type of nonregular polygon instead of regular polygons. One example is the so-called “Cairo Tiling” (MacMahon 1921), which is the result of the dualization of the semiregular tiling of three regular triangles and two squares. This is characterized by 3-3-4-3-4, or the sequence of triangles and squares in each vertex.

## Tiling with Regular Pentagons

Caris’ artistic style is mainly interested in the pentagon as a systematic fundament for his creations.

## Pentagrid as Art Repertoire

In the ETX, Caris refers to the concept of vectors forming a star, which is called well-arranged, or eutactic, if the orthogonal projection from a higher-dimensional space is placed onto a subspace (Coxeter 1951). In this case, the projection moves from the three-dimensional space into the two-dimensional space. This background can explain spatial interpretations of two-dimensional works.

## From the Pentagrid to the Kite-Dart-Grid

## Spatial Structures with Dodecahedra

Finally, Euclid proved that there were only these five regular convex polyhedral. He described the properties of the Platonic solids and their construction.

Most researchers-architects-students focus on the polyhedral (dodecahedral) elevations of the buildings, but actually the geometric strategy of the project is much more complex. It is related to the form of the site and the phenomena of the Golden Section manifested in the arrangement of seeds in sunflowers and in the geometry of the pentagon. The general plan of the Ramot Housing is reminiscent of the palm of an open hand – its five fingers are retaining walls ‘supporting’ the slope of the hill. Each ‘finger’ is composed again of five boomerang–like buildings assembled in such a way as to create interior courtyards, kind of pedestrian paths reminiscent of the Old City of Jerusalem. The buildings’ inclination towards the interior of the courtyards provides a protected shadowy exterior. The geometry of the buildings follows a stereometric dense space packing of cubes inscribed into dodecahedrons (as dodecahedrons do not pack densely). In this project the elevation pentagons are the smallest in scale elements based on number five. (Hecker 2018).

## Spatial Structures with Rhombohedra: Golden Diamonds

## Geometry and Art: Reflections on Aesthetics

Now in every work of art the basis of its composition is geometry or in other words the means of determining mutual relationship of its component parts either on plane or in space. (…) It must not be supposed that an art based on the principles of mathematics, such as I have just adumbrated, is in a sense the same thing as a plastic or pictorial interpretation of the latter. (Bill 1949, pp. 7–8)

Let us start with the extreme case: a plane is covered with a uniform distribution in the sense in which this is understood in statistics; or a uniform network extends into space. This is an order which could be uniformly extended without end. Such an order we here call a structure. In a work of art, however, this structure has its limits, either in space or on the plane. Here we have the basis for an aesthetic argument in the sense that a choice has to be made: the possible, aesthetically feasible extension of the structure. Actually, it is only through this choice to limit the arbitrarily extensible structure on the basis of verifiable arguments that a discernible principle of order becomes comprehensible. (…) This means that art can originate only when and because individual expression and personal invention subsume themselves under the principle of order of the structure and derive from it a new lawfulness and new formal possibilities. (…) Such lawfulness and such inventions manifest themselves as rhythm in an individual case. Rhythm transforms the structure into form; i.e. the special form of a work of art grows out of the general structure by means of a rhythmic order. (Bill 1965)

How can aesthetics be substantiated in relation to order structures, rhythm, and as Bill characterized it, in the individual creative decisions that constitute the difference between art and mathematics? Some fundamental assumptions from Information Aesthetics can give criteria for aesthetic measures and evaluations. Information Aesthetics was developed in the 1960s, mainly in Germany and France, as an aesthetic theory on a rational mathematical fundament. The theory was developed by Max Bense (1965), a professor at the University of Stuttgart, and Abraham A. Moles (1966), a professor at the Université de Strasbourg. Today, the term “information aesthetics” is used with a different meaning: the display of huge quantities of data (Nake 2012). Frieder Nake, a protagonist of Information Aesthetics himself, profoundly summarized the theory, its applications, and its critics in his article. Information is seen as the key concept to understand aesthetic processes, and its aim was to create an opposing position to the classical tendencies of aesthetic theory using formalizations (Gianetti 2004).

The typical aesthetic experience may be regarded as compounded of three successive phases: (1) a preliminary effort of attention, which is necessary for the act of perception, and which increases in proportion to what we shall call the complexity (C) of the object; (2) the feeling of value or aesthetic measure (M) which rewards this effort; and finally (3) a realization that the object is characterized by a certain harmony, symmetry, or order (O), more or less concealed, which seems necessary to the aesthetic effect. (Birkhoff 1933, p. 3)

The aesthetic measure of an artwork could be calculated out of order and complexity as numeric quantities. Bense supplemented Birkhoff’s numeric aesthetics with Information Aesthetics. Bense defined the aesthetic state as the relation of an ordered to a not-ordered state. The artwork gives aesthetic information; it is a material carrier of the aesthetic state. Information is always transmitted by signs. Therefore, information theory is based on semiotics, the science of signs for information transmission. Moles (1966) explained the difference between semantic and aesthetic information. Semantic information refers to the meaning of what appears in the message, which is dependent of conventional signs. On the other hand, aesthetic information is how it appears, the way it is expressed, and thus bound to individual signs. The artworks of Caris show us this difference very clearly; they give us aesthetic information.

A perfect innovation in which there were only new states as in chaos, would not be recognizable. A chaos is finally unidentifiable. The recognizability of an aesthetic state requires not only the recognizability of its singular innovation, but also their identifiability based on their redundant order characteristics. (Bense 1965, p. 356, translated from German by the author)

Therefore, the interplay of redundancy and innovation – order and chaos – have to be in an optimal relation to achieve an aesthetic state. Although the goal of information aesthetics to offer totally objective methods for evaluating aesthetic objects has not been reached, these categories of redundancy and innovation remain important for an aesthetic evaluation. The recent developments in art and architecture show more and more complex, asymmetric, and chaotic trends with the tendency to arbitrariness. Often it is even expressed that the artist or architect wants to create complex works. Unfortunately, these aesthetic considerations make obvious that then the perceptibility is no longer reachable, and an aesthetic state cannot be established. Only with the help of redundant order features, which can be achieved by geometric order structures, perceptibility can be reached. More details on the aesthetic theory and the relationship to geometry have been explained by the author in “Prolegomena zu einer geometrischen Ästhetik” (Leopold 2011).

By applying these considerations to the art of Caris, we are able to determine that aesthetic states are identifiable by their geometric order, without identifying geometry with art. His presented artwork transmits the pentagonal system by compositions of material elements, in the form of aesthetic information, the way it is expressed in paintings and sculptures as well as individual selections from the repertoire as it is formalized in the Pentragrid.

The grid and its three-dimensional version, known as a lattice, is considered as the principle structuring element of reality as well as a leading principle in my personal exploration of art in which an entirely new form of development, self-coined as ‘Pentagonism’, has come into existence. This reveals creations never envisioned before in art, in which an interlinking of art and mathematics becomes self-evident for me and everyone else to see, as well as bringing about an aesthetical appreciation dependent on the prior package of conditioned aspects of the individual.

Geometric as well as non-algorithmic elements involved in the creation of these new works evoke a sense of Unity in the viewing process in which aesthetics and mathematical logic merge together as one. (Statement of Gerard Caris, 29 September 2018)

## Conclusion

The pentagon evokes its own manifold structures in plane and in space. Compared to the mostly common orthogonal system that is based on the square, the pentagonal system reveals many unique geometric relationships. Through the artworks presented, we can follow the development of the pentagonal system and the related creations of art. The derived geometric structures can serve as fundamental order characteristics in the described meaning of redundancy according to Information Aesthetics. The work of the artist Gerard Caris shows in an extraordinary way how the pentagonal and dodecahedral structures can be a repertoire for so many different art expressions in drawings, paintings, and sculptures, each with its aesthetic information. It is apparent that neither geometry nor mathematics can be directly identified with art, but aesthetics and mathematical logic merge together for Caris. The analysis of pentagonal and dodecahedral structures goes back to the early history of geometry (i.e., Plato, Euclid, Dürer, and Kepler), but still today new relationships have been found in recent research. Monohedral tiling with convex pentagons or topological interlocking of dodecahedra, for example, is a new research topic connected to advanced digital possibilities.

Caris’ art expresses the stimulating role of geometric structures in the creation processes. The viewer of his works can take part in his structural thinking processes, which are visualized in methodological tools like the Pentagrid, for example. The manifold surprising and fascinating patterns as well as the spatial configurations in Caris’ art juxtapose innovation and redundancy, a balance that guarantees perceptible aesthetics.

## Cross-References

## Notes

### Acknowledgements

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 paper.

## References

- Bense M (1965) Aesthetica. Einführung in die neue Ästhetik. Agis, Baden-Baden, 2nd expanded edn 1982Google Scholar
- 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
- 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
- Birkhoff GD (1933) Aesthetic measure. Harvard University Press, CambridgeCrossRefGoogle Scholar
- Bourbaki N (1948) Éléments de mathématique. Paris 1939 ff, L’Architecture des MathématiquesGoogle Scholar
- Caris G (2018) Pentagonism. http://www.gerardcaris.com. Accessed 10 Oct 2018
- Conway JH, Lagarias JC (1990) Tiling with polyominoes and combinatorial group theory. J Combin Theory Ser A 53:183–208. Figures available https://commons.wikimedia.org/wiki/File:Penrose_vertex_figures.svg. Accessed 10 Oct 2018MathSciNetCrossRefGoogle Scholar
- Coxeter HSM (1951) Extreme forms. Can J Math 3:391–441. https://doi.org/10.4153/CJM-1951-045-8MathSciNetCrossRefzbMATHGoogle Scholar
- Dürer A (1525) Underweysung der Messung, mit dem Zirckel und Richtscheyt, in Linien, Ebenen und gantzen corporen. Nürnberg, p 66–69. Online Edition: digital.slub-dresden.de/werkansicht/dlf/17139. Accessed 10 Oct 2018Google Scholar
- 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
- El-Said I, Parman A (1976) Geometric concepts in Islamic art. World of Islam Festival Publishing Company Ltd, London, p 82ffGoogle Scholar
- Estrin Y, Dyskin A, Pasternak E (2011) Topological interlocking as a material design concept. Mater Sci Eng C 31(6):1189–1194CrossRefGoogle Scholar
- Euclid (300 BC) Elements Book XIII. English Version by David E. Joyce, 1996. https://mathcs.clarku.edu/~djoyce/java/elements/bookXIII/bookXIII.html. Accessed 11 Oct 2018
- Ghyka MC (1977) The geometry of art and life, 2nd edn. Dover, New YorkzbMATHGoogle Scholar
- Gianetti C (2004) Cybernetic aesthetics and communication. Media Art Net. http://www.medienkunstnetz.de/themes/aesthetics_of_the_digital/cybernetic_aesthetics. Accessed 14 Nov 2018
- Groß D (2007) Planet “Goldener Diamant”. In: Leopold C (ed) Geometrische Strukturen. Technische Universität Kaiserslautern, Kaiserslautern, pp 28–33Google Scholar
- Grünbaum B, Shephard GC (1987) Tilings and patterns. W. H. Freeman, New York, pp 537–547zbMATHGoogle Scholar
- Hecker Z (2018.) http://www.zvihecker.com/projects/ramot_housing-113-1.html and corrections, sent by email. Accessed 27 Oct 2018
- Hegel GWF (1835) Vorlesungen über die Aesthetik. In: Hotho HG (ed) Duncker & Humblot, Berlin, p CXVIIIGoogle Scholar
- Jansen G, Weibel P (eds) (2007) Gerard Caris. Pentagonismus/Pentagonism. Walther König, KölnGoogle Scholar
- 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
- Kant I (1783) Prolegomena zu einer jeden künftigen Metaphysik, die als Wissenschaft wird auftreten können. Johann Friedrich Hartknoch, Riga. http://www.uni-potsdam.de/u/philosophie/texte/prolegom/!start.htm
- Kepler J (1619) Harmonices Mundi. Lincii Austriae, Linz. Online Edition https://archive.org/details/ioanniskepplerih00kepl. Accessed 10 Oct 2018Google Scholar
- Kuperberg G, Kuperberg W (1990) Double-lattice packings of convex bodies in the plane. J Discrete Comput Geom 5:389–397. https://doi.org/10.1007/BF02187800MathSciNetCrossRefzbMATHGoogle Scholar
- Leopold C (2011) Prolegomena zu einer geometrischen Ästhetik. In: Kürpig F (ed) Ästhetische Geometrie – Geometrische Ästhetik. Shaker, Aachen, pp 61–65Google Scholar
- 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
- 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
- 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. https://doi.org/10.1007/978-3-319-93949-0CrossRefzbMATHGoogle Scholar
- MacMahon MPA (1921) New mathematical pastimes. University Press, Cambridge, p 101zbMATHGoogle Scholar
- Miyazaki K (1986) An adventure in multidimensional space: the art and geometry of polygons, polyhedra, and polytopes. Wiley, New YorkzbMATHGoogle Scholar
- Moles AA (1966) Information theory and esthetic perception. Urbana, University of Illinois Press. French original 1958Google Scholar
- Nake F (2012) Information aesthetics: an heroic experiment. J Math Arts 6(2–3):65–75. https://doi.org/10.1080/17513472.2012.679458MathSciNetCrossRefGoogle Scholar
- Plato (360 BC) Timaeus. Translated by Jowett B. Online Edition https://www.ellopos.net/elpenor/physis/plato-timaeus. Accessed 12 Oct 2018
- Pöppe C (2015) Unordentliche Fünfeckspflasterungen. Spektrum der Wissenschaft 11/2015, pp 62–67. https://commons.wikimedia.org/wiki/File:PentagonTilings15.svg. Accessed 12 Oct 2018
- Rao M (2017) Exhaustive search of convex pentagons which tile the plane. Manuscript: 16, Bibcode: 2017arXiv170800274R. https://perso.ens-lyon.fr/michael.rao/publi/penta.pdf. Accessed 10 Oct 2018
- Shannon CE (1948) A mathematical theory of communications. Bell Tech J 27:379–423; 623–656MathSciNetCrossRefGoogle Scholar
- Valéry P (1895) Introduction à la méthode de Léonard de Vinci. La Nouvelle Revue Française, ParisGoogle Scholar
- 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
- Viana V (2018a) From solid to plane tessellations, and back. Nexus Netw J. 20:741–768 https://doi.org/10.1007/s00004-018-0389-5CrossRefGoogle Scholar
- 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
- 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
- Weisstein EW “Dual Tessellation”. From MathWorld – a Wolfram web resource. http://mathworld.wolfram.com/DualTessellation.html. According Williams R (1979) The geometrical foundation of natural structure: a source book of design. Dover, New York, p 37. Accessed 10 Oct 2018