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Geometric and Aesthetic Concepts Based on Pentagonal Structures

  • Cornelie LeopoldEmail author
Living reference work entry

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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

The foundation of art can be seen in configurations of elements on the image plane or in space to create sculptures, in which elements are brought in relation to each other. Mathematics as a structural science was developed in the 1960s by the Bourbaki project (Bourbaki 1948): This provides a new foundation of art. Mathematical structures of order can especially allow to explore and describe possible geometric arrangements and configurations of artistic elements. By expressing such geometrically ideal structures of order in perceptible materialized configurations, these mere thinking categories become perceptible. We can refer to the fundamental work of Kant (1783) in which forms of intuition, space, and time are necessary conditions for all sensual experiences, and therefore also aesthetics. Kant understood the notion of aesthetics in its original Greek meaning “Aisthesis,” as the theory of the sensual perceptions. After further development, aesthetics has been defined as the theory of beauty or art, though the sensual perceptibility of an idea remains essential to this modern definition of aesthetics. In his “Vorlesungen über die Aesthetik,” Hegel described the task of art as:

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)

According to these characteristics of art proposed by Hegel, the task of art can be said to make an idea perceptible to our senses. But how can an idea be captured and considered in the creation process? The philosopher Max Bense developed a new definition of aesthetics (Bense 1965) starting with Hegel’s description of art. In that definition, the aesthetic state of an object is related to distributions of elements or schemata of order in the meaning of arrangements. Elisabeth Walther described the role of this new definition of aesthetics:

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.

With the help of the mighty tool of dualization, the center of each polygon as a vertex can be joined to the centers of adjacent polygons in order to form dual tessellations (Weisstein). The number of lines in one point (vertex) produces the number of vertexes in the resulting polygon (Fig. 1).
Fig. 1

Regular and semiregular tessellations and their duals (http://mathworld.wolfram.com/DualTessellation.html)

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.

Mathematical research on pentagonal tiling later continued. Casey Mann, Jennifer McLoud, and David Von Derau discovered a 15th monohedral tiling convex pentagon in 2015 (Pöppe 2015). In July 2017, Michaël Rao (2017) completed a computer-assisted proof showing that there are no other types of convex pentagons that can tile the plane (Fig. 2).
Fig. 2

The 15 possible monohedral tilings by convex pentagons (https://commons.wikimedia.org/wiki/File:PentagonTilings15.svg)

The artist Caris started with an intuitively perceived irregular pentagon in his works “Birth of Forms” (Fig. 3, left) and “Creation of the Pentagon.” For Caris, these works answered the question: “How to imagine something from nothing?” These spontaneous compositions were his keyworks. Pentagons are not obvious in these works, but they can be derived by connecting their vertices, extending and adding lines. The configuration in Fig. 3 is reminiscent of an overhand knot created with using a paper strip (Fig. 3, right). This astonishing fact shows the self-evident occurrence of the pentagon, which was not considered at that time by the artist.
Fig. 3

Gerard Caris, Birth of Forms, 1968. Knot construction of a regular pentagon, 2017. (© VG Bild-Kunst, Bonn 2018)

After some explorations into tiling the plane with variations of irregular pentagons and hexagons (Fig. 4), Caris was fascinated/captivated by the systematic consequences of the universe found in the regular pentagon. Therefore, his art explorations were later guided by the universe of the regular pentagon, or “Pentagonism.”
Fig. 4

Gerard Caris, View of the universe 1, 1969. Cosmic motion series, 1971. (© VG Bild-Kunst, Bonn 2018)

Tiling with Regular Pentagons

Early attempts to tile the plane with regular pentagons were unsuccessful. Rhomboidal gaps nevertheless remained between pentagons and could be placed at various positions. Figure 5 shows three examples of Caris’ work and the aesthetic results he discovered from those variations. The use of bright-dark colors supports spatial interpretations of the works, which he continues in his later reliefs.
Fig. 5

Gerard Caris, Structure 1C and 2C, 1974, and Structure 6 C 2, 1975. (© VG Bild-Kunst, Bonn 2018)

Such studies on tiling with regular pentagons have been conducted by Albrecht Dürer (1525, pp. 66–69) and Johannes Kepler (1619, p. 77), as shown in Fig. 6. It is interesting to see that Kepler had tried to achieve a subdivision of the pentagon, though Caris is able to more thoroughly explore the complex structures that arise from the pentagonal tiling in their aesthetic variations.
Fig. 6

Configurations of regular pentagons by Dürer (left) and Kepler (right)

In recent times, mathematical studies on pentagon packings in order to find the closest pentagon packing have been conducted by Greg Kuperberg and Włodzimierz Kuperberg (1990) based on double-lattice packings (Fig. 7).
Fig. 7

Closest pentagon packing with double-lattice packings, according to Kuperberg

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

Pentagrid as Art Repertoire

Explorations in tiling with regular pentagons led Caris to develop the Pentagrid (Fig. 8), which became the structural basis for his art works in the plane. It contains manifold relationships and configurations from the pentagon and forms and is thus the repertoire for all potentially derivable art works.
Fig. 8

Gerard Caris, Pentagrid, 1994. (© VG Bild-Kunst, Bonn 2018)

The Pentagrid reflects the pentagonal system. Caris describes the Pentagrid as a grid with five degrees of freedom. The five lines in one node form its basic structure (Fig. 9, left). The regular pentagon has a side-to-diagonal relationship that follows the golden section; therefore, the grid has angles of 36° and 72°, and their sum, 108°, is the magnitude of the interior angle of the pentagon. The two possible golden triangles are part of the pentagon. An infinite continuation of subdivision according to the golden section would result in fractal structures of the Pentagrid. In this way, the Pentagrid explains the structures depicted in Fig. 9: the pentagram, golden triangles, and two kind of rhombuses (one of them is divided by the grid lines in kite and dart figures) (Fig. 9, right).
Fig. 9

Five-dimensional grid of the pentagon and figures in the Pentagrid

Pentagonal structures in the Pentagrid are the basis for various painting series by Caris such as Structure, PC (Pentagon Complex), and ETX (Eutactic Star Series). During a visit to his atelier, it was interesting to see that he used the Pentagrid on his drawing table during the creation process. Figure 10 shows some diverse examples from Caris’ creation period. Possible spatial interpretations become increasingly more obvious.
Fig. 10

Gerard Caris, PC 26, 1995. ETX 21, 1998. PC 34, 2016. (© VG Bild-Kunst, Bonn 2018)

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.

Patterns have been especially developed in the tradition of Islamic ornaments. In the ninth century, references to patterns based on the golden section had already been made, in addition to root two and three proportional patterns. Figure 11 illustrates how an Islamic ornament can be constructed from the regular pentagon according the elementary studies of El-Said and Parman (1976). By overlaying two pentagons that have been rotated by 180°, the decagon is created, and the ornament is further developed (Ghyka 1977, p. 34). Two works of Caris (Fig. 11, right) related to these ornaments were also developed out of the Pentagrid.
Fig. 11

Constructing Islamic ornament from pentagon. Gerard Caris, ETX 36, 1999. ETX 145, 2014. (© VG Bild-Kunst, Bonn 2018)

From the Pentagrid to the Kite-Dart-Grid

Kites and darts are present within the two rhombuses that form the structure of the Pentagrid (see Fig. 9, right). They form the basis for the quasiperiodic tilings, also known as Penrose tiling, found by Robert Ammann and Roger Penrose in the 1970s. In 1971, Caris showed that kites and darts are part of the Pentagrid, and he derived the Kite-Dart-Grid from it (Fig. 12).
Fig. 12

Gerard Caris, Pentagrid with kite and dart, 1971. Kite-Dart-Grid, 2015. (© VG Bild-Kunst, Bonn 2018)

There are seven ways to arrange kites and darts around a node. Conway and Lagarias (1990) called them the Star, Ace, Sun, King, Jack, Queen and Deuce (Fig. 13). Each kite and dart is composed of two golden triangles, also called Robinson triangles (Grünbaum and Shephard 1987), which are marked in Fig. 13.
Fig. 13

The seven possible arrangements of kites and darts around a node (https://commons.wikimedia.org/wiki/File:Penrose_vertex_figures.svg)

Caris worked extensively on the arrangements of kites and darts, especially in his Kites and Dart series 2015–2017 (Fig. 14). In these works, Caris creates figurative works from the arrangement of the kites and darts, symmetrical or asymmetrical, and reveals the golden triangles by coloring them. In a new work of 2017, he returns to the original figure, the regular pentagon, with the help of the kite and dart configurations (Fig. 14, right). In this way, the circle of his substantial pentagon studies closes again.
Fig. 14

Gerard Caris, Kites and Darts series #26, 2015. #29, 2015. #98, 2017. (© VG Bild-Kunst, Bonn 2018)

Spatial Structures with Dodecahedra

During his working process, Caris created spatial pentagonal structures in parallel to those in plane. Twelve regular pentagons form the dodecahedron, which is one of the five Platonic solids. The Platonic solids are built using congruent regular polygons so that the same number of polygons abut each vertex. This condition leads to five regular polyhedra. Plato mentioned these polyhedra in his dialogue Timaeus and assigned each a cosmological meaning: cube (hexahedron) – earth, tetrahedron – fire, octahedron – air, icosahedron – water, and dodecahedron – universe. About the dodecahedron, Plato remarked, “There was yet a fifth combination which God used in the delineation of the universe” (Plato 360 BC, Timaeus 55c). Kepler illustrated the Platonic solids with these assignments in his Harmonices Mundi (Kepler 1619, p. 80) (Fig. 15).
Fig. 15

Platonic solids with cosmic assignments by Kepler

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

In searching for regular tessellations of space via tessellations in the plane, regular tessellations of space were found only to be possible for cubes. This may be the reason for the dominance of the orthogonal structures in the built environment. Caris looked to pentagonal structures as a new system of order to establish a pentagonal universe of art in the hopes of escaping overwhelming orthogonality. He started to arrange spatial configurations with dodecahedra, for example, by extending the edges of the dodecahedron. The result was a spatial dodecahedra grid. These attempts, along with other criteria for the distances, led to sculptures of Caris like Monumental Polyhedral Net Structure (Fig. 16). He also experimented with spatial packings of dodecahedra in accordance with the tessellation attempts of pentagons in plane. The resulting sculptures and reliefs (Fig. 17) show various gaps between the dodecahedra dependent of the kind of packing, or configurations, with dodecahedra.
Fig. 16

Dodecahedron grid and Gerard Caris, Monumental Polyhedral Net Structure, 1977. Polyhedral Net Structure #1, 1971. (© VG Bild-Kunst, Bonn 2018)

Fig. 17

Gerard Caris, Reliefstructure 1K-1 Detail, 1988. Reliefstructure 1E-2 Detail, 1985. Helix 2–2 branching, 2002. (© VG Bild-Kunst, Bonn 2018)

Euclid focused in his Elements, Book XIII, Proposition 17 (Euclid 300 BC) on construction possibilities packing dodecahedra. Because packing is possible with cubes, the dodecahedron can be truncated so that a cube is left, or the hipped roofs on the cube faces can make the dodecahedron arise. Kepler illustrated the idea (Fig. 18, left) in his Harmonices mundi (Kepler 1619, p. 181). With this idea, a concave solid can be constructed by taking a cube and putting the six, hipped roofs inwards. This concave solid can be used to fill the gaps between the dodecahedra for a space filling arrangement. Caris used this idea for series of reliefstructures (Fig. 19).
Fig. 18

Truncating a dodecahedron or putting hipped roofs on the faces; a concave solid is created by turning them inside

Fig. 19

Gerard Caris, Reliefstructure 1D-3, 1985. Reliefstructure 2M-1, 1989. Reliefstructure 1O-2, 2002. (© VG Bild-Kunst, Bonn 2018)

Besides these more geometric sculptures and reliefs, Caris strove for applications of these spatial packings for housing designs. Plates with conceptional drawings by Caris illustrate possible housing creations with the appropriate truncated dodecahedra. The 12-sided dodecahedron and truncation results in the 14-sided tetracaidecahedron shown in Fig. 20.
Fig. 20

Gerard Caris, Tetracaidecahedron, 1975. (© VG Bild-Kunst, Bonn 2018)

These conceptional ideas led him to experiments with models of such housing or sculptural designs (Fig. 21).
Fig. 21

Gerard Caris, Model D House, 1985. Polyhedra sculpture 3 (truncated), 1979. (© VG Bild-Kunst, Bonn 2018)

There are few architects which have used dodecahedra for housing structures. In the 1970s, Zvi Hecker created a modular structure composed of dodecahedra in his Ramot Polin housing project in Israel. The structures were constructed with prefabricated pentagonal concrete panels. Hecker was commissioned by the Israeli Ministry of Housing to plan a complex of 720 housing units, northwest of Jerusalem. The project was also based on the truncated dodecahedron shown in Fig. 18. The concept is described by Zvi Hecker (Figs. 22 and 23):

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).

Fig. 22

Zvi Hecker, Ramot Polin housing project, 1971 ff. (© Zvi Hecker)

Fig. 23

Zvi Hecker, Dense space packing of cubes inscribed into dodecahedrons. Assembly System of the pre-fabricated elements. (© Zvi Hecker)

Topological interlocking of convex regular polyhedra has been a subject of recent geometric and experimental research in structural design (Viana 2018a). Topological interlocking is defined as: “Elements (blocks) of special shape [arranged] in such a way that the whole structure can be held together by a global peripheral constraint, while locally the elements are kept in place by kinematic constraints” (Estrin et al. 2011). The topological interlocking of dodecahedra was found to differ in their relation to a tessellation at a half-section of the dodecahedron. The first possibility is deducible from the regular tessellation outlined by their hexagonal cross-sections following Dyskin et al. (2003) (Fig. 24, left). The second is based on a regular decagonal cross-section, also devised by Dyskin et al. (2003; Kanel-Belov et al. 2008) (Fig. 24, middle), and the third, described by Viana (2018b, p. 257), refers to the half-sections of dodecahedra as regular hexagons that tesselate the plane together with equilateral triangles (Fig. 24, right).
Fig. 24

Three types of topological interlocking of dodecahedra, TI 1, TI 2, TI 3. (© Vera Viana)

It seems that Caris also worked on those arrangements of dodecahedra in reliefstructures. The shown reliefstructures in Fig. 25 are similar to TI 2 and TI~3.
Fig. 25

Gerard Caris, Reliefstructure 1I-1, 1986. Reliefstructure 1E-2, 1985. Reliefstructure 1R-1, 1993. (© VG Bild-Kunst, Bonn 2018)

Spatial Structures with Rhombohedra: Golden Diamonds

The Kite-Dart-Gridthat developed out of the Pentagrid led to the Penrose-tiling. The spatial counterpart can be found in the rhombohedra forming the so-called golden diamonds (Miyazaki 1986). There are two rhombohedra necessary for spatial tessellation that corresponds to the types of rhombuses in the plane (see Figs. 9 and 12). They play an important role for aperiodic space fillings in quasi-crystals (van de Craats 2007). In the rhombohedra, the diagonals cut each other in a golden ratio. The rhombohedra can be developed out of the axes of the dodecahedron. The six possible fivefold rotational axes in the dodecahedron, through the faces and the center of the dodecahedron, build the spatial structural grid, or eutactic star (Fig. 26, left). Choosing three of the six axes generates the two different rhombohedra with vertices along these axes. Each rhombohedron consists of six congruent rhombi as faces. Two of each different rhombohedra together form a rhombic dodecahedron (Groß 2007), which can grow into larger rhombic polyhedral spatial structures (Fig. 26, right).
Fig. 26

Axes of the dodecahedron – eutactic star. Two types rhombohedra and compound rhombic polyhedral solids

These axes of the dodecahedron enable the dodecahedron to be used as a universal node for the node-edge-model of a related lattice structure (Fig. 27). A hole is placed in the middle of each face of the dodecahedron into which rods can be stuck. This forms a structural node system (Groß 2007). In this way, the relationship between dodecahedra and rhombohedra again becomes apparent.
Fig. 27

Dodecahedral node for the rhombohedral structural node system

These spatial rhombohedra structures are applied by Caris in various reliefstructures and sculptures (Fig. 28).
Fig. 28

Gerard Caris, Reliefstructure 13V-1, 2003. Sculpture 2X-1, 1996. Rhombohedra sculpture # III, 2017. (© VG Bild-Kunst, Bonn 2018)

Geometry and Art: Reflections on Aesthetics

Are these examples by Caris just geometric models or artworks? This question leads to reflections about the relationship between geometry and art as well as aesthetic categories. As mentioned in the introduction, Paul Valéry compared the role of ornamental drawing in art to the role of mathematics in the sciences. Geometry is the groundwork for ornaments, and the found geometric structures drive explorations in art. However, art cannot be equated to geometry. Artwork gives the chance for aesthetically perceptible, materialized geometric structures. Max Bill (1949) explained in “The Mathematical Way of Thinking in the Visual Art of Our Time” that geometry is the primary element of every artwork, or the relationship of the components on the surface or in space. Nevertheless, geometry and art are not identical.

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)

Bill studied the relationship between structures and art. In his opinion, rhythmical order is the creative act of the artist producing an artwork, starting with a general structure.

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).

There are two roots of this new aesthetic theory: information and aesthetic measures. Information as a root was introduced by Claude E. Shannon (1948) during the rise of communication theory and communication technology. His mathematical information model integrated the stochastic nature of news. The possible states of a system can be described in combination with a set of transition probabilities going from one state to the next. Bense applied Shannon’s information theory to aesthetics. Successive emergence of structures is achieved by stochastic selections of unstructured material as concrete, perceivable realizations. Aesthetic realizations are seen as part of a communication process. The second root is Georg David Birkhoff’s aesthetic measure (Birkhoff 1933). He defined the aesthetic measure as the function of the order and complexity grade of the viewed configuration: M = O/C, where O is the measure number of order relations, symmetries, and harmonies and C is complexity. Birkhoff described an aesthetic experience as follows:

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.

Birkhoff’s aesthetic measure M is interpreted by Bense as aesthetic information. The order relations O correspond to redundancy in finding order relationships. Redundant features are necessary for innovations to become recognizable:

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.

Caris’ preference for the pentagon is not due to its form, nor is the form of the pentagon declared an aesthetic object. The preference is rather due to the manifold geometric structures derived from the pentagon. According Bense (1965, p. 43), the geometric form, here the geometric element pentagon, is not identical to the aesthetic element; only in its materialization in a composition with the individual decisions of the artist does it become the aesthetic element, which is seen as the smallest aesthetic unit and aesthetic structure. Caris refers in his art to the geometry of pentagonal structures in plane as well as in space. He uses the geometric structures to experience them in aesthetic compositional processes. In this way, he explores many geometric relationships in aesthetic expressions, without reducing the art to these geometric order structures. But structural thinking turns out as an adequate method to consolidate design and creation processes (Leopold 2012). Bense used later the term “generative aesthetics”; he defines this as “the summary of all operations, rules, and theorems, through whose application to a set of material elements that are able to function as signs, can deliberately and methodically generate in these aesthetic states (distributions and/or configurations)” (Bense 1965, p. 333, translated by the author). He compared generative aesthetics to the generative grammar for words. These attempts led to first computer generated artworks and texts in the 1960s by students of Bense like Nake. Although the works of Caris are not computer generated, we can interpret the investigated geometric-mathematical structures, as they are expressed in the Pentagrid, as the underlying generative aesthetics that he calls also “principle structuring element” or “leading principle.” Caris explained the role of these generative mathematical principles in the creative process for his artworks with high importance:

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.

Caris’ “Pentagonism” should be understood as a radical structural fundament for art and design. With this claim, he works out all consequences for an alternative to orthogonal structures, showing a more complex, fascinating, and perceptible structural system. Exploring the pentagonal and dodecahedral structures through his artworks leads him again back to the fundamental starting element: the pentagon. He recently created in a new work, through the radiation of lines forming the pentagon or in a sculptural work by polyhedral net structures, a pentagon by arranging of dodecahedra (Fig. 29). Aesthetics and mathematics merge together as one for him:

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)

Fig. 29

Gerard Caris, Pentagonal Radiation # 2, 2017. Polyhedral Net Structure #2, 1972. (© VG Bild-Kunst, Bonn 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.

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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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