Math and Art in Sculptures

Math and Art in Sculptures The objective of this work is to examine the connection between abstract sculpture and abstract mathematics and to investigate the connection between mathematics and art. As well this work will examine the artist sculptors George W. Hart and Charles O. Perry and discuss their incorporation of math and art in their works and their beliefs regarding the connection between math and art.

In the time that Plato lived, the thought concerning art and mathematics was in all likelihood that "never the two shall meet" because they were viewed as such different fields of study and philosophy.

Plato states of geometry that it is "…pursued for the sake of the knowledge of what eternally exits, and not of what comes for a moment into existence, and then perishes" and that geometry "must tend to draw the soul towards truth, and to give the finishing touch to the philosophic spirit." (Field, 1997) The work of Pinney and Thomas (2001) entitled: "Beyond Aesthetics: Art and the Technologies of Enchantment" states that modernist art…was born out of advanced mathematical and scientific thinking of the time." It is related that a new branch of mathematics was developed recently which utilizes modeling as imaging technology.

Pinney and Thomas state that computer-driven mathematicians have "for long drawn on art to enrich the awareness techniques. Artists likewise have visualized 'intuitions' which are result of abstract reasoning, sometimes surprisingly paralleling mathematical made visible in computer imagery." (Pinney and Thomas, 2001) Mathematics and art have the common interest of visualizing the dimension however this only became possible toward the "end of the nineteenth century" with the "development of a graphic system known as 'hypersolids'." (Pinney and Thomas, 2001) Visualization of the fourth dimension was realized by mathematics in computer modeling.

The figural know it used to express the cross-over between arts and mathematics and this is not a new idea.

In Plato's view there exists such a contrast between the "falsity of art and the truth of geometry" and because of this is held by Ivins (1964) in the work entitled: "Art And Geometry: A Study In Space Intuitions" that during the time in which Plato existed that "…there was little chance that the similarities between the two would be recognized." (Ivins, 1964) However, the situation "has vastly chanced." (Ivins, 1964) Ivins states that that the relationship "between logic and mathematics has brought about a much deeper understanding of what mathematics is.

Today geometry has ceased to be The Truth and become a form of art marked by lack of contradictions above rather a superficial level." (Ivins, 1964) G.H. Hardy is reported by Ivins to refer to "the real mathematics, which must be justified as art if it can be justified at all." (1964) I. GEORGE W. HART The work of George W. Hart states that "as a sculptor of constructive geometric forms" that Hart's work "deals with patterns and relationships derived from classical ideals of balance and symmetry.

Mathematical yet organic, these abstract forms invite the viewer to partake of the geometric aesthetic. I use a variety of media, including paper, wood, plastic, metal, and assemblages of common household objects. Classical forms are pushed in new directions, so viewers can take pleasure in their Platonic beauty yet recognize how they are updated for our complex high-tech times.

I share with many artists the idea that a pure form is a worthy object, and select for each piece the materials that best carry that form." (1999) Hart states of his work that they "…invite contemplation, slowly revealing their content; some viewers see them as meditation objects. A lively dancing energy moves within each piece and flows out to the viewer.

The integral wholeness of each self-contained sculpture presents a crystalline purity, a conundrum of complexity, and a stark simplicity." (Hart, 1999) Hart states in regards to his "Leonardo Project" that the work of Leonardo Da Vinci "illustrated Luca Pacioli's 1509 book "De Devina Proportione" (The Divine Proportion). The illustration shown in the following figure is one of the illustrations in that book. This illustration is titled "Ycocedron abscisus Vacuus." Hart is recreating models in wood that are somewhat similar to those of Da Vinci's.

Figure 1 Ycocedron abscisus Vacuus Source: Hart (1999) The following figure is a picture of Hart's conception of the Ycocedron abscisus Vacuus of Da Vinci show just above in Figure 1. Figure 2 George W. Hart's Conception of Ycocedron abscisus Vacuus Source: George W. Hart (1999) Another example of the sculptures of George W. Hart is the sculpture in the following figure labeled Figure 4 which Hart has titled 'Battered Moonlight'. Figure 3 Battered Moonlight Source: Hart (1999) II. CHARLES O. PERRY The work of Charles O.

Perry is that of a creator and artist "…of many dimensions who ponders the wonderful mysteries of the universe." (www.charlesperry.com, 2009) Perry's works are said to "...celebrate and question the laws of nature.

It is his intuitive investigation of nature's variables that provides the springboard for many of Perry's concepts." (www.charlesperry.com, 2009) It is the belief of Perry in regards to sculpture that "…sculpture must stand on its own merit without need of explanation; Perry's work has an elegance of form that masks the mathematical and scientific complexity of its genesis." (www.charlesperry.com, 2009) Perry is a worldwide lecturer on mathematics and art.

The following work is an example of Perry's sculptures and is 'titled' 'The Guardian." Figure 4 The Guardian by Charles O. Perry Source: Perry (2009) The following sculpture of Perry is titled 'Infinity'. Figure 5 Infinity by: Charles O. Perry Source: Perry (2009) III.

DIRAC'S PRINCIPLE OF MATHEMATICAL BEAUTY The work of Alexey Stakhov (nd) entitled: "Dirac's Principle of Mathematical Beauty, Mathematics of Harmony and Golden Scientific Revolution" states that 'Dirac's Principle of Mathematic Beauty' can be understood by studying the contents of Vladimir Arnold in his lecture "The Complexity of Finite Sequences of Zeros and Units, and the Geometry of Finite Functional Spaces." Arnold is an eminent Russian mathematician and academician.

Arnold states: (1) In my opinion, mathematics is simply a part of physics, that is, it is an experimental science, which discovers for mankind the most important and simple laws of nature; and (2) We must begin with a beautiful mathematical theory. Dirac states: "If this theory is really beautiful, then it necessarily will appear as a fine model of important physical phenomena.

It is necessary to search for these phenomena to develop applications of the beautiful mathematical theory and to interpret them as predictions of new laws of physics." Thus, according to Dirac, all new physics, including relativistic and quantum, develop in this way." (Stakhov, nd) From the view of Dirac "A physical law must possess mathematical beauty." (Stakhov, nd) Stakhov states that it is generally agreed among mathematicians that 'beauty in mathematical objects and theories…exist." (Stakhov, nd) Stakhov examines this and states in regards to 'Platonic Solids' that "We can find the beautiful mathematical objects in Euclid's Elements.

In Book XIII of his Elements Euclid stated a theory of 5 regular polyhedrons called Platonic Solids And really these remarkable geometrical figures got very wide applications in theoretical natural sciences, in particular, in crystallography (Shechtman's quasi-crystals), chemistry (fullerenes), biology and so on what is brilliant confirmation" of the Principle of Mathematical Beauty as posited by Dirac.

(Stakhov, nd) Figure 6 Platonic Solids: Tetrahedron, octahedron, cube, Icosahedron, dodecahedron Source: Statkhov (nd) The work of Field (1997) entitled: "The Invention of Infinity: Mathematics and Art in the Renaissance "states that regardless of the period of history being discussed "mathematics seems to share with art the capacity for generating works that can be enjoyed irrespective of their historical context." Field states that art, "in general seems to be in advance of the mathematics, in the sense that, for instance, what are now the most admired examples of the art of ancient Greece almost all date from before the most admired mathematics." (1997) IV.

ARTISTIC EXPRESSION IN GEOMETRY The work of Clifford Singer (1999) entitled: "The Conceptual Mechanics of Expression in Geometric Fields" states that he has taken "a historical perspective in my art works to represent and reflect geometries throughout time. Invariably, in this methodology, much of what is integrated into my work has a strong historical foundation in the classical construction methods of the ancient Greek geometers.

As I research and integrate into the work, more complex elements of the natural world, such as the structures and color usages, coexist within the realm of these properties. Geometrical mechanics and constructions with color form the fundamental components in the building process of my work.

Geometry is thus defined in pictorial space." (1999) Singer states that collinearity or a set of points lying in the same straight line are a "recurrent theme" and one that is "never coincidental" but instead is an "Collinearity, (of a set of points lying in the same straight line) a recurrent theme is never coincidental but is an inevitable outcome, forced in the same way that it occurs in the classical Theorems of projective geometry, such as those of Pascal, Brianchon, Desargues, and others.

These mathematicians recognized that spatial situations which produce collinearity were invariably the result of deep underlying geometric truths. The incidence of a point on a line is invariant under the projective rules. If three or more points are collinear along a line, then incident with a straight line, the images will thus be collinear.

Therefore, the characteristics of incidence, collinearity, and concurrence are principle requisites of my work." (Singer, 1999) According to Singer the world has entering on many levels 'geometrical rules, axioms, methods and carefully thought-out plans…" (1999) However, the method that tends to be the most "imaginative and creative" is the geometrical method which is stated by Singer to have "provided a sustained interest and impetus" for him in his art.

(Singer, 1999) It is interesting to note the statement of Singer that "…many of the elements, straight lines, arcs, elliptical sections, parabolas, hyperbolas, epicycloids, and spirals, accelerate the speed of the viewer's eye through the picture, thus spatial experience is altered in time. Color vibration heightens the speed and expanse for the viewer to complete the picture plane." (1999) Singer states that this makes the provision of visual instruments to the viewers which they are able to "consider proportionately in relations to themselves and in the image.

An image mathematically abstract in origin and represented as nonobjective in context will change experientially through the picture plane, enabling the viewers' apperception and organization of mind set." (1999) The curvature of line, whether inward or outwardly and both positive and negative in nature "are aesthetic choices combined with reasons. It is the idea complex that reaches the positive and negative ranges in the approach.

Although the work may seem to have little direct human or emotional content (in the most superficial sense), there are metaphors and connections to the human condition in the various dynamical paths of particles and objects to which we are all subject to." (Singer, 1999) It is possible, according to Singer (1999) to express these themes in a more literal manner however, the method employed by Singer is one that creates images "…of contemplation and thought.

So the viewer experiencing my work, visually is subject to the same universal and impersonal truths, whether the solutions of differential equations with differing initial conditions are known.

As it is, they are bound within the light cones of Minkowski space, whose intersections with planes yield the same conic sections of the ancient sage Apollonius." (Singer, 1999) Singer relates that there has yet to be a "…complete integration of all the phases of monumental works (geometrical) and "spanning time from Pythagoras, through Kepler and Newton, and on to Bolyai, Lobachevskii, Riemann, Minkowski, and Einstein.

Most have documented individual systems of modeling along established aphorism." (Singer, 1999) George Hart (1999) writes in the work entitled: "Polyhedra and Art" that polyhedra have "through history…been closely associated with the world of art. The peak of this relationship was certainly in the Renaissance." (Hart, 1999) The polyhedra for some artists represented "challenging models to demonstrate their mastery of perspective." (Hart, 1999) However, for other artists the polyhedra were symbolic representation of "deep religious or philosophical truths." (Hart, 1999) Hart relates that for others the polyhedra make provision of both an.