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 "inspiration and a storehouse of forms with various symmetries from which to draw on. This is especially so in twentieth century sculpture, free of the material and representational constraints of earlier conceptions of sculpture." (1999)
Bruter (2002) states in the work entitled: "Mathematics And Art: Mathematical Visualization In Art And Education" that mathematicians and artists "are captivated by polyhedra" and the starting point of the study of these individuals is a very critical piece of the works of Hart and Perry because Hart and Perry began with known polyhedra and then graduated to the learned control of deformations to render objects "full of power, of dynamism and of novelty." (Bruter, 2002)
Hart is stated by Bruter (2002) to deal with "partial but regular simplificial subdivisions on the edges of nested polyhedra. And the result is the creation of "new local groups of symmetry" which serves to contribute to the enlargement of the theory "of the 230 classical crystallographic groups by introducing, over that basis, kinds of algebraic fibers." (Bruter, 2002) Bruter states that Hart "…enriches the motives of the spatial tilings by the use of methods belong to static mathematicians." (2002) The 'curves…trajectories…the threads without thickness which close on themselves are called knots of topological dimension.' The diversity and interwoven nature of these shapes of "infinite variations of shapes immerse the mind into reveries, or on the contrary fix it on perfection…" inspiring "the most impressive works of the sculptors Nat Friedman, Charles Perry and John Robinson." (Bruter, 2001)
SUMMARY & CONCLUSION
In recent years the term 'sacred geometry' has emerged as that which attempts to properly address the connection between mathematics and art and mathematics and music because the tenants of sacred geometry hold that there is something inherently powerful in the various formations that are possible with the realm of geometric construction. Now that mathematics and art are beginning to truly acknowledge that silver cord that binds the two it is likely that an even greater understanding will emerge and continue to emergent as the barriers are removed and art and mathematics claim each the other and through integration with one another serve the function that only art and mathematics -- together may potentially negotiate not only in terms of creativity in art and architecture but also in terms of the visualization of math via visual methodologies.
Bruter, Claude Paul (2002) Mathematics and art: mathematical visualization in art and education. Springer, 2002
Field, Judith Veronica (1997) The invention of infinity: mathematics and art in the Renaissance. Oxford University Press, 1997.
Hart, Gary (1999) Leonardo Project. Online available at: http://www.mi.sanu.ac.yu/vismath/hartex/leonardo-project.html
Hart, Gary (2001) Geometric Sculpture. Online available at: http://www.georgehart.com/sculpture/sculpture.html
Hart, George W. (1999) Leonardo Project. Online available at: http://www.mi.sanu.ac.yu/vismath/hartex/leonardo-project.html
Ivins, William Mills (1964) Art & geometry: a study in space intuitions. Courier Dover Publications, 1964
Perry, Charles O. (2009) Sculptor. Online available at: http://www.charlesperry.com/
Pinney, Christopher and Thomas, Nicholas (2001) Beyond aesthetics: art and the technologies of enchantment. Berg Publishers, 2001.
Singer, Charles (1999) The Conceptual Mechanics Of Expression In Geometric Fields New York, N.Y. Online available at: http://www.mi.sanu.ac.yu/vismath/clif/index.html