The visual field is scientifically exact as far as measure, and as a universal platform for individual expression that is erected as a whole and complete structure for this creative enterprise. As a visual mathematician I recognize the existence of a naïve spirituality applied through creative process. Where the process begins and ends is open for much discussion. Geometry being empirical narrows the framing of the image to an immediate experience to the picture plane. The science of geometry and its modern developments may be one of the most original creations derived from the human spirit.
The language of geometry is abstract and formal that I have re-invented for my artworks. It has been widely thought that the certainty of the characteristics of number, and structure in mathematics is reason for the certainty of our geometrical knowledge of the space of the physical universe. The question of geometry as mathematical object remains a major concern. Demonstration of collinear points distinguishes the visual format and formulation for the basis of the work.
Is geometrical art an applied science? There reigns a certain pre-established harmony in geometry, what is required in one line of thought is supplied in another line, so that there appears to be a logical necessity, independent of our individual disposition. It is not in the formalist’s mode of thought to deal mainly with the skilful formal treatment of a formal question to devise an algorithm. But, it is rather for the intuitionists who give particular emphasis and stress on geometrical intuition in general and not to give a complete account of a specific subject area. An additive construction as a supplement to a range of mathematical views that I find prevalent through geometrical thinking as to gain insight into the shape of curves as far as general classification and an enumeration of all fundamental forms are concerned.
All this suggests, is the question whether it would not be possible to create an abridged system of mathematics that is adapted to the needs of the applied sciences, without passing through the whole realm of abstract mathematics. It is my general aim to gain, in the course of time, a complete view of the whole field of mathematics with particular regard to the intuitional or in the highest sense of the term: geometrical standpoint.
What is the distinction between a naïve and refined intuition? It must be stated that the root of the matter is that naïve intuition is not exact. While the refining of intuition is not properly intuition at all, however, arises through a rational development of axioms considered as being perfectly correct. To further describe our naïve intuition, for example when thinking of a point we do not picture in our mind an abstract mathematical point, but rather substitute something visual and concrete in its place. In imagining a line, we do not picture in our minds a length without breadth, but however a strip of a particular width. We can always imagine a tangent as a straight strip having a small portion or point in common with a curved strip; similarly with respect to an osculating circle or variable curvature.
The curves and straight lines are of mathematical origin. There is an aesthetic selection for a particular mathematical curvature for a composition itself and are not necessarily mathematically driven or having a mathematical thrust (as a research mathematician may solve as particular problem) although the overall structure can be broken down line by line into mathematical expressions – geometric proofs and partial differential equations. As an artist using geometrical elements for the work I rely on classical construction methods. I have developed methods of construct for epicycloids (such as found in my painting Limacon), spirals (such as my painting The Geometry of the Circle, Square and Heart), parabolas, hyperbola and other curvatures as may be needed while doing my research. The process to plot a parabola by computing the plotting of nodes exponentially or any other creative measure as it presents itself. Creative thinking becomes asymptotic.
In many of my earlier works, I would start with a square panel and began drawing with compass and ruler. Quartic, a painting I created in 1999 was used for the front cover of FOCUS, The Mathematical Association of America, December 2000 featuring an article by Ivars Peterson on the exhibition Art & Mathematics 2000 where I was Curator at The Cooper Union. Quartic was quite difficult to construct as well as The Geometry Lesson done in 1998. In my paintings of recent years, I usually begin with a specific curvature, ellipse, etc. and start working with it planning space. There is much trial and error in this process. I find the intersections of three, four, and five fold as both intuitive and of rational process as I build the image. Of course, I go through many algorithms both numeric equation and visual as I build the picture plane. The steps that I go through visually are in finding the straight line, parabola, or composition of curves (combining two curves as one) that will work within my system of collinear plotting. As the composition is built within my visual system, I find a minimum of three points of intersection for a given line, which requires three, four, or five fold per intersection. This is how I achieve the interweaving of the lines for my compositions. The lines are drawn in a sequential order although it is not the final result. The final result is discovery of the geometries from the invisible. I have been doing geometric paintings, prints and sculptures since 1972 and find that working with geometry over decades has become second nature.
In order to validate the work of art we must give attention not only to the numerous complex aesthetic issues but also to a real underlying nature supportive to the picture plane. In this matter we are fundamentally concerned with the intrinsic network of thought determined by a direct uniform property. Such is a coplanar family of lines that pass through a common vertex. To every property in relation to each other, which points can have, there corresponds some locus, which consists of all the points possessing the property.
The picture plane is the resolve for these matters. Of course one may perceive through a classical viewpoint of analysis that encompasses symmetrical, approximate, asymmetrical, and radial properties as well as the elements and principles of design that I am visually suggestive of as a unified and harmonious composition. Based on physical law, one can reach approximate zero gravity while traveling along a parabolic curvature within the earth’s atmosphere and gravitational pull. Fast and slow curves accelerate speed of eye while looking at the painting. Can visual perception reach zero gravity within the picture plane for the viewer? Artistic expression is found in personal vision and realms of consciousness. Issues of aesthetics may not necessarily be entirely rooted in beauty and judgment, but may fall into the realm of praxis.
I have distinguished in this module, the mechanics of geometrical expression in Non-Euclidean space, which parallels and brings beauty, visual order, and mathematical realities into view. These properties are operative on a universal scale. Change, adaptation, and reassigning values for solutions to problems are key to geometrical demonstration. As a geometer artist having mechanical drawing instruments I might compare and contrast in some respects to the archer with the use of the bow where the target is the resolve. Aesthetic and technical experience in drawing increases my understanding and appreciation of the picture plane, thus the completion of the idea complex. The quantification of nature as expressed on the picture plane may define physics such that the state of reality explicitly and tangibly varies according to the mathematics through which one notices or apprehends it. Acquired dexterity, sensitivity, and mastery of technique have taken control of the medium.
The tendencies thus characterized have exerted a wide-felt influence, and give a distinctive character to a large part of our present geometrical investigations into visual art.
