Abstract
To reify the significance of computational aesthetics in this age of booming technology, it is essential to take notice of our early history, when geometry was the connection between craftsmanship and natural science.
"The Parametric Courante" [1], describes exploratory research and experiments conducted on the cardioid equation. It first examines the parameters of the equation as static numbers in a geometrical context. The paper then assigns the static numbers as variables to constrain strings of floating points (irrational numbers), which generate seamless patterns of geometrical visualizations. Finally, it selects a certain result from the visualizations: one that contains a particular combination in its linear frequency and geometrical movement that exhibits craftsmanship and depicts a unique, lifelike "movement."
This paper is conducted to inspire the creative use and interdisciplinary assessment of parametric functions. It also attempts to articulate an infant stage of integrating media art, mathematical visualization, and the history of geometry.
Key words: parametric equation, visualization, archeology, art
Introduction
Throughout history, mathematicians who contributed to the study of curvature and parametric geometry were drawn to those subjects because of their inclusion of shapes reminiscent of natural forms. [2]
The Parametric Courante Project originates from the cardioid equation discovered in the 18th century. [3] It embodies the quest of mathematician Ian Stewart's question as to whether there is a mathematics of life. [4] To understand life forms by its gradated geometry, Stewart is fond of the 19th century biologist Darcy Thompson, who hypothesized that "life is founded on the mathematical patterns of the physical world." [5] To integrate the rationale between natural forms and pure mathematics is intellectually intensive and complex. The Parametric Courante project tries to reframe the topic of mathematical process and organic growth to trigonometry and artificial movements. It attempts to artfully experiment with the cardioid equation by asking whether there is a geometry of sensuous visualizations.
The project studies the cardioid function in the broader context of parametric equations. First, I created a series of particular mathematical visualizations that describe movements like those of an elegant baroque dancer. Through these visualizations, cardioid trigonometry and mathematical structure are explored in the environment of the computational language tool known as Processing. [6] In this context, computer codes are not only the keys to plot the cardioid function, but also reveal its dynamic geometric potential. This hybrid craftsmanship integrates the rationale of trigonometric functions with artful visualizations. Thus, it generates depictions of various charismatic movements using traditional cardioid patterns. I then quantified the artificial visualizations by their likeness to organic or metaphysical motions such as breathing, dancing, diving, soaring, loving, agitating, and so on.
Explanation
In mathematic structure, the cardioid equation is often expressed as:
x = r∙[2∙sine(t)+sine(2∙t)]
y = r∙[2∙cosine(t)+cosine(2∙t)] ➀
In geometry, the expression of the cardioid equation is often a trace mark of one circle rotating around another circle with an identical radius. [7] Using polar coordinates, point (x,y) is located by radius r and the increment of angle (theta) or time (t).
According to its mathematical structure and parametric geometry, I examined the cardioid equation using the following methods.
1. Examine parameters that signify particular frequencies
Using the cardioid formula, I was able to experiment with constant 2's position and numerical value. This experiment determines particular frequencies as individual wave forms. A parametric geometry is identically revealed when the set of wave forms is translated from linear geometry to polar coordinates (see Chart 1). However, static identical geometry does not always consist of identical frequencies.

Chart 1. Numerical studies. 2013, Guang Zhu, Mathematical visualizations © Guang Zhu. 
2. Assign parameters that can be used to visualize seamless geometrical movements
x = r∙[ 2  a∙sine(1 b∙t)+sine(2 c∙t) ]
y = r∙[ 2  a∙cosine(1 b∙t)+cosine(2 c∙t) ] ➆
a = 2∙cosine(t)
b = 1
c = map(2∙sine(t), [2, 2], [0, 2π]) ➃ ➄ ➅
In this circumstance, constant 2 does not represent one static number as it does in the original cardioid formula. Rather, it represents two arrays of values ➆. One array is assigned by the trigonometry function cosine, labeled a; the other is associated with the sine function, labeled c. c is positioned inside of a trigonometrical function; therefore, this variable is best visualized if its value is modified to the arrangement [0, 2π].

Chart 2. Numerical studies. 2013, Guang Zhu, Mathematical visualizations © Guang Zhu.
The three columns match the three rows from Chart 1, as [➀➃] [➁➄] [➂➅]. 
Mathematically, Chart 2 describes the involute/evolute curvature of the cardioid function and its variations on symmetrical quality. It also expresses the differences among all the geometrical movements. Artistically, Chart 2 evinces the craftsmanship of numbers and shapes. It formalizes a certain connection between digital aesthetics and parametric functions.
This method accentuates the probabilities of parametric cardioid function, transforms static patterns to "fluid" movements, and magnifies the equation's original geometry.
3. Select what examined parameters will be used to present a complete courante dance
The processes of Method 1 and Method 2 provide visualizations, some of which appear more "natural" or familiar than others. According to Walter Zimmerman, some visualizations can be intuited to have a certain imitation of life, while others cannot.
Mathematical visualization is not math appreciation through pictures. The intuition that mathematical visualization seeks is not a vague, superficial substitute for understanding, but the kind of intuition that penetrates to the heart of an idea. It gives depth and meaning to understanding, serves as a reliable guide to problem solving, and inspires creative discoveries. [8]

Figure 1. Two Snow People, 2012, Guang Zhu. Aesthetic computer programming. © Guang Zhu. 
The process of considering the results of all examined parameters fosters an internal reaction to the images, much in the same way that we synchronize with rhythm while listening to music. Thus, the equations and computer codes were selected for inclusion in the final presentation according to the seeming naturalness of their visualizations.
x₁=r∙[2∙sine(t₁)+sine(2∙t₁)]
y₁=r∙[2∙cosine(t₁)+cosine(2∙t₁)] ➇ t₁=t+θ₁ θ₁ =map[sine((t), (1,1), (0, 2π)] ⑩
x₂=r∙[2∙sine(t₂)+sine(2∙t₂)]
y₂=r∙[2∙cosine(t₂)+cosine(2∙t₂)] ➈ t₂=t+θ₂ θ₂=map[cosine(t),(1,1), (0, 2π)] ⑪
As a variation of the cardioid function➀, Two Snow People is primarily constructed using a set of equations [➇ ➈]. Each equation uses a different construction of parameter (t). In Two Snow People, the time variable is associated with two values. One is an equally increased parameter (t) while the other is a periodically increased parameter (θ), driven by the trigonometry functions of sine ⑩ and cosine ⑪. This difference is visualized and magnified by identical mathematical structures ➇ ➈. The final crafted piece exhibits a romantic courante dance of the sine and cosine functions according to the "sheet music" of the cardioid function.
Conclusion
The Parametric Courante Project is an exploration of trigonometry on the cardioid function and the resulting sensuous, lifelike imaginations. It precipitates inquiries into the intuition shared between the abstract language of mathematics and the organic craftsmanship of artmaking. Its goal is to convey the sense of abstract communication, much as the kind described by artist Engel Jules, who states, "My work is abstract, but it contains an organic element that brings people close to their inner feelings. It doesn't'explain'; within feeling, one can discover answers." [9]
Afterthoughts
Working on this project has been a tremendously diverse and inspiring experience. The research materials and experiments in production suggest that certain expressions of mathematics, certain perceptions of art, and certain metaphysical senses are connected on a micro scale. This subtle understanding among art and mathematics feels validated through my personal experience of this project. However, such comprehension has yet to be defined and explored formally in individual study or larger, institutional research.
According to Professor Antoine Picon from Harvard University, since the 18th century, the importance of geometrical methods in architecture has been gradually discarded by the development of calculus. [10] Thus, geometry has endured in absence through this transformation of formalizing computation in architectural design and digital art. Before the 16th century, geometry was the major engineering method for constructions. Its use demanded acuity and a broad knowledge of mathematics, philosophy, and craftsmanship. Mathematics from the understanding of geometry concerned the compositions of logistic proportioning rather than the arithmetic of numerical measurements. [11] Numbers were used mainly for indicating processes and marking proportional segments. This suggests that successful artisan achievements in construction and composition followed well established and systemized rules given the absence of binary/arithmetic based computation.
Oswald Veblen and J.H.C. Whitehead’s (1932) study The Foundations of Differential Geometry inquired why geometry has become a branch of mathematics instead of an independent area of science. In early art history, geometry for artists was considered as a source of continuous perception in time and space. [12] Because of its significance through the history of architecture, mathematics, and digital art, a new kind of study is yet essential to be formed. It should consider mathematical visualizations as an artificial continuous environment which efficiently expresses and quantifies the geometry of movement as a segment of its continuation. In this context, the process of mathematical visualization will be no longer a tool for plotting equations, but a medium that allows an indepth and diverse investigation of geometry as a subject and an application from its early history to this digital era.
The Parametric Courante Project attempts to conduct interdisciplinary inquiries on the study and the experiment of geometrical movement in relation to the history of art and mathematics. Prior to the Enlightenment Age of the 18th century, were there common methods formally established between the mathematics of parametric geometry and the craftsmanship of artmaking? [13] What information will be revealed by the study of the mathematics of parametric geometry from an archaeological perspective? How can we adapt modern wave theory to analyze parametric functions in wave forms? Those inquiries will inform the research and exploration pursued by this project in the coming years.
