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The Parametric Courante

Guang Zhu

The Parametric Courante is an exploratory research project that consists of a research paper on Cardioid function in the broader context of parametric equation, and a series of particular mathematical visualizations that moves like an elegant Baroque Dance.

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The research component is inspired by the 19 century discovery of The Cardioid Equation. In the paper I study the history and the function of The Cardioid Equation. I analyze the parametric potentials in its geometry.











The visualization component is significant because of the infinite mathematical visual dynamics that The Parametric Equation is capable of generating. Each data set of different parameters determines a particular frequency that brings the equation to life. Using the Processing Language, I intuitively crafted visually and mathematically interesting sets of spirited parameters, so that the visualization shall present itself as a Courante Dance!

Background
Is there mathematics of life? Ian Stewart raised the question the "life 's other secret-the new mathematics of living world"
If there is mathematics of life,what is its life like. In 1917 DArcy Thompson writes that : life is founded on mathematical patterns of the physical world. Ian Steward explains the hypothesizes by the in-depth mathematical methodic reasoning. He examines the the mathematical nature of biological world by the foundation of symmetry, probability and set theory..etc.

"Mathematical functions are a fundamental and prevalent ingredient in the endeavors of scientists and engineers. The conclusions, predictions,and analyses of such professionals are most often concisely contained in these abstract relations, which are commonly referred to as "curves" when illustrated. "The Parametric Courante project rely on The Parametric Equation as a crafting resource to be studied first. The result of its study can be generated from any computer. Based on its summery, just like an artist, it starts "hand crafting" with different numerical set of variables to came up with one "montage" composition that satisfy the artistic goal of its aesthetic. Therefore, the final piece does not only speak of its visually Fine Art presence but also delivers its mathematical symbolic and dynamic functions in its "visual thinking"-motion.

Audience
The general goal of The Parametric Courante project is to transcend the beauty of Parametric Equations by presenting its mathematical visualization in an accessible and friendly manner. It hopes to inspire mathematicians' visual imagination and visual thinking. Because of its dynamic contains infinite potentials that creates complex organism-like visual compositions, the project hopes to encourage mathematical thinking for designers and artists.

Implementation
The research component is inspired by the 19 century discovery of The Cardioid Equation. In the paper I study the history and the function of The Cardioid Equation. I analyze the parametric potentials in its geometry.
The visualization component is significant because of the infinite mathematical visual dynamics that The Parametric Equation is capable of generating. Each data set of different parameters determines a particular frequency that brings the equation to life. Using the Processing Language, I intuitively crafted visually and mathematically interesting sets of spirited parameters, so that the visualization shall present itself as a Courante Dance!

Conclusion
The Parametric Courante project tries to follow the significant essential practice of Fine Art. It unfold itself by searching for the scientific methods to resonate and logicalize to build its solid foundation. Based on this foundation, it crafted the algorithm to generate a particular curve by intuition just like an artist. The result can be generated from any computer. Therefore, the final piece does not only speak of its crafted visually presence but also delivers its mathematical symbolic and dynamic functions in its "thinking"-motion.The project discoveres that by using programming language, from 19 century some individual named static curves can be expressed as one continue curve motion.