Week 2 – “Mathematics, Perspective, Time, and Space” by Derek Spitters

It is clear that mathematics is intricately intertwined with art, nature, and beauty. For example, the concept of perspective allowed painters to portray a more realistic and believable representation of the real world. On the other hand, abstract art often ignores these principles. This does not invalidate the importance of mathematics in art, but simply illustrates that a mathematical approach to art is one possible avenue towards creating a masterpiece.

A more compelling argument for the importance of math in art is its apparent prevalence in nature. Fractals are an example of the presence of math in nature. A fractal is a geometric shape that is made up of increasingly smaller parts, each of which approximates the shape of the figure as a whole. These types of shapes are common in nature. It was once thought that many of the beautiful shapes found in nature were simply random occurrences. It is now known that many of these phenomena can be represented by iterative equations. In fact, great advancements in creating computer models of nature have been made thanks to the use of fractals. An illustration of this is that equations that model fractals can be used to render more accurate landscapes.

Examples of fractals found in nature:
A cross between broccoli and cauliflower
Landscape

An additional instance of mathematics in nature is the golden ratio. This proportion, which is approximately 1 : 1.618, is widespread in both geometry and nature. For some reason, this ratio has an intrinsic aesthetic appeal, and is therefore commonly used by both architects and artists. However, more importantly, this ratio can be found in many places in nature. The branching of tree limbs and veins in animals follows this ratio. Even human bone structure is in accordance with this proportion.

Mathematics also appears in music. Even a cursory study of harmonics illustrates how math governs music. Conversely, it is still surprising that sequences of numbers, such as the Fibonacci Sequence, can be use to create an attractive musical composition. One would expect a series of random sounds to result from such an approach. The interconnectedness of these two seemingly unrelated fields demonstrates the overlap between the “two cultures.” Under a rigorous analysis, it is impossible to completely separate one culture from the other. There will always be some overlap.

The fourth dimension is an interesting concept both in art and science, and is an idea that has evolved over time. Originally, artists of the Cubist movement attempted to portray objects in four dimensions by depicting them simultaneously from many different angles. These artists were attempting a spatial representation of the fourth dimension. Einstein’s theory of relativity postulated that the fourth dimension was time. Some of the artists in the Surrealist movement attempted to adopt this new theory into their portrayals of the fourth dimension. One way of doing this was through movement since movement shows the passing of time. Other artists felt that the two concepts of the fourth dimension (space and time) were not mutually exclusive. As the scientific definition of the fourth dimension began to change, artists depicting four-dimensional objects began to change their methods as well. This is a perfect example of the integration of the arts and sciences.

Fractals:
http://mathworld.wolfram.com/Fractal.html

Using fractals to model nature:
http://www.miqel.com/fractals_math_patterns/visual-math-natural-fractals.html

The Golden Ratio:
http://mathworld.wolfram.com/GoldenRatio.html

–Derek Spitters

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