Mathematics in Nature: How We Use Math Outside of School

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A science that dates back to the beginning of the 6th century BC when used by the Babylonians to record history and keep track of lunar cycles, mathematics is a necessity that is part of our daily lives due to the many applications it has from the most simple functions to the complex computational problems. 

Pure Mathematics becomes Applied Mathematics when connected to various fields such as physics, medicine, finance, and architecture. People easily see the basic relationship between mathematics and engineering, or mathematics and finance, but it seems difficult for some to perceive that there is a very close relationship between mathematics and non-STEM subjects like arts, such as painting and music. Although some might see music as a pure form of art that has nothing to do with science or mathematics, they are very closely intertwined, and as the great Pythagoras once said, “there is geometry in the humming of the strings, there is music in the spacing of the spheres” (qtd. in “The Connection”). Music serves as another example of how we use math outside of school.

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There are certain melodies, chords, or notes that are played together, which are pleasing to the ear. In addition to this discipline, the application of mathematics is utilized once our brains begin to process sound waves which are not disturbed by the frequency of human hearing When a note is played, the sound that reverberates on our eardrums is actually a frequency of a sound wave that aids our brains to indicate the pitch of the note. That sound coming from an instrument or amplifier causes little hairs in our ears called stereocilia to vibrate at different frequencies. Different notes have different frequencies, for example: C - 261.6 Hz E - 329. 6 Hz G - 392.0 Hz When forming up a chord, the similarity of wave frequencies is of utmost importance. For instance, if we take the notes E and C, the ratio of their frequencies is around 5/4ths meaning that every fifth wave of E matches with every fourth wave of C. The same procedure applies for C and G, in which their ratio is 5/4ths too, indicating a similar sound pattern and melodic chord This mathematical relationship explains why some notes sound so pleasing and why a more abstract ratio or fraction would produce a more dissonant sound. As indicated above, music, in essence, is mathematics and calculations which are just translated as repetitive frequency patterns transforming into a melody. In principle, mathematics is considered as the study of patterns and music is concerned with creating patterns. 

As a result of mathematical perspective including geometry, trigonometry, calculus, and series, everything in music, from the tones between notes to the timbres of sound to melodies and harmonies, can be explained and analyzed. Even the electronic tools that are used nowadays to make music, are fundamentally based on the mathematical understanding of sound patterns. When you’re making a beat by tapping your fingers on the table, in principle it is a period function - a wave. However, without misinterpreting it, music is not mathematics, nor is mathematics music; they are two different fields that happen to have a very strong bond with one another. An example of achieved mastery in this mathematical sense of music is the prominent and genius, Ludwig Van Beethoven. At the age of 31 he began to lose his hearing and later at 46 he went completely deaf; yet, during this period of his life, Beethoven composed some of the most well-known, successful and appreciated pieces such as the famous “Symphony No. 9”. 

But how was he able to compose such masterpieces without his ability to hear? The answer lies behind the mathematical point of view of music, behind the patterns and series in the notes. His “Moonlight Sonata”, is a sufficient example of an intricate and complex piece which seems not that difficult to the naked eye. The piece starts with the D major triplet, composed of the notes A4, F#4 and D4, each with their own individual frequencies and series. If played together at the same time, they create a previously mentioned pattern - consonance, which sounds pleasant to our ears. Beethoven’s true genius lays behind the fact that he was able to feel the effect of notes and melodies played together, in the absence of his sense of hearing. Apart from music, another aspect of art in which mathematics finds great application is painting. During the course of history, there have been many artists such as Leonardo Da Vinci, Durer, Escher, that have not only been painters or sculptors but also polymaths. 

There seems to be a certain barrier between different disciplines which causes artists or scientists to not bother with other fields that do not concern them or rather do not interest them. However, as previously mentioned, this is a fairly new ideal as in previous times, artists in addition to their artistic outlook on life, incorporated their mathematical knowledge in their artworks to create masterpieces. In the 4th century BC, the perfect nude male was said to have proportions based on the ratio 1:√2. This was called the golden ratio and it was used later on during the years by famous sculptors and painters, for example, during the Italian Renaissance by Leonardo Da Vinci in most of his artwork such as in “Mona Lisa”, “The Last Supper” and “The Annunciation”. 

Another example is in the book on mathematics “De Divina Proportione” (The Divine Proportion), written in 1497 by Luca Pacioli and illustrated by Leonardo Da Vinci. Pacioli writes about the connection between mathematics and the artistic discipline and focuses on the golden ratio. Alongside his descriptions are Da Vinci’s illustrations of three and two-dimensional geometric shapes, architectural designs and sketches of line intersections. In his work, Pacioli states that “his intent is to reveal to artists the secret of harmonic forms through the use of the Divine proportion”(qtd. in Meisner et al.). Besides the golden ratio, there are many other mathematical concepts that have been used and have influenced art, such as symmetry, linear perspective and even the camera lucida, an optical device which helped artists duplicate key points of a scene in perspective. Beyond mathematics influencing artists and finding application in it, “the similarity goes far beyond immediate relationships such as the fact that geometry plays a significant role in both”, as they are both examples of “the human consciousness striving to comprehend reality”(Jensen, “Mathematics is Painting”). Both disciplines take in and analyze abstract elements and patterns of nature, whilst attempting to present them to form a concrete function. 

Many artworks convey abstract emotions or feelings with the use of a diverse range of colors, shapes, lines, and patterns. For instance, in Edvard Munch’s “The Scream” the psychological mood that is captured is horror. From the disoriented lines to the overdramatic face features, to the simplistic background in order to emphasize the front scene, Munch takes an abstract element of nature and turns it into something that is real, empirical. In a similar way, mathematicians analyze a mathematical property by “extracting that specific arithmetical property of the natural numbers and studying [it] in its pure form in the context of abstract group theory”(Jensen, “Mathematics and Painting”). 

Since there is such a strong connection between painting and mathematics, why should there be a barrier between them? Why should there be a distinction between art and science when in principle they are intertwined? They are both used to understand the world around us and according to Jensen they “share a programme aimed at developing symbolic, concise, and often very abstract, representations of reality” and because of these similarities we consider them “simply as two different but complementary ways of visualizing aspects of the concrete or abstract reality in which we are embedded” (“Mathematics Is Painting”; ”Mathematics and Painting”). In the same way that mathematics is used to explain musical phenomenons, artistic patterns, and elements, throughout the course of history, it has also been used to describe the physical properties of the universe. 

However, many scientists such as Max Tegmark, have posed the question: “Is the universe itself mathematics?”. According to Tegmark, physicist and cosmologist, everything in the universe is mathematics and humans too “[are] part of a mathematical structure” (Lewis). He states that everything has properties, and space has dimensions but in essence, all are mathematical structures. The notion is that if you look closely enough, mathematical patterns are scattered all around nature: from the systematic sequence that the Artichoke leaves follow, to the parabolic trajectory that a ball trails when hit or thrown. On a larger scale, even in outer space, planets and other spacial bodies follow an elliptical orbit. 

As stated by Tegmark “[t]here's an elegant simplicity and beauty in nature revealed by mathematical patterns and shapes, which our minds have been able to figure out” (qtd. in Lewis). In his controversial Mathematical Universe Hypothesis, Tegmark argues that the physical universe isn’t just explained or analyzed by mathematics, but in principle it is mathematics. Even though not supported by all, his ideas have been presented since earlier times by great philosophers and mathematicians such as Pythagoras who believed that the universe was built from mathematics and then later by Galileo Galilei who stated that “[nature] is a grand book written in the language of mathematics” (qtd. in “The Universe”). 

In conclusion, different disciplines consisting of art and science aren’t that fundamentally contrary as they are similar in ways of visualizing and trying to make sense of the abstractions that is our universe. Mathematics is the intermediary language between the universe and us and it is embedded on all disciplines and practices of life. It could stand out as “one of the greatest and most ingenious inventions ever made”(“The Universe”), an invention that describes the way the universe is constructed and functions. Contrary to people’s beliefs, mathematics isn’t just a tool we use to solve equations or problems, it’s not a mental exercise to make us smarter, but it is a universal language that we use to communicate with the universe and all that is yet to be discovered. 

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