Math and Art

So far I have mostly talked about the pre-existing laws of mathematics that govern everything in our universe. This post I will talk about how artist and architects use geometry in their works. We sometimes forget that math is more than just formulas and logic. The objective beauty in some artwork and architecture is really derived from the underlying sacred geometry the artisan employed. I talked about how ratios determine how we differentiate and determine what things are aesthetically pleasing to the eye. So it should come as no surprise that artist and architects would take advantage of mathematical concepts such as patterns, symmetry, structure, and shape. No one took more advantage of the secrets of geometry more than Leonardo DaVinci. DaVinci utilized proportions and perfected the art of linear perspective. Linear perspective is the technique of representing a three dimensional object or a particular volume of space on a flat surface. For this to create the illusion of depth, essentially another dimension, all of the lines in the painting must converge to a single, invisible point on the horizon. The Last Supper is a perfect example of this method and it also contains specific proportions that directly relate to harmonic balance. The Mona Lisa’s facial proportions contain an almost perfect Golden Ratio, and this is why so many have found the painting to be so beautiful. It truly is amazing that DaVinci would be able to incorporate such complex mathematics into his work while hiding it from the public for centuries. This is an excellent example of how the beauty of mathematics can be right in front you without knowing to the naked eye. Below is a picture of DaVinci’s Last Supper, if you examine it closely you can see how the proportions make the picture appear so balanced and complete.

Sounds and Matrices

When listening to music, we usually don’t think about the actual process that is taking place. In my earlier posts on String Theory, I talked about how everything in the entire universe is vibrating strands of energy. Every vibrating strand can take on an infinite range of frequencies that determine the properties of that object, force, or in our case sound waves. However, our brains are only capable of sensing and deciphering a certain range of frequencies. For instance wild animals typically have much sharper senses and are able to take in a wider range of frequencies needed for their survival. Communication devices send and receive frequencies we as humans are unable to physically sense because they are beyond the narrow range that our brains can interpret. The way in which we actually send and receive messages through man-made communication devices is essentially all mathematics. We take a sound or image and represent it using a matrix of numbers we sometimes call wavelets. Wavelet Transformations use linear algebra to compress the digital media in order to send it over computer networks, and then reconstruct them by undoing the mathematical operations. Our brains have been naturally programmed to receive extremely complex wavelets and decode them into what we ultimately experience through our five senses. Our brains are able to process insane amounts of information effortlessly and almost instantaneous. We don’t really think of our brains as supercomputers, but whether we know it or not they are constantly decoding a divine matrix we call reality.

Mathematics of Music

I feel it is safe to say that all people enjoy at least some genre of music, and I personally find music to be one of the most pleasurable sensations we are given in life. However, few people truly understand the direct correlation between mathematics and music. Music is an excellent example of how the human brain can receive a wide range of frequencies and interpret them from both objective and subjective points of view. Putting aside an individual’s specific tastes, most people can agree upon the need for certain harmony and musical theory to exist in order for the sound to be considered pleasing to the ear. But what is it that determines whether a music piece is amazingly beautiful or absolutely awful? The answer lies within the mathematical language of sound and harmony. The music of nature first serenaded humans, but it didn’t take long for man to learn how to produce every sound imaginable. Pythagoras was the first to really apply his knowledge of sacred mathematics and ratios to the art of music. For any stringed instrument, the position at which the string is pushed down determines the note produced by the different frequencies. For each depressed string, the ratio at which the string is divided governs how well the sound is to the ear. By recording every different ratio and its respected sound, Pythagoras was able to find the most delightful ratios and create the Diatonic Scale. Pythagoras was fixated on the Golden Ratio and the Fibonacci Numbers and wanted to see how they behaved in the musical spectrum. Not surprisingly, the perfect ratios yielded the most beautiful notes and chords in the entire scale. For these reasons, many musicians including Beethoven, Mozart, Chopin, and Schubert implemented the Fibonacci ratios into some of their entire compositions. Even the timing and measures at which the different instrumental sections such as strings, percussion, and cellos entered the symphony were deliberately based on the Fibonacci intervals. Next posts, I will continue discussing the fundamentals of mathematics that create the frequencies we perceive as musical harmony.

Zero Destroys the Number Line

Nearly everyone with a high school diploma has been forced to learn at least the basic laws of Algebra that are the foundations for all other mathematics. However, very few people ever really question the underlying theory from which we form our logic. We no longer carry the Renaissance approach to learning in which an individual would study a wide range of liberal arts to acquire a well rounded education. However, most of our early mathematicians were also philosophers and influential figures that shaped politics and religion. As a result, zero brought with it a threat to our understanding of logic. If we imagine the real number line as an elastic rubber band that can stretch and shrink, we are able to visually see how zero can create problems. Multiplication can be thought of as stretching the rubber band by a scalar. But when any integer is multiplied by zero, the entire number line collapses into an infinitely small point. So if multiplication crushes the number line, then division in theory should undue to destruction. This wishful thinking is anything but true. Dividing by zero, even one time, destroys the entire framework of mathematics. It is difficult to show without a simple proof, but by multiplying and then dividing any number by zero one can show that the ratio of zero to zero [0:0] is equal to anything and everything. One can imagine the problem this would have created among philosophers and mathematicians. Eventually, mathematicians came up with a clever solution to the problem by saying that the answer is “undefined”. This in itself is an oxymoron for in the very act of defining the unanswerable as “undefined”, we have in fact given it a definition.

Zero Nothing or Everything?

Zero is by far the most fascinating and misunderstood mathematical concept ever unearthed. It is only now that we are beginning to appreciate the power of such a seemingly innocent number. Looking back throughout history reveals just how feared this abstract concept was to ancient civilizations. Zero was first invented by the Babylonians, as it proved to be not only practical when doing calculations, but necessary to the very framework of logic. The history of zero and its acceptance in the scientific community was anything but stable. Ancient Greece banned the number and Aristotle himself was so threaten by the idea that he strongly lectured against any notion of such a profane conjecture. Because early mathematics and religion were closely correlated, the Catholic Church deemed the concept blasphemous due to it threatening the church’s narrow understanding of God. Few people truly understand and appreciate the existence of zero and how it allows for everything that is and ever will be. Zero is such an exception to all mathematical logic that most people simply learn and accept its unique properties that overpower all other mathematics. For millennium, society had functioned perfectly fine without the use of zero. After all, we don’t need a number to express the lack of something. Most people can respond to the absence of matter by using expressions such as “I don’t have anything” or “There is nothing”. An early farmer would not say “I have zero sheep.” It is for these reasons that zero had been pushed under the rug of everyday thinking and was only brought into existence when conveniently needed for computational purposes. Next posts, I will talk about the unrivaled properties that sets zero apart from all other numbers.

Unpredictable Science

Like any school of thought, Chaos Theory has been shaped from the historical database of past human knowledge. It is from the countless contributions that mankind has shared and past down over time that has allowed for new growth and the continuous evolution necessary for survival. The internet is now the conduit that makes this rapid exchange of knowledge possible. Therefore, it should come as no surprise that our overall understanding of life is growing at an exponential rate. Old school Western Science is being forced to loosen up their pride belts and embrace some of the Ancient Eastern “mysticism” that most modern scientists scoffed at over the last centuries. Much of the belief in Chaos Theory requires an open mind to other possibilities beyond the tight framework of science that has limited our expansion of knowledge. The picture of God throwing dice has scared scientists and prevented many from breaking free of the physical laws that bound us to earth. Now that we have mathematical systems to model chaos, we are able to see the beauty in chance that makes possible the unpredictable relationships needed for evolution to continue. The primitive notion that the universe is a cosmic clock winding down due to thermodynamics no longer holds. Rather than the watch-smith winding up the clock and sitting back, Chaos believes that this Strange Attractor is pulling all of life together as well as creating it along the way.

Chaos Attractors

“At any given moment, life is completely senseless. But viewed over a period, it seems to reveal itself as an organism existing in time, having a purpose, trending in a certain direction.” This quote from Aldous Huxley is an excellent description of the philosophy of Chaos Theory. Chaos Theory is very multifaceted and requires a great deal of prior knowledge to fully understand. It believes in the existence of four forces or “attractors” that bring order out of a seemingly chaotic cosmos. Each of these attractors are rather abstract concepts that entail a significant amount a time and research to comprehend. Therefore, I will briefly define each attractor and its role in Chaos Theory. The Point Attractor is the simplest of the four forces and its domain is the first dimension (a line). It is what draws or repels objects to other objects. It is the pull that influences our choices/actions based on our likes and dislikes. The next is the Cycle Attractor, as the name suggest, is responsible for the cyclical patterns that govern literally everything. The Torus Attractor is similar to the Cycle Attractor in that it is based off of repetition. However, it has much more irregularity and exists primarily in the long run with respect to time. The last and most important of the four forces is the Strange Attractor. It is this bizarre force that allows for self organization which is the basis of Chaos Theory. It feeds off the numerical concept of zero which allows for the spontaneous unpredictability of life. It is responsible for gluing all matter together as it evolves over time. This oversimplified description of the chaotic forces is nowhere near enough information to logically make sense of Chaos Theory. I would recommend researching this rapidly growing school of thought in order to appreciate its scientific components. The title of this post is linked to a site that focuses on bridging the gap between science and religion. On the site, you can find all kinds of information on Chaos Theory including a more in depth explanation of the Four Attractors.