The Great Pyramid

The Pyramids of Egypt have been the study of many scientists since they were built over 4,000 years ago. These seventh wonders of the world also contain many of the sacred mathematical constants that I have talked about in earlier post. Three of the most mystical and puzzling numbers in all of mathematics have actually been found incorporated in the same structure being The Great Pyramid. The three constants that combined to form the backbone of complex math are Phi (The Golden Ratio), Pi (3.141), and the base logarithm e (2.718). The chance that all three of these numbers are found in the dimensions of the pyramid is unlikely a coincidence. But this would imply that the ancient Egyptians were so advanced in mathematics that they were able to actually apply such intricate patterns into a massive construction such as the pyramids. The purpose of the pyramids is still somewhat a mystery and most Egyptologists are not convinced that they were simply the tomb for famous pharaohs. Perhaps they were built to teach and unlock mathematical secrets that scientists have been searching for years.

Platonic Solids

Since I have been writing about geometry and the specific qualities that determine a figures shape, I will now touch on the Platonic Solids. The Platonic Solids are the most recognizable three dimensional objects that we encounter in our world. There are exactly five Platonic Solids that possess the unique properties to set them apart from all other three dimensional shapes. In order to be classified as a Platonic Solid, the faces, edges, and angles must all be congruent. Their perfect symmetry has given them the aesthetic beauty that has attracted geometers for thousands of years. Greek philosopher Plato, for whom the solids are named after, first made the relationship between the solids and the four classical elements (earth, air, water, and fire). His argument was primarily philosophical, and was mostly based off of intuition. However, scientists have found that these solids due in fact manifest themselves in nature. At the microscopic scale, many viruses and bacteria take on the shape of Platonic Solids, and multiply by building off of the repeating patterns. The image below shows the five Platonic Solids associated with their respected elements. This topic can go very in depth and anyone interested can find further information by clicking the title that is linked to an excellent web-site.

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.