Looking into the past, through the millennia, we can see the use of these proportions and ratios emerge. The use of mathematical tools from the straightedge, to the more sophisticated computer technology, has given us the beautiful perspective, symmetry, and patterns seen in art. These proportions have been shown to us through the use of these tools from the paintings in the time of Greek culture, to the fractal geometry of modern technology in the present. We can also see that many of these ideas and mathematical uses have, and are being used in architectural design. Ratios such as the Golden Mean have been seen everywhere in architecture of the past, and even in the design of modern buildings. This Golden Mean proportion is a ratio of numbers, which has fascinated

The book Exploring the Invisible, by Lynn Gamwell, explores how Pythagoras discovered a mathematical basis to music, and how mathematics is used in the creation of abstract art. Pythagoras found the fundamental musical laws, which relate to musical proportions, such as an octave, a fifth, and a fourth (Gamwell, 149). These ratios of tones have been found to give a satisfying feeling to the listener which is created when mathematics are applied. This discovery was the first scientifically well grounded theory of musical harmony. The Golden Mean ratio, which is rounded to 1.618: 1, and its inverse .618: 1, has also been known to show up throughout musical history. Explanations of the Golden Mean can be seen with a brief description of the Fibonacci sequence. The Fibonacci sequence comes up everywhere in nature from the branches and leaves on trees, to the reproductive cycle of certain animals. Numerically shown as 1, 1, 2, 3, 5, 8, 13, 21… with each number being the sum of the two previous numbers going on into infinity. On many trees, branches come forth as you go up the tree in this sequence. You start with 1 trunk, then one 1 branch, then 2 branches, then 3, and so on. We find that the ratio of one number in the sequence to the previous number in the sequence approaches the Golden ratio. The Golden Mean is a special relationship between a part and a whole that acts as a formula for harmony and beauty. Awareness of the Golden Mean’s importance has been instrumental in the acceleration of culture throughout history, and is only partially conveyed through musical artwork. Pythagoras is known for his knowledge of arithmetical, geometrical, harmonic proportions, and also for the law of the golden section, otherwise known as the Golden Mean (Stakhov). The Golden Mean proportion began to come up when Pythagoras related his musical theories to the planets of the solar system. He believed that each planet had its own individual tone, which was related to its velocity around the sun (Gamwell, 149). He, and other philosophers and scientists of the time, such as Johannes Kepler looked into this idea of the planets, and how they relate to the golden section.

“Number is the within of all things.” -Pythagoras of Samos

Looking back into Ms. Gamwell’s world of mathematics, art, and music, we can begin to understand how the three interrelate. We can see that from the time of ancient Egypt, through the eras of Greece and the Renaissance, to modern times, that mathematics has been an essential element in the creation of many forms of art and music. It is interesting to find the use of mathematics, particularly the Golden Mean ratio, in so many of the mythical cultures throughout the millennia. It makes us wonder if there is a deeper understanding in this mathematical relationship, which goes far beyond its use in art. Many mathematical, and artistic geniuses of the past have grasped the meaning of the Golden Mean, and applied it to their life and artwork. It has been the foundation of spiritual understanding for many cultures and people such as Plato, Pythagoras, and possible the entire Egyptian and Greek cultures.

The pyramids are yet another example of how mathematics were applied to the creation of architecture. Egyptians used the Golden Ratio in creating structures that had a base to height ratio of 1.6, which approximates this ratio (Zechar). It may be coincidence that these were the dimensions, or perhaps the Egyptians were aware of the Golden Mean, and realized its aesthetic appeal. Whether the Golden Mean proportion was used to convey beauty, or for another more universal reason is still controversial. Millions travel to see the wonder and beauty of the Pyramids every year. Whether or not we understand the creation of these huge monuments, or how they were created with such perfection, we do know that for one reason or another, the Golden mean and many other mathematical proportions were displayed in every millimeter

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