Some of the earliest references to the aesthetics of mathematics are made by Pythagoras (569-500 B.C.), who observed certain patterns and number relationships occurring in nature.

The Five Platonic Solids

According to Pythagoras , there are five, regular convex solids, each of which could be circumscribed by a sphere: the tetrahedron, cube, octahedron, icosahedron and dodecachedron. Flat shapes such as, the square , the equilateral triangle and the pentagon are used to construct three dimensional volumes. The dodecahedron was given special significance by the Ancient Greek. It’s twelve regular facets correspond to the twelve signs of the zodiac. Many mathematicians during the Middle Ages and the Renaissance were interested in the golden mean ratio. In 1509, Luca Pacioli, published De Divina Proportione, which was illustrated by Leonardo da Vinci.

The Pentagram Star

The Phi proportion lends itself to the generation of the five point star. The Golden Means number is discovered through the square root of five. It is considered self-similar, which means there is unity of form across various scales in size. Again, the parts mirror the whole.

The Fibonacci Series

Leonardo Pisano, (c1170-1250),called Fibonacci, that is, son of Bonaccio, is considered one of the great mathematicians of the Middle Ages. He received his academic education among the Mohammedans of Barbary, where he learned the Arabic, or decimal, system of numbering as well as Alkarismi s teaching of algebra. At around age 27, he published Liber Abaci (the book of Abacus). Because of him we now use the arabic system of numbers and not Roman numerology. "The mathematical work of Leonardo Pisano was innovative, creative, and high in quality. He introduced into Europe the numerals, calculation, and algebra of the Orient. Yet he was, after all, a profound scholar of Greek mathematics, a worthy successor of Euclid, Archimedes, Heron, and Diophantus. He stands above all other mathematicians from classical times to the Renaissance. He joined the theoretical tradition of the Hellenes and the algebraic tradition of the Arabs and established them in Europe. Leonardo Pisano is the great European mathematician of the Middle Ages."(L.E. Sigler,1987)

The infamous Fibonacci number pattern starts with 1 and 0 and grows through the process of addition. If you add the first number with the second, by it’s sum you get the third. The fibonacci phenomena appears in many unexpected places, such as the laws involved with the multiple reflections of light through mirrors, generations of honeybees and rabbit breeding patterns.

As an example:
0+1=1,
1+1=2,
1+2=3
2+3=5 and so on.
The sequence continuing to infinitum…. 0, 1, 1, 2, 3, 5 8, 13, 21, 34, 55, 89, 144, 233, 377…

The Rabbit Question:A Mathematical Puzzle

There are one pair of rabbits in the months of January which breed a second pair in the month of February and they produce another pair monthly, and that each pair of rabbits produce another pair in the second month following birth and thereafter produce one pair per month. Find the number of pairs at the end of the following December.

1. # of pairs of breeding rabbits at the beginning of month.
2. # of pairs of non-breeding rabbits at the beginning of the month.
3. # of pairs of rabbits bred during the month.
4. # of pairs of rabbits living at the end of the month.

The Golden Rectangle

To produce a more harmonious rectangle, one only has to use the golden mean ratio.

Another method to achieve the golden rectangle is through additive squares.(similar to the Fibonacci series) you can obtain closer and closer approximations to the golden rectangle. The longer the series continues the closer the approximation to the golden section and the golden rectangle. By adding an arc within each square, the formation of the golden Mean spiral emerges.

Spira Miribilis

Spira Miribilis, the equiangular spiral, has been noticed mostly in nature’s flora and fauna. You can see this curve in the florets of the daisy. Also, it’s very distinctive in the sunflower. These seem to have two equiangular spirals superimposed on each other, both going in opposite directions. And, you can also notice that Fibonacci numbers appear to us in the
sunflower specimen. There are usually 21 clockwise spirals and 34 anti-clockwise spirals. These opposing spirals are also found in pinecones and the pineapple. Shells also offer a variety of examples of spira miribilis. The chambered nautilus is a supreme example of the beauty of mathematical design.

The Fibonacci series can also be noticed in the distribution of branches, leaves and seeds (phyllotaxis). This is the angular arrangement of leaves and their attachment to the stem forming a type of vertical helix.

 

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