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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