Here’s Why I Love Google
Say what you will about Google. I have certainly said myself that Google is becoming like Disney: it is evil in its apparently benign offerings of goodness, generosity of spirit, and cooperative worldview; however, I believe its ultimate intent is to lure us humans into depending on its offerings for its own nefarious devices.
Image Credit: Bob MacMurray. http://blog.kidinc.com/ |
But oh, what wonderful devices it offers.
This morning, I went to research something (I forgot what; such is the power of Google), and a chalkboard popped up in place of the usual Google logo.
Being the self-inflicted ADHD victim I am, I clicked on the link; consequently, I learned something I never knew. At 6:45 a.m., I have a brand new bit of knowledge in my head that I never had before. That’s why I love Google.
It is Pierre De Fermat’s 410^{th} birthday today. This morning, I learned that his work as an amateur mathematician led to the very same number theory that we use today. Sir Isaac Newton and Gottfried Leibniz used Fermat’s theorems, and together with Blaise Pascal, they made discoveries in the properties of numbers that would lead to probability theory.
But wait…there’s more! Fermat’s Last Theorem (1637) states: in the equation a^{n} + b^{n} = c^{n}, if a, b, and c are positive numbers and not equal, then no integer greater than 2 for n can satisfy the equation. No one could prove this was true until 1995, when Andrew Weil used technology and math theory that finally caught up with Fermat’s work.
All that is very gratifying to learn at 6:45 a.m. However, what really excited me was this:
This spiral (also known as a parabolic spiral), which follows the equation……reflects mature disc phyllotaxis in nature (plant growth patterns) like sunflowers or daisies.
When a plant is developing, its angle of succession follows a single spiral pattern that is reflective of the Fibonacci numbers approaching the Golden Ratio.
Let me explain.
The mathematician Leonardo Fibonacci in his 1202 book, Liber Abaci, first presented the Fibonacci numbers:
The first two Fibonacci numbers are 0 and 1, and each subsequent number is the sum of the previous two, like so: 0, 1, 2, 3, 5, 8, 13, 21, 34….that goes on forever.
The Golden Ratio is a little harder to explain, so I’ll let Wikipedia do it for me.
“In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.6180339887.
Image source: Wikimedia Commons. |
“The golden section is a line segment divided according to the golden ratio: The total length a + b is to the length of the longer segment a as the length of a is to the length of the shorter segment b^{ 1}.”
This ratio goes back to the building of the Parthenon in Greece. The Parthenon is a wonder because there are no right angles in it; it looks perfect in its imperfection. That’s the same as nature. There are no right angles in nature, just imperfection that looks perfect!
The Fibonacci Numbers and the Golden Ratio are interconnected2. The Golden Ratio shows up everywhere, from the formation of galaxies to the formation of isosceles triangles. The latter has led to all sorts of geometric calculations, which led Euclid, Socrates, Aristotle, Plato, and a wide host of Hermetic philosophers to philosophical and mystical speculations as to the structure of the universe – both known and unknown.
There are proportions of the body that come very close — if not exactly correlate with — the Golden Ratio (again, imperfection that appears perfect)! As a chiropractor and amateur philosophical aesthetician, I find this all very, very interesting.
Image of a human body in a pentagram from Heinrich Cornelius Agrippa’s Libri tres de occulta philosophia. Symbols of the sun and moon are in center, while the other five classical “planets” are around the edge.
Image Source: Heinrich Cornelius Agrippa’s Libri tres de occulta philosophia. Scanned by Jörgen Nixdorf; originally at en:Image:Pentagram3.jpg |
So, let’s get back to our Birthday Boy. Fermat’s spiral represents the mature plant’s spiral progression, as opposed to the developing one. In the mature progression, one could trace the finished spiral pattern from either end…it spirals toward the middle following the Fibonacci sequence, does a turn, and spirals out again, still using Fibonacci’s sequence.
“In mature-disc phyllotaxis, when all the elements are the same size, the shape of the spirals is that of Fermat spirals—ideally. That is because Fermat’s spiral traverses equal annuli in equal turns. The full model proposed by H Vogel in 1979^{3} is…
…where θ is the angle, r is the radius or distance from the center, and n is the index number of the floret and c is a constant scaling factor. The angle 137.508° is the golden angle, which is approximated by ratios of Fibonacci numbers.^{4}”
This mature expression of the Golden Ratio is reminiscent of ancestral labyrinth walks and its patterns are in carvings and structures throughout the so-called primitive archeological world, from India, to Ireland, to the Americas.
Image credit: http://wapedia.mobi/nl/Doolhof |
The spiral is inherent throughout nature and in the intuitive heart of man. We divined its meaning just as we were developing our sense of spirituality and sensibility on every level of contemplation: the mystical, the observational, the aesthetic, the mathematical, and the scientific. It is yet another example of humankind remembering where we come from, and from that knowledge, deducing where we’re going.
These thoughts definitely keep me going. Thanks, Google, for giving me a completely new level of insight to my personhood…all before breakfast on a Wednesday morning!
Notes:
- From http://en.wikipedia.org/wiki/Golden_ratio#cite_note-quadform-0: ^ ^{a} ^{b} The golden ratio can be derived by the quadratic formula, by starting with the first number as 1, then solving for 2nd number x, where the ratios (x + 1)/x = x/1 or (multiplying by x) yields: x + 1 = x^{2}, or thus a quadratic equation: x^{2} − x − 1 = 0. Then, by the quadratic formula, for positive x = (−b + √(b^{2} − 4ac))/(2a) with a = 1, b = −1, c = −1, the solution for x is: (−(−1) + √((−1)^{2} − 4·1·(−1)))/(2·1) or (1 + √(5))/2.
- ibid., scroll to “Relationship to Fibonacci Sequence.”
- http://www.nndb.com/people/768/000087507/, Copyright ©2011 Soylent Communications
- http://www.geek.com/articles/geek-cetera/google-doodle-celebrates-pierre-de-fermat-and-his-last-theorem-20110817/
Sources:
- http://en.wikipedia.org/wiki/Fermat%27s_spiral
- Vogel, H (1979). “A better way to construct the sunflower head”. Mathematical Biosciences 44 (44): 179–189. doi:10.1016/0025-5564(79)90080-4
- Prusinkiewicz, Przemyslaw; Lindenmayer, Aristid (1990). The Algorithmic Beauty of Plants. Springer-Verlag. pp. 101–107. ISBN 978-0387972978. http://algorithmicbotany.org/papers/#webdocs.
http://en.wikipedia.org/wiki/Golden_ratio#cite_note-quadform-0
Posted on 08/18/2011, in Uncategorized and tagged Andrew Weil, Blaise Pascal, Fermat's Last Theorem, Fibonacci, Fibonacci number, Golden Ratio, Google, Gottfried Leibniz, Isaac Newton, Liber Abaci, Pierre, Pierre de Fermat. Bookmark the permalink. Leave a comment.
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