fibonacci

In my last Fibonacci post I wrote about how Fibonacci set himself a question and then went about answering it.  The resulting sequence has been intrinsically linked to the golden ratio ever since.  I am very grateful for Denise’s input in the comments section of that particular post because I was still confused.  How on earth did Fibonacci know that this sequence would turn up in his rabbit problem.  She set me straight, however, by stating ‘there really wasn’t any “why” behind Fibonacci’s rabbit story. It was just an excuse to practice with the new Hindu-Arabic number system and to show other Europeans how to calculate stuff. It’s a fun number pattern to play with, but the fact that it connects to a ratio that shows up in nature is an unintended coincidence.’  Thank you Denise, that makes much more sense to me now!

In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities.   The Golden Spiral is simple mathematics beginning with the Golden Ratio, expanding to Golden Rectangles, and then by drawing curves through the corners of the Golden Rectangles to reflect the Golden Spiral. The Golden Spiral was first discovered by Pythagoras in the 5th century B.C. as a mathematical expression but it has been recognised by man in nature for 4000 years and has been around since the beginning of time.  It can be seen in galaxies, hurricanes, shells, sunflowers and all aspects of nature including DNA.

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During the Renaissance artists used the Golden ratio, or the Divine Proportion (a term coined first by Da Vinci in his book ‘De Divina Proportione’), extensively in their paintings and sculptures to achieve balance and to make the art immediately aesthetically beautiful to the eye.

Leonardo used it in his painting The Last Supper:

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And in his Mona Lisa:

monalisaspiral

And again with a spiral drawn between opposite corners of each rectangle:

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The children watched a lovely relaxing video demonstrating extensively how Da Vinci used golden geometry in his painting of the Mona Lisa:

I found particularly this video interesting and it introduced the children to the idea of subliminal messages.  It was particularly fascinating to see the faces which appear in the back ground when one turns the picture on its side!  Who knows whether Da Vinci purposefully painted the faces in as a subtle cipher to be found.  Somehow it wouldn’t surprise me.  He was known for his sense of humour and practical joke playing.  Maybe as he painted he smiled at all he had hidden for those in the future to find.  Or maybe they were there entirely by accident.

Maybe he measured everything to reflect the golden ratio, but I wonder if he knew just how far mathematicians would one day dissect his paintings.  Maybe, just maybe his head was so filled with what he called the divine proportions he painted using them without the need for measurements.  Maybe it all happened very naturally.  Or maybe the reason it took him so long to complete the Mona Lisa is because he measured every angle, every length.  Who knows?  One thing is for sure.  The world remains as fascinated today with his painting as it has done in the years gone by, and this fascination shows no signs of abating any time soon.

This was a good introduction, especially for L11, who had chosen to study Da Vinci for her project this term.