In case you don’t plan to read anything else in this powerpoint………. There is an activity you must do, somewhere hidden in this slide show, in preparation for next lesson!

Aaah bunnies! Did you know that rabbits reproduce like, well rabbits! Suppose a newly-born pair of rabbits, one male, one female, are put in a field. The rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits (that is, gestation takes one month). Suppose that our rabbits never die and that the females always produce one new pair (one male, one female) every month from the second month on. How many pairs will there be in one year?

Recognise the pattern? The sequence is 1, 1, 2, 3, 5 What’s next? 8, 13, 21, 34, 55… This is actually a really old puzzle set (and answered) by

Leonardo Pisano Bigollo, of course Aka - Fibonacci Italian Mathematician (1170 – 1250) 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597,... (btw he neither invented nor discovered it – he nicked it!) The pattern is of course interesting but the (supposed?) link to nature is more interesting

What happens if you divide one number by the next one? Give it a go It’s dead easy if you have Excel! You get numbers converging on about Which is as we all know

The Golden Ratio (phi)

And….. (does this get a bit too convoluted?) If you take the reciprocal of the Golden Ratio (1/1.618) and multiply that by 360 you get the Golden Angle (approximately 137.5°) which pops up in nature all the time (pine cones, pineapples, leaf buds and the arrangement of seeds in a tomato – only joking!)

Fibonacci and Art Look at the 10 following pieces of Art and just by sight decide if the Golden Ratio, Spiral, Rectangle etc is present A couple of these pieces of art were created very definitely with the Golden Ratio in mind and some very definitely not (the artists said so) Can you tell which is which? Pick two, print them out and draw in the ratios, rectangle, spirals etc

Leonardo da Vinci’s “Vitruvian Man”

Leonardo da Vinci’s “La Gioconda” Or just the Mona Lisa to you and me

Salvador Dali’s “Sacrament of the Last Supper”

Brent’s “Untitled”

Jacques Louis David’s “Death of Marat”

Two abstract watercolour paintings by unknown artists

“Bathers” by Paul Seurat

“Composition in Red, Yellow, and Blue” by Mondrian

Pablo Picasso’s “Weeping Woman”

It’s the same in nature The shell of the nautilus

Violet, 5 petals Mayweed, 13 petals Pyrethrum, 34 petals Lily, 3 petals (bottom 3 are sepals!) And in flowers

Dame’s rocket – 4 petals Lily – 6 petals Starflower – 7 petals Well almost!

Notre Dame in Paris And in Architecture The CN Tower in Toronto Taj Mahal in Agra

The Parthenon in Athens

Great Pyramid of Giza (built around 2500BC) The length of the side of the pyramid is feet (2a = ) The triangular height (s) = ft s / a = / = which is remarkably close to φ!

But, is there any historical documentation that supports the hypothesis that φ was intentionally used in the construction of the pyramid? Supposedly, Herodotus ( Greek historian ca. 485 – 425 B.C.) wrote in his book “History” that φ was used as a proportion in the pyramids. However, this belief stems from an inaccurate interpretation of Herodotus’ text first made in 1859 by John Taylor in his book “The great pyramid, why was it built and who built it?” Ever since, the claim has been accepted as true by several authors without any scrutiny of the original statement in Herodotus. In any case, the figures of the pyramid’s dimensions mentioned by Herodotus are wildly off; its height is only 481 feet, not 800 feet as stated by him in his text!