The Golden Section (sectio aurea), attributed to Leonardo Da Vinci

how Leo used it everywhere in his art,

in lovely Mona Lisa,

in his Vitrivian man

Phi , attributed to Mark Barr, American mathematician

Now, make these ratios: A / B = B / C = 1.61803398874989 . . . .

Phun Phrolics with Phi

- the Fibonacci Series

0 1 1 2 3 5 8 13 21 34 55 89 144 . . . .

1 0+1 = 1

1 1+1 = 2

2 2+1 = 3

3 3+2 = 5

5 5+3 = 8 and 5/3 = 1.66666

8 8+5= 13 and 8/5 = 1.60

13 13 + 8 = 21and 13/8 = 1.625

21 21+13 = 34

. . . . . and 144/89 = 1.61797

. . . . . and 1012, 334, 155 / 63, 245, 986 = 1.61803988749895

. . . .and all the rest, ratios = PHI

Sharon has some beautiful (if geeky) phi moments:

from the math tricks, I mused inwardly "there's something about 5 and square roots that reappears ..."

on another site I find I am right!

( 5 1⁄2 + 1 ) / 2 = 1.6180339... = Phi

from the Golden Section diagram of lines:

A/B = B / C = 1.61803398874989 . . . .

also C/B = 0.61803398874989 . . . .

molecular orbital #1 = 0.618 x C1 + 1.618 x C2 + 1.618 x C3 + 0.618 x C4

molecular orbital #2 = 1.618 x C1 + 0.618 x C2 - 0.618 x C3 - 1.618 x C4

Then I found the special relationship that Phi +1 = (Phi)2

which can be re-written as x2 - x -1 = 0, of which the roots are Phi = 1.618.. and 0.618...

well, that explains the chemistry results ( for me...)

meanwhile . . . I rearrange x2 - x -1 = 0

as (x - Phi) (x + p) = 0

and then as (x - A/B) (x + B/A) = 0

now doesn't that look symmetrical, in a way??

More cool phi-geeky relationships to chemistry: DNA (this on'es for Alanna)

On to human beauty: do YOU measure up?

Phi relationships in the body:

the hand

the body

the face