12 Pentagons Make a Dodecahedron

Or 2 pentagons each having 6 smaller pentagons in it make a dodecahedron.

Polyhedron like dodecahedron will be explored with Excel to use polygon like pentagon as building blocks.

I have designed and drawn a chain of 12 pentagons to cut and fold to make a dodecahedron dice so we can play with our kids. Excel Trigonometry.

Martin Gardner Challenges us in "In the beginning God...."

"1) In the beginning God created the heaven and the Earth.

2) And the earth was without form, and void; and darkness was
upon the face of the deep.
And the Spirit of God moved upon the face of the waters.

3) And God said, Let there be light: and there was light."

In "Wonders of Numbers" (Clifford A. Pickover, 2000), Mathematician Martin Gardner challenges the reader to find out that:

Start with any word in 1) above ('In' to 'Earth'), and count the number of (length of) the word (say 3 in 'the'): "n1";

and Proceed to the word n1 after the first word (say 'created'), and count the number of (length of) the word (7): "n2";

and Proceed to the word n2 after the second word ('created'),
and so forth, and the two 'God's (one each in 2) and 3)) will thus end up just n* after the preceeding words.

My Excel spreadsheet will help find out about this.

Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning

Correct? "phi - (((1 / phi) * (1 - ((1 / phi)^70))) / (1 - (1 / phi))) = 0"

phi - (((1 / phi) * (1 - ((1 / phi)^70))) / (1 - (1 / phi))) = 0 is not verified by what I figure out with my Excel Trigonometry.


06/12/12

((1 / phi) * (1 - ((1 / phi)^70))) / (1 - (1 / phi)) = 1.61803399, more precisely, means that:

1. 1/phi + 1/phi^2 + 1/phi^3 + .... + 1/phi^70 is 1.618033988..
   


2. 1/phi + 1/phi^2 + 1/phi^3 + .... + 1/phi^70 + .... + 1/phi^n
   becomes indefinately closer to phi as 'n' increases to infinity.
   
   Phi to 20,000 places

Note:

1. 1/phi, 1/phi^2, 1/phi^3, ... forms Fibonacci Numbers


2. Formula for the geometrical series for this:
   (((1 / phi) * (1 - ((1 / phi)^n))) / (1 - (1 / phi)))


3. phi+1, (phi+1)^2, (phi+1)^3, ... also forms Fibonacci Numbers