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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,585

A while ago I made a 12 sided figure out of paper, by

drawing touching pentagons, and cutting and taping it up.

(I did that because I wrote each of the 12 notes of the chromatic scale on each side to make a die (dice) for thinking about random musical intervals)

I thought today, if you make a dot at the center of the 12

pentagons, and connect them together to form edges of

another 3-D figure, you get the twenty sided one made out

of triangles. Then I noticed if you repeat that, and put dots

in the middle of the triangles, you revert back to the original

shape (dodecagon), but it is a reduced size dodecagon.

I wonder how much smaller it is?

**igloo** **myrtilles** **fourmis**

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**MathsIsFun****Administrator**- Registered: 2005-01-21
- Posts: 7,558

If you ever need to do another one: http://www.mathsisfun.com/geometry/dode … model.html

What you have discovered is called "Dual Polyhedra", one of those magical things about the world.

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,585

Dual Polyhedra. I wondered what it would be called. I bet there aren't triple polyhedra! Maybe if you deal with unsymmetrical things like the continents on the Earth and draw a dot in middle of each and connect... I think that would just be a jumbled mess, I'm not sure. I suppose a 3-D computer program could do the process of finding the next shape and displaying it. Doesn't a golf ball have 236 indents? I thought that was a Trivial Pursuit question, but it was a long time ago, so I might have the # wrong.

**igloo** **myrtilles** **fourmis**

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**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,908

Something like this?

*Last edited by krassi_holmz (2006-03-03 09:29:20)*

IPBLE: Increasing Performance By Lowering Expectations.

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**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,908

Here's another(sorry, I'm not good at graphics stuff in Mathematica):

sorry and for the space but it's better to understand:

*Last edited by krassi_holmz (2006-03-03 09:49:22)*

IPBLE: Increasing Performance By Lowering Expectations.

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,585

That's what I was thinking, thanks for the pictures.

How come in the pictures, above the light blue surfaces,

there are only four lines, not five eminating from a center.

Probably just a small error. Really cool though!

**igloo** **myrtilles** **fourmis**

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**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,908

Ya, this is coz you can't rotate pictures in Mathematica.

For this I used external program: JavaView, that connects with Mathematica and translates the Mathematica's lanuage graphics to it's own. That's why sometimes there's some line, which is "eaten".

IPBLE: Increasing Performance By Lowering Expectations.

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