Math Is Fun Forum

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

dodecagon 3-d shape

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?

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igloo myrtilles fourmis

MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

Re: dodecagon 3-d shape

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.

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

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: dodecagon 3-d shape

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.

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igloo myrtilles fourmis

krassi_holmz

Real Member

Registered: 2005-12-02

Posts: 1,905

Re: dodecagon 3-d shape

Something like this?

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

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IPBLE:  Increasing Performance By Lowering Expectations.

krassi_holmz

Real Member

Registered: 2005-12-02

Posts: 1,905

Re: dodecagon 3-d shape

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)

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IPBLE:  Increasing Performance By Lowering Expectations.

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: dodecagon 3-d shape

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!

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igloo myrtilles fourmis

krassi_holmz

Real Member

Registered: 2005-12-02

Posts: 1,905

Re: dodecagon 3-d shape

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".

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IPBLE:  Increasing Performance By Lowering Expectations.