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#1 2006-03-01 07:31:25

John E. Franklin
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Registered: 2005-08-29
Posts: 3,562

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?


igloo myrtilles fourmis

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#2 2006-03-01 09:35:38

MathsIsFun
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Registered: 2005-01-21
Posts: 7,550

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.


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

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#3 2006-03-01 10:16:56

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,562

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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#4 2006-03-03 09:27:26

krassi_holmz
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Registered: 2005-12-02
Posts: 1,908

Re: dodecagon 3-d shape

Something like this?

View Image: 1in2.JPG

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


IPBLE:  Increasing Performance By Lowering Expectations.

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#5 2006-03-03 09:38:25

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

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:

View Image: s.001.001.gif View Image: s.001.005.gif View Image: s.001.010.gif View Image: s.001.015.gif View Image: s.001.020.gif View Image: s.001.025.gif View Image: s.001.030.gif View Image: s.001.035.gif View Image: s.001.040.gif View Image: s.001.045.gif View Image: s.001.050.gif View Image: s.001.055.gif View Image: s.001.060.gif View Image: s.001.065.gif View Image: s.001.070.gif View Image: s.001.075.gif View Image: s.001.080.gif View Image: s.001.085.gif View Image: s.001.090.gif View Image: s.001.095.gif

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


IPBLE:  Increasing Performance By Lowering Expectations.

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#6 2006-03-03 14:55:31

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,562

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!


igloo myrtilles fourmis

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#7 2006-03-03 20:11:23

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

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".


IPBLE:  Increasing Performance By Lowering Expectations.

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