mersenneforum.org Platonic solids and the Golden ratio (r)
 Register FAQ Search Today's Posts Mark Forums Read

 2010-07-03, 19:44 #1 davieddy     "Lucan" Dec 2006 England 2×3×13×83 Posts Platonic solids and the Golden ratio (r) As an erstwhile 3D-engine programmer (among other things) I read a beautiful article about simple starting points for the vertices of the five regular solids. Before my video(visio?)spatial ability goes completely bonkers let me try to remember them: (1,1,1), (1,-1,-1), (-1,1,-1), (-1,-1,1) tetrahedron (+/-1,+/-1,+/-1) cube (+/-1,0,0) etc octahedron (+/-r,+/-1,0) (permute cyclicly) icosohedron HELP! David Last fiddled with by Prime95 on 2010-07-03 at 21:40 Reason: Watch the language please
2010-07-04, 13:57   #2
ccorn

Apr 2010

32×17 Posts

Quote:
 Originally Posted by davieddy (+/-r,+/-1,0) (permute cyclicly) icosohedron HELP!
Use the duality.

P.S.: And post the result

2010-07-04, 17:14   #3
davieddy

"Lucan"
Dec 2006
England

145128 Posts

Quote:
 Originally Posted by ccorn Use the duality. P.S.: And post the result
I thought of that, but I'm not sure that taking
the centre of the 20 triangles gives the simplest
orientation for the vertices of the dodecahedron.

This is "puzzles" - not "homework help"

David

PS Apologies for my French in the first post.

2010-07-04, 17:48   #4
ccorn

Apr 2010

32×17 Posts

Quote:
 Originally Posted by davieddy I thought of that, but I'm not sure that taking the centre of the 20 triangles gives the simplest orientation for the vertices of the dodecahedron.
I get face-center coordinates such as [s,s,s] and [s,-s,s] with s = (1+r)/3. This can be scaled to [1,1,1] etc. Should be simple enough. Will follow up.

2010-07-04, 17:52   #5
ccorn

Apr 2010

32×17 Posts

Quote:
 Originally Posted by ccorn This can be scaled to [1,1,1] etc.
Which suggests that we have cubes in there. Correct.

2010-07-04, 18:14   #6
ccorn

Apr 2010

32·17 Posts

Quote:
 Originally Posted by ccorn I get face-center coordinates such as [s,s,s] and [s,-s,s] with s = (1+r)/3. This can be scaled to [1,1,1] etc. Should be simple enough. Will follow up.
Here is it:

(+/-r-1, +/-r, 0) (permute cyclicly), (+/-1,+/-1,+/-1) dodecahedron

(I have used r = (1+sqrt(5))/2, hence 1/r = r-1, but the above scheme can be used with r's conjugate as well.)

Last fiddled with by ccorn on 2010-07-04 at 18:55 Reason: Explain r

2010-07-04, 19:35   #7
davieddy

"Lucan"
Dec 2006
England

194A16 Posts

Quote:
 Originally Posted by ccorn Here is it: (+/-r-1, +/-r, 0) (permute cyclicly), (+/-1,+/-1,+/-1) dodecahedron (I have used r = (1+sqrt(5))/2, hence 1/r = r-1, but the above scheme can be used with r's conjugate as well.)
Sounds right to me.
The edges of one are perpendicular to those of the dual.
I've just remembered why I brought this up:
World cup football!
In Mexico 1970 they first used a truncated icosohedron,
(20 white hexagons and 12 black pentagons).
Better known these days as C60 or Buckminsterfullerine.
I can't see why he found it so difficult to think of
a structure with 60 vertices.
I made one out of cardboard at the time, also
the great(?) stellated(?) dodecahedron which
makes a beautiful Christmas decoration.

David

Last fiddled with by davieddy on 2010-07-04 at 19:47

 2010-07-04, 21:41 #8 XYYXF     Jan 2005 Minsk, Belarus 24×52 Posts
2010-07-04, 21:48   #9
ccorn

Apr 2010

32×17 Posts

Quote:
 Originally Posted by XYYXF http://en.wikipedia.org/wiki/Dodecah...an_coordinates
Well, now I can confirm that statement in Wikipedia

2010-07-04, 22:21   #10
ccorn

Apr 2010

32×17 Posts

Quote:
 Originally Posted by XYYXF http://en.wikipedia.org/wiki/Dodecah...an_coordinates
I'd like to add a link to Wolfram's Mathworld. Particularly interesting for me is the stuff beginning with equation (28).

 Similar Threads Thread Thread Starter Forum Replies Last Post ONeil ONeil 12 2018-04-17 09:15 xilman Reading 2 2015-11-07 02:18 davar55 Puzzles 2 2008-02-13 03:20 mfgoode Puzzles 1 2007-01-31 16:26 mfgoode Math 28 2004-05-31 10:40

All times are UTC. The time now is 20:09.

Sun Jun 26 20:09:13 UTC 2022 up 73 days, 18:10, 1 user, load averages: 0.86, 1.04, 1.08