20100703, 19:44  #1 
"Lucan"
Dec 2006
England
2·3·13·83 Posts 
Platonic solids and the Golden ratio (r)
As an erstwhile 3Dengine 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 20100703 at 21:40 Reason: Watch the language please 
20100704, 13:57  #2 
Apr 2010
2×3×5^{2} Posts 

20100704, 17:14  #3 
"Lucan"
Dec 2006
England
2·3·13·83 Posts 
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. 
20100704, 17:48  #4 
Apr 2010
2·3·5^{2} Posts 
I get facecenter 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.

20100704, 18:14  #6  
Apr 2010
2·3·5^{2} Posts 
Quote:
(+/r^{1}, +/r, 0) (permute cyclicly), (+/1,+/1,+/1) dodecahedron (I have used r = (1+sqrt(5))/2, hence 1/r = r1, but the above scheme can be used with r's conjugate as well.) Last fiddled with by ccorn on 20100704 at 18:55 Reason: Explain r 

20100704, 19:35  #7  
"Lucan"
Dec 2006
England
1100101001010_{2} Posts 
Quote:
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 20100704 at 19:47 

20100704, 21:41  #8 
Jan 2005
Minsk, Belarus
2^{4}·5^{2} Posts 

20100704, 21:48  #9  
Apr 2010
2×3×5^{2} Posts 
Quote:


20100704, 22:21  #10  
Apr 2010
2·3·5^{2} Posts 
Quote:


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