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 
[QUOTE=davieddy;220524](+/r,+/1,0) (permute cyclicly) icosohedron
HELP![/QUOTE] Use the duality. P.S.: And post the result :big grin: 
[quote=ccorn;220564]Use the duality.
P.S.: And post the result :big grin:[/quote] 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":smile: David PS Apologies for my French in the first post. 
[QUOTE=davieddy;220573]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.[/QUOTE] 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. 
[QUOTE=ccorn;220577]This can be scaled to [1,1,1] etc.[/QUOTE]
Which suggests that we have cubes in there. [url=http://www.chiark.greenend.org.uk/~sgtatham/polypics/dodeccubes.html]Correct.[/url] 
[QUOTE=ccorn;220577]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.[/QUOTE]
Here is it: (+/r[sup]1[/sup], +/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.) 
[quote=ccorn;220579]Here is it:
(+/r[sup]1[/sup], +/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.)[/quote] 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 
[url]http://en.wikipedia.org/wiki/Dodecahedron#Cartesian_coordinates[/url]

[QUOTE=XYYXF;220594][url]http://en.wikipedia.org/wiki/Dodecahedron#Cartesian_coordinates[/url][/QUOTE]
Well, now I can confirm that statement in Wikipedia :smile: 
[QUOTE=XYYXF;220594][url]http://en.wikipedia.org/wiki/Dodecahedron#Cartesian_coordinates[/url][/QUOTE]
I'd like to add a link to [url=http://mathworld.wolfram.com/GoldenRatio.html]Wolfram's Mathworld[/url]. Particularly interesting for me is the stuff beginning with equation (28). 
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