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-   -   Platonic solids and the Golden ratio (r) (https://www.mersenneforum.org/showthread.php?t=13578)

davieddy 2010-07-03 19:44

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

ccorn 2010-07-04 13:57

[QUOTE=davieddy;220524](+/-r,+/-1,0) (permute cyclicly) icosohedron

HELP![/QUOTE]
Use the duality.

P.S.: And post the result :big grin:

davieddy 2010-07-04 17:14

[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.

ccorn 2010-07-04 17:48

[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 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.

ccorn 2010-07-04 17:52

[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/dodec-cubes.html]Correct.[/url]

ccorn 2010-07-04 18:14

[QUOTE=ccorn;220577]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.[/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 = r-1, but the above scheme can be used with r's conjugate as well.)

davieddy 2010-07-04 19:35

[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 = r-1, 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

XYYXF 2010-07-04 21:41

[url]http://en.wikipedia.org/wiki/Dodecahedron#Cartesian_coordinates[/url]

ccorn 2010-07-04 21:48

[QUOTE=XYYXF;220594][url]http://en.wikipedia.org/wiki/Dodecahedron#Cartesian_coordinates[/url][/QUOTE]

Well, now I can confirm that statement in Wikipedia :smile:

ccorn 2010-07-04 22:21

[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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