Machine Platonic shapes

[corrosionman]

I want to machine from a solid brass casting the five platonic shapes. The first three are easy but please can anyone out there offer any guidance on how to start with a 3 inch solid ball and make a icosahedron then a dodecahydron.
Any comments much appreciated.
David Whitlock.

 

[josephv]

Hello David,

I am thinking that you may want to create some planes for the faces, and then cut the ball using these planes. You will need to look into the math behind the platonic solids to see the best way to create the planes.

happy holiday!

 

[gatz]

David,
Your question intrigued me, so I did some online searching to find information about Platonic solids.

Among the sites that had pertinent info, and which turned out to be the most informative was...

http://en.wikipedia.org/wiki/Dodecahedron

(There's also an article about the Icosahedron on Wiki.)

After a while, I figure out how to construct the planes that josephv alluded to. Then, it's a matter of modifying the values of Phi, & phi and 1 which are given on those pages for a UNIT Dodecahedron and which result in a dimension of sqr(3) as the distance from any vertex to the origin.
There are very good explanations as to how the values were obtained.
The new values for the 3" sphere are decreased by the ratio of the radius of your sphere (1.500") to the sqr(3) or 1.732+; giving a factor of .866025404
Using Excel, I set these up so that I could easily copy and paste the values into the point co-ordinates in a 3-D sketch in SW.
After placing a 3" sphere in a new Part, and all 20 points (vertices) were in the 3-D sketch, I Inserted planes that connected sets of 5 points to form a pentagon, then Extrude-Cut outward from the sphere and at a 32 angle outward.
(This has to do with the "dihedral angle" that is formed between any adjacent interior faces of the pentagons.)
It may be easier to just use the Plane for cutting; however, see below.
By the way, none of the Cartesian coordinates that are given actually gives the 1.5" dimension in Top, Front, or Right planes, but it is in fact, an indirect result of placing the coordinates given, and can be varified before proceeding with all the vertices.

One thing that became obvious, is that only 4 adjacent pentagons were actually needed; just have establish an Axis and Circular Pattern them around it. See pic.

Also, how do you plan on machining these? Looks like it might be a challenge.

Good luck, Gatz

 

[gatz]

Here is the Icosahedron (from a 3" sphere) which proved to be much more involved than the Dodecahedron in solving for the 3-D sketch points. Again, I set up some formulae in Excel and copy/pasted the values into the coordinate boxes when making the 3-D sketch.
For this model, I constructed Planes by connecting sets of adjacent 3 Points and then did a Cut From Surface.
This also only required 4 Cuts, then a Pattern/Circular of 5 equal instances about Axis1

Gatz