Platonic Solid Vertices

by WilliamAAdams, published

Platonic Solid Vertices by WilliamAAdams Jun 16, 2011
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I always have need for a good/better vertex to construct space frames.

This derivative thing just makes a few 'improvements' to the original design.

First of all, I've modularized the code somewhat. This doesn't make any particular improvement, but it will make it easier to see what's going on, and how to add to it in the future when more vertices are needed.

Second, being the slave to phi (1.618) that I am, I changed things like the thickness of the connector to be proportional to the radius of the rods that you're using (instead of a fixed value).

I've also made a change such that the length of the connector is automatically calculated proportional to the length of the rods that you're using. The base size is 10mm, and it grows by an appropriately 'phi' influenced length. So, the ones for a 36" rod, for example, are closer to about 20mm, rather than 10mm.

With these changes, it becomes rather trivial to make good new connectors for any size, by just changing the single diameter value, and the rod length.

I've played with a lot of vertex designs over the past few months. This is one of the most compact, and easy to utilize. I give props to Sjoerd de Jong for the simplicity of the design.

In the picture, the struts are 36" long. The overall height of the structure is about 8'. With this design, the fit is strong enough that you can actually assemble the thing with one person. If you've ever done dome development, you might recognize this is a 'good thing'.


1) Download the .scad file, and choose which vertex type you want to construct.
2) Put in the appropriate rod radius (in millimeters)
3) Print out enough vertices to achieve the job
4) Slip rods into the appropriate places
5) Rejoice!

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I just made something similar where there connectors and the bars are printed for an icosahedron.

Thanks you William for your kind reply. My "little" problem is that first, I don't know at all how to use OpenScad for 2 kind of angles : I shall try to understand it ! My second problem is a conception problem : One cubooctaedral cage is good, but I must think to join in 3 dimensions X, Y, Z some adjacent cages, and like in cristallography, it would be fine to prepare for each summit (atoms ) not 4 links (chemical bonds) but to
prepare more (8 links ? ) and I have to concieve the mathematical connection of the translationnal cells. It is very funny to vizualize molecular sieves to separate and/or modify chemical molecules... In Art, I like also "Piranese", and these cages look like jails ... for Plutonium atoms !
(Faujasite or NaY zeolites).

The vertices for Platonic solids are a very good concept for spherical symmetries of small molecules Ih like Pt13 etc... It would be very fine to have the same type of vertice connectors also for crystallographic (translationnal ) symmetries. I think to CuboOctahedra connectors with 3D angles of 60, 90, 60, 90 degres for each (4) bindings of the polyhedron summit.. I like the hole in the center of the connector which permits to screw a diameter to the origin of the centrosymmetric solid. This cell can be repeated to build up a complete metal structure (Nickel, Palladium, Platinum and so on !). I shall be enjoyed to find the same type of building up with balls and rods for Zeolithe solids (Faujasite).

I got into 3D printing originally so that I could make such thing with my daughter. Originally I wanted to use drinking straws to make connections, but rods turned out to be much more useful.

Going for other geometries should be a fairly easy task using the OpenScad code here. It's fairly modular for that. If you know your angles, just plug them in and you get a new connector.

I also like the hole in the middle. In my case, it gives a place to tie in bunny cords.

Would this be able to do Archimedean solid's like the cube octahedron? any idea how I could modify your code? I'm confused a bit by this line
"acos((1 1 + 1 1 + 1 0) / (sqrt(3)sqrt(2)))"

Any help would be great.

Well, I haven't looked in a few years, but I can imagine I was trying to generalize the equation, and didn't use variable names, but hard coded the values instead. I'm not sure right off hand, I'll have to look at the entirety of the code and reconstruct my thoughts on the matter. It might actually happen soon as I wanted to modify one of these vertices for a new purpose.

really beautiful work; need to print out a set of these! i like how you integrate the golden ratio into your designs for pleasing proportions. beyond phi, some other good ones i know of are 1.414
amp; 1.732

This design works fairly well. With that big one in the picture, the oak dowels have enough bend that the thing is 'bouncy', but it does not fall apart.

If you had some bracing strut/chord thing, it would become very stable.

At smaller sizes, and using aluminum or carbone fiber, I'm sure it would be very stiff, if still weak from a vertical loading perspective.

Thanks William for this great update. You really enhanced the scad file; now I can even understand my own code ;)

Maybe it's important to say that, except for the tetrahedron connectors, it's easiest to print the connectors upside down.

I did some math on connectors for http://en.wikipedia.org/wiki/Archimedean_solidhttp://en.wikipedia.org/wiki/A.... The angles for these connectors are a lot more complicated though. Can you figure out how to make a http://en.wikipedia.org/wiki/Truncated_tetrahedronhttp://en.wikipedia.org/wiki/T... for instance? It's a nice math challenge! O:-)

I think I need a good book on doing vertex math for solids.

My non-math solution thus far has been to construct simple vertices which have the ability to rotate their joints. That saves my brain from having to steam up.

One thing I really like about your design work is that you did in fact do the maths. I was thinking it would be good to add in the '2v'
form as well, just to make structures that much more interesting.