Five Interpenetrating Tetrahedra

The definition uses the Polyhedron and Parakeet plugins, although this could be easily avoided. (27.2 KB)

The magnet holes were tight, and needed reaming with an 1/8inch drill bit, but better this way than too large. Magnets 1/8 inch cylinders, super-glued in. These need to be pressed in firmly to break the air-lock behind the magnet.



Hello @bob.h.mackay ,

Very interesting post, congrat.
A few questions :

  • I have trouble downlading required plugins. Any idea how to proceed ?
  • Maybe it does not exist for mac ?
  • did you 3D print it ?

thks in advance

Beautiful shape! I love platonic solids. The can be combined in so many ways.
Once I modeled a star from the same shape: five Interpenetrating Tetrahedra. I just modeled it as a material test to print in stainless steel:


Hi Thomas,

I believe that the Polyhedra plugin should be shown in the Package Manager. I’m not sure about Parakeet, which I think I got from food4Rhino: I see nothing on the web site about not working on a Mac, although perhaps this is the case.

That said, I am using both of these in a truly simplistic way, and both could easily be eliminated. I have them, so went the lazy way - I was not thinking about publishing!

I used the Polyhedron plugin just to make an initial tetrahedron, which is about the simplest possible thing you could ask it to do. [As a software engineer, I find it hysterical that the input argument to Polyhedron is “Tetrahedron” as a string! It will do a “Rhombic Triacontahedron” just by asking! Capital letters for extra class.]

The Parakeet plugin does some amazing patterns, but in this case I am using it just to take a line and to find the point on the line that will divide it into two parts, having the golden ratio of their lengths (1 : 1.618…). This should also be easily done using native components.

And yes, these are actual 3-d prints. I used a print service, which charged nearly $10 per piece, but if you have your own machine, it would be cheaper. Shapeways uses some sort of layered fusing technique for their basic plastic, which comes out with very high quality.

If you just want to print the pieces, here is the .stl file: FiveTetrahedra3.stl (89.8 KB)


Er, actually, I see that Parakeet is for Windows. Sorry!

OK, If I claim it to be easy, it better be! (25.3 KB)
Here is a version using only standard Grasshopper components. Let me know how it goes!


Hi @bob.h.mackay , thank-you for your post, but it is still requesting a plug-in “Parakeet”

There is a problem (I do not know if it is related) that I do not know how to solve

Thank-you for your help

Woops - I missed a second Golden Point component.

Could you try: (33.2 KB) ?

I hope that once the last component is deleted, it forgets the dependency.

Sorry about that,


got it! thks!

I like to drawing you have at the start (see below). I do not know the relation to the end product, but it is sure also a nice geometry.


The problem is the same as the sculptor trying to carve an elephant from a block of marble - you have to keep knocking off all the bits that don’t look like an elephant!

That shape is the surface of the major tetrahedra. Some parts of the tetrahedra are ‘inside’ other tetrahedra; some parts can be seen. I need to find the points where the visible parts vanish inside. These are then projected to the centre, to form nine invisible faces where one eventual corner piece will touch its neighbours.

Unfortunately, the positions of all these points are real numbers, related to Tau, the diagonal of a pentagon, since ultimately these five tetrahedra live inside a dodecahedron. There are actually ten ways to put a tetrahedron inside a dodecahedron, but this is only half of them.

Exercise for the reader: there are also ten ways to put a cube inside a dodecahedron. What could we do with that?



“real” → “irrational”, I guess!