Designs that have relatively few degrees of freedom have always been attractive to me. Perhaps one of the most rewarding aspects of designing the truss model of this illusion is that it almost completely designs itself. There are very few arbitrary choices for the engineer to decide. In fact, there are exactly as many choices for the truss model of the Penrose triangle as there are for geodesic domes, which makes this triangle some sort of kindred object to the domes of the playa.
When building a geodesic dome, there are thee variables you need to decide:
- Dome diameter
- Beam type and size
- Level of subdivision
The dome diameter is continuously variable; you can make it any size you want, provided you can find beams long enough. The beam type and size is limited to your supply resources. The level of subdivision is an integer. Here is an example of the subdivision incrementing :
Images derived from the dome calculator at Desert Domes.
Similarly, for the Penrose triangle realized by a truss structure, there are three variables
- Size of triangle (continuously variable)
- Beam type and size (as supply allows)
- Level of subdivision (integer)
And here is how the subdivision increments for the triangle:
From here, the location of the vertices is fully determined if you decide that you want the arms to be made out of arcs (up to handedness). Maybe I’ll walk through how to build it in 3D in a future post.
 Note that for the purpose of illustration I’m skipping odd geodesic subdivisions, for which the typical hemisphere domes don’t get divided down the middle.