Tetrahedral Planetoid by M.C. Escher 3D Model

All Four Faces 2023-12-22

I created original designs for the far two sides. You can download the 3D printable Blender file here This is of my imagination of what the unseen far two faces of the planetoid look like.

Tetrahedral Planetoid Completed by dudecon on Sketchfab

3D Print

I had a color SLS 3D print made of the tetrahedral planetoid, here are some photographs.
Tetrahedral Planetoid 3D print in bell jar. Novel bridge face on the left, classical palm face on the right.

Despite thickening all of the thin features for printing, a few of them were still too delicate and broke off. The center spire of the pagoda is one example, with the broken stub seen on the left.
Tetrahedral Planetoid 3D print in bell jar. Classical pagoda face on the left, novel bridge face on the right.

Here is an edge-on view of one of the novel faces I invented for this print. I could have re-used the visible faces from Escher's woodcut, but chose to concept and model two totally novel faces instead.
Tetrahedral Planetoid 3D print. Novel Waterfall Face angled view.

Here's a macro close-up of the waterfall face. You can clearly see the individual grains of nylon which were laser-sintered to create this print.
Tetrahedral Planetoid 3D print. Novel Waterfall Face angled closeup.

Here's the bridge arch edge between the two classical faces (the palm face and the pagoda face). While it is at the center of the original woodcut, the arch itself was not shown, so I came up with something plausible. I kind of want to put a water-gate in the side of the arched tunnel. Perhaps at some future time. Or some other artist.
Tetrahedral Planetoid 3D print. Classical Palm Face closeup.

A good overview of the waterfall face. I had originally intended to cover this side with waterfalls and channels, but the design ended up with just one, seen on the right. The hole in the center also has 5 main waterfalls and 30 smaller ones, but I left them out so the hollow center could have a sufficiently sized access hole.
Tetrahedral Planetoid 3D print. Novel Waterfall Face closeup.
I had the 3D model up for purchased 3D prints on Shapeways for almost a year before they declared bankruptcy. I never sold a single one, so I probably won't go through the trouble again, but I still think they would make a really cool desk ornament.

Development Process

These are some sequential images from my project to make a 3d model based directly from the picture "Tetrahedral Planetoid" by M.C. Escher. It took a long time, so I hope you enjoy the results as much as I have.

The process of creating the geometry was much more grueling than I anticipated. Getting all of the odd faces to match up, while maintaining the beautiful regularity of the model has proven extremely challenging. Even thinking about modeling some of the shapes has taken a while, and I have yet to develop a consistent mode of thought which can adequately address all of the issues I've encountered. I encourage anyone wishing to stretch their spacial perception and 3d modeling skills to tackle this simple yet elusive object.

This is the 3D model of Escher's original vision only. The link to the Blender file of the complete planetoid with the far two sides designed and modeled by me, can be found above at the top of the page.
If you'd like to download the model, here is the Blender file, complete with five previous revisions of the model. Choose between .7z (1.3MB) and .zip (4.2MB)
Tetrahedral Planetoid 3D model, Blender format, 7z archive
Tetrahedral Planetoid 3D model, Blender format, zip archive

If you have comments or feedback, feel free to leave it here. If you'd like to commission a custom 3D model, I can help you with that.

Tetrahedral Planetoid by dudecon on Sketchfab