Have you heard of an omnitruncated dodecaplex?

On January 25th 2019, Benoit Charbonneau and a team of volunteers attempted to construct a three-dimensional shadow of the omnitruncated dodecaplex, a four-dimensional geometric shape, built using Zometool pieces. Over the course of 14 hours, volunteers built and supported the prototype structure, which was ultimately disassembled as it could not support its own weight. Today, a projection of a dodecaplex, the non-truncated version of the shape, hangs in the atrium in M3.

The polytope barn-raising project will return! Stay tuned for future announcements on this project, so you can volunteer to help build the second version of the omnitruncated dodecaplex. Sign up for the mailing list to receive an email when a date for the rebuild has been set.

14,400 vertices
28,800 edges
17,040 polygonal faces(10.800 squares, 4,800 hexagons, 1,440 decagons)
2,640 cells(120 great rhombicosidodecahedra, 720 decagonal prisms, 1200 hexagonal prisms, and 600 truncated octahedra)
21,360Zometool parts (7,200 balls and 14,160 struts)
35 kgtotal weight

What is a Omnitruncated Dodecaplex?

A three-dimensional object can be seen in two dimensions by shining a light at it and looking at the shadow it casts on a surface behind it. More generally, this process, known as projectioncan be used to view an n-dimensional object in any lower dimension. The omnitruncated dodecaplex is a 4-dimensional solid, and what is being constructed is a three-dimensional shadow of it obtained by projecting through a light source placed "infinitely" far away.

The omnitruncated dodecaplex possesses symmetries through an arrangement of four mirrors in four-dimensional space that form a kaleidoscope generating it. Moreover, it is the largest four-dimensional solid possessing this property. The name comes from the fact that it may also be viewed as a uniform truncation of all the faces of a dodecaplex, the four-dimensional analogue of the dodecahedron. The dodecahedron and dodecaplex have no analogue in five or higher dimensions, making the omnitruncated dodecaplex a remarkably complex and beautiful object.

omnitruncated dodecaplex projects

The projection we will construct this object in is orthogonal projection of its skeleton through a rhombicosidodecahedral cell. This projection reveals a four-, six-, and ten-fold axes of symmetry in the construction, and this is the most symmetric projection that can be constructed in the ZomeTool system being used.

While more symmetric constructions cannot be physically built, it is possible to create computer images of them. Below is an image of the omnitruncated dodecaplex in a Coxeter plane, generated using a Maple package built to work with objects like the omnitruncated dodecaplex.

What goes into the model?

As a four-dimensional object, the "faces" of an omnitruncated dodecaplex are three-dimensional elements called cells, and there are a total of four kinds: hexagonal prisms, decagonal prisms, truncated octahedra, and rhombicosidodecahedra.But projection does not treat all of these faces equally: only a single rhombicosidodecahedron will be faithfully represented in the model, and each other cell will be deformed in some way, to ensure that everything fits in three dimensional space. In total, there are around 30 different cell shapes after projection. In fact, a number of cells will be squished all the way down into two dimensions!

Take a look at what the projected cells look like below. Each cell is given with its angle in degrees from the light source; notice that as this grows larger, the cell becomes more "squished", with an angle of 90 degrees corresponding to being perfectly flat, and an angle of 0 degrees meaning no distortion at all. Think about how this relates to shadows of 3-dimensional objects!

Cell 00

Cell 312.918
Cell 525.367
Cell 937.951
Cell 1247.663
Cell 1452.959
Cell 1860.834
Cell 2268.143
Cell 2372.471
Cell 2879.272
Cell 3083.394
Cell 3390

Cell 29.125
Cell 627.981
Cell 1144.402
Cell 1756.921
Cell 1963.797
Cell 2574.163
Cell 2780.290
Cell 3190
Cell 415.839
Cell 834.747
Cell 1553.518
Cell 2168.441
Cell 2676.873
Cell 2981.931
Cell 3290

How is it being built?

The model will be constructed in the ZomeTool system, and suspended with string.

ZomeTool is a plastic building system constructed about the three-dimensional icosahedral symmetry system, making it the ideal tool to construct the omnitruncated dodecaplex, which has this symmetry.

Stay tuned!

Currently, Professor Charbonneau and his team are planning the second build of the omnitruncated dodecaplex. When it is ready, they will need 30-40 volunteers to help with the build in 30- or 60-minute intervals.

After we complete the build, once completed, the model rise to its permanent spot - suspended from the ceiling in the M3 Atrium.

If you're interested in volunteering, sign up for the mailing list to receive an email when a date has been set.



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