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.

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.

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!

0

12.918

25.367

37.951

47.663

52.959

60.834

68.143

72.471

79.272

83.394

90

9.125

27.981

44.402

56.921

63.797

74.163

80.290

90

15.839

34.747

53.518

68.441

76.873

81.931

90

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.

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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