Park City Mathematics Institute
Secondary School Teacher Program
Summer 2007
Zome: Truncated 120-cell

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The giant Zome model pictured here is a "truncated 120-cell."


click on the image to view a larger image

A 120-cell is a four dimensional polytope related to the dodecahedron. In the same way a polyhedron has polygon faces, a 4d polytope has polyhedra "hyperfaces" -- here, 120 dodecahedra. In the first week we made a projection ("shadow") of a 120-cell in just white & warm colors.

The new object is what happens when each of the 330 nodes is replaced by a tetrahedron. Shaving down a polyhedron node gives a new polygon face; truncating a polytope node gives a new polyhedron.

There are beautiful symmetries to be seen from various points of view: 2, 3, 4, 6-fold symmetries, 'tunnels,' and much more.

Construction notes: 1260 nodes, 780 B1 struts, 800 Y1, 480 R1, and 600 RO (projects like this prompted Zome to make these shorter reds). It helps to have a 120 cell as a guide (see the book Zome Geometry). Study the nodes from the center outward, which partitions them as 330 = 20 + 20 + 30 + 60 + 60 + 60 + 20 + 60. Each node will be replaced by one of eight types of tetrahedra, detailed in step #3 of Zome: Model of the Month - April 2007 (for a different project). Make the tetrahedra first -- this will use all the nodes, so that everything else is connecting these with struts. Conveniently, the tetrahedra types are numbered 1 through 8 corresponding to their order from center to boundary. Build from the inside out. We succeeded with only a few missteps; the wonderfully engineered Zome system forces many of the decisions.


all the type 1 tetrahedra


all type 1 & 2 tetrahedra


all type 1, 2, & 3 tetrahedra


all type 1, 2, 3, and 4 tetrahedra


the completed object

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IAS/Park City Mathematics Institute is an outreach program of the Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540.
Send questions or comments to: Suzanne Alejandre and Jim King

With program support provided by Math for America

This material is based upon work supported by the National Science Foundation under Grant No. 0314808 and Grant No. ESI-0554309. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.