Ed Pegg Jr., December 18, 2006

A recent column by Ivars Petersen, *Quark
Park*, showed various pictures
of a 4-dimensional sculpture by Marc Pelletier. This sculpture, a 120-cell,
was dedicated to mathematician John Conway. Below is a photo from the event. The
120-cell is to the dodecahedron as a *tesseract* (MathWorld, Wikipedia, Cut
the Knot, Geometry
Center) is to a cube -- it's the 4-D extension. 4 lines make a square,
6 squares make a cube, 8 cubes make a tesseract. 120 dodecahedra make a 120-cell.

Figure 1. John Conway reaches into a 4-dimensional object, the 120-cell, as Marc
Pelletier watches.

Marc's history with the 120-cell goes back to when he was 17 years
old, when he discovered that an early version of the Zome
construction system could
make an accurate model of the 120-cell. While still a teenager, he co-founded Zome
with Paul Hildebrandt. His sculpture above was partially inspired by Paul Donchian
(1895-1967), a dealer of oriental carpets who provided all of the wire frame
models for H. S. M. Coxeter's 1949 book *Regular Polytopes*.

Figure 2. Donchian's 1940-era display with a 120-cell as a centerpiece.

Another exploration program by Roice Nelson allows you to try many different visualizations of the 120-cell. Below, one of the many visualizations peels away some of the dodecahedra. Each dodecahedron touches exactly 12 others. Four dodecahedra meet at each corner.

Figure 3. From the
*Explore the 120-Cell* program.

Another 120-Cell visualizer is the beautiful Jenn3D
program by
Fritz Obermeyer and Willy Winkel, with subtitle "For visualizing
Coxeter Polytopes." According to Jenn's *Catalog of Uniform
Polytopes*, the Coxeter notation for the dodecahedron is ,
for the icosahedron is ,
and for the 120-cell is .
Jenn allows a full exploration of Coxeter notations.

Figure 4. Coxeter's 120-Cell ()
as rendered by
Jenn3D.

The 120-Cell has been shown on TV.
In this scene from *Numb3rs* episode "The OG," actor Peter MacNicol is holding
a metal 120-cell made by Bathsheba
Grossman. The original design is by George
Hart.

Figure 5. A 120-cell by Bathsheba
Grossman, as shown on *Numb3rs*.

A viewer of the slices of 3-D stellations
of the 120-cell is available as a Java applet at Mark Newbold's *Hyperspace
Star Polytope Slicer*. One image from shell 15 of the stellation is shown
below.

Figure 6. A polytope slice of
a stellation of a 120-cell, by Mark Newbold.

So far, I haven't ventured into hyperbolic space. The PoincarĂ©
disk; also called the Lobachevskian, L2, or hyperbolic plane; allows either
4 pentagons or 5 quadrilaterals to meet
at a corner. These images were made with *Mathematica*, with assistance from
notebook "Tessellations
of the Euclidean, Elliptic and Hyperbolic Plane."

Figure 7. Hyperbolic tilings with pentagons and quadrilaterals.

When hyperbolic tilings are brought into the next dimension, 3-manifolds
are the result. The program *Curved
Spaces 3* by Jeff Weeks allows the visualization
of what happens when 8 dodecahedra meet at each corner. Notice that hyperbolic
pentagon tiling above in figure 7 is repeated in figure 8 to make curved planes.
Another view is available as a part of *Not Knot*.

Figure 8. Hyperbolic dodecahedra, as seen by *Curved Spaces
3*.

Plato (*Timaeus* 55), after describing the other four Platonic
solids, said "There still remained a fifth construction, which the god
used for embroidering the constellations on the whole heaven." Other philosophers
assigned the dodecahedron to the cosmos. I wonder how Plato would have liked the
visualization programs I mention above.

George Hart, "4D Polytopes and 3D Models of Them," http://www.cs.sunysb.edu/~cse125/notes/08-4D-Forms.ppt.

George Hart, "4D Polytope Projection Models by 3D Printing," May 3, 2002. http://www.georgehart.com/hyperspace/hart-120-cell.html.

Paul Hildebrandt, "Zome-inspired Sculpture," http://www.lkl.ac.uk/bridges/Zome-Hildebrandt.pdf.

Roice Nelson, "Explore the 120-Cell," July 1, 2006. http://www.gravitation3d.com/120cell/.

Mark Newbold, "Hyperspace Star Polytope Slicer," Feb 1, 2003. http://www.dogfeathers.com/java/hyperstar.html.

Fritz Obermeyer, "Jenn 3d," http://www.math.cmu.edu/~fho/jenn/.

Marc Pelletier, "Paul Donchian, Modeler of Higher Dimensions," Fields Institute Presentation, February 15, 2002. http://www.fields.utoronto.ca/audio/01-02/sculpture/pelletier/.

Ivars Petersen, "Math Trek: Quark Park," Nov 11, 2006. http://www.sciencenews.org/articles/20061111/mathtrek.asp.

Miodrag Sremcevic, Radmila Sazdanovic, and Srdjan Vukmirovic, "Tessellations of the Euclidean, Elliptic and Hyperbolic Plane," Wolfram Information Center, March 23, 2003. http://library.wolfram.com/infocenter/MathSource/4540/.

John Stillwell, "The Story of the 120-Cell," *Notices
of the AMS*, Jan 2001. http://www.ams.org/notices/200101/fea-stillwell.pdf.

Jeff Weeks, Curved Spaces 3, Dec 2006. http://www.geometrygames.org/CurvedSpaces/.

Andrew Weimholt, "120-Cell Foldout," http://www.weimholt.com/andrew/120.html.

Comments are welcome. Please send comments to Ed Pegg Jr. at ed@mathpuzzle.com.

Ed Pegg Jr. is the webmaster of mathpuzzle.com.
He works at Wolfram Research, Inc. as an associate editor of *MathWorld*.
He is also a math consultant for the TV show *Numb3rs*.