Here [PDF] is a preprint of a paper I prepared from a 20 page handwritten manuscript I found in Klarner's files entitled "Satterfield's Tomb."
Satterfield's Tomb is closely related to the geometry of 20 cannonballs stacked in a four layer pyramid inside an enclosing and tangent regular tetrahedron. Satterfield Tomb Animals are certain connected unions of five Voronoi cells of the tomb, each cell enclosing a cannonball. Klarner and Wade Satterfield were interested in which tomb animals could be used to pack the tomb if you're allowed to make copies of them (or their mirror images).
Click and drag it to see the animal perform tricks!
The Little Bear.
Four copies of this five-celled animal will pack the tomb in three non-isomorphic ways.
Wade Satterfield sent me a photo of a model he and Klarner made of the tomb in the 1980s.
Here's another (larger) photo of the tomb model taken from a different viewpoint [Photo].
Wade sent some email on the same subject [Email].
Note on LiveGraphics3D images
The images on this page are Java applets that use the LiveGraphics3D package by Martin Kraus. By clicking and dragging your mouse over an image, you can rotate the object. If you release a click while dragging, you can start the object spinning about a fixed axis. And by holding down the shift key and dragging up or down, you can move the object closer or farther away.
Visualizing the Tomb
Satterfield's tomb is a regular tetrahedron dissected into twenty polyhedral cells. Each cell can be thought of as enclosing one of 20 cannonballs stacked as a four-layer pyramid.
Each cell is one four types:
The complete tomb contains 4 vertex cells, 12 edge cells, and 4 face cells. These numbers stand in the ratios 1:3:1. A Satterfield Tomb Animal is a connected subset of five cells of the tomb that contains 1 vertex cell, 3 edge cells, and 1 face cell.
In early 2002, I made paper models of the tomb cells.
We usually identify animals that are identical with respect to one of the twenty-four symmetries (rotations and reflections) of the tomb. However we also sometimes have reason to distinguish right- and left-handed versions of animals that themselves have no line of symmetry.
The Little Bear Packings
One Satterfield Tomb Animal is the Little Bear.
The Little Bear.
Four identical copies of the little bear will pack Satterfield's Tomb in two non-isomorphic ways. And if we allow ourselves to use both right- and left-handed versions of the Little Bear, then there are 3 non-isomorphic ways the bear will pack the tomb.
To visualize these packings, it's useful to consider the following subset S of 10 cells.
A subset S of 10 cells isomorphic to its complement S' in Satterfield's Tomb T
The subset S can be decomposed as a union of two little bears in two different ways (stare at it!). In fact, either two "left-handed" or two "right-handed" bears can be used to pack both S and its (isomorphic) complement S'. Suppose first we limit ourselves to one handedness category in packing S and S'. Then the resulting packings of T are always identical under one of the 24 available reflections and/or rotations of the tomb (ie, the same isomorphism class of packings is obtained). But if we choose to pack S with right-handed bears, and S' with left-handed bears (or the reverse), then a second, non-isomorphic packing is obtained.
But there is still one more way to pack the tomb with four little bears!
Another self-complement subset S'' of ten cells packable by two little bears.
If you're good at visualizing this stuff, you'll see why S'' only adds one more isomorphism class of little bear packings to the ones described already.
Counting Satterfield Tomb Animals
The little bear is just one Satterfield Tomb animal. Klarner and Satterfield's original paper shows that there are nineteen distinct Satterfield tomb animals if we treat as identical two animals that can be superimposed by rotating or reflecting the tomb tetrahedron.
You can find pictures of all nineteen animals in the original paper (link at the top of the page). Here are some highlights.
It packs the tomb in two distinct ways.
It also packs the tomb in two distinct ways.
It packs the tomb in one way only.