Shape Reference

Building Blocks

In Geometry Examples I used a standard set of blocks to build arches and spheres and so forth, and then later at the end of More Geometry Examples I introduced a compact set of blocks that could be used to do the same things. Here's a nice table from the latter that shows how the pieces correspond.

block3  diag3  ceil3
block4  diag4  ceil4   extra4

block3  half3  rtetra
block4  half4  wedge4  rpenta

The first few shapes are all completely straightforward.

• block3 is a cube and block4 is a tesseract.
• diag3 and diag4 are diagonal pieces.
• half3 and half4 are what you get when you cut block3 and block4 in half diagonally.

The remaining shapes need a bit more explanation.

• rtetra is a right (right-angled) tetrahedron with vertices at the origin and at the endpoints of the unit vectors.
• Similarly, rpenta is a right pentachoron.
• ceil3 is a ceiling piece that can be used to complete solid domes in 3D. It fits nicely onto rtetra.
• ceil4 is a prism of ceil3. It cannot be used to complete solid domes in 4D. Because there's an extra dimension, you need an extra piece.
• extra4 is the extra piece you need. It fits nicely onto rpenta.
• wedge4 is a prism of rtetra. It plays the same role in the compact set of blocks as ceil4 plays in the standard set.

These last two shapes aren't in the table, but I think they count as building blocks anyway.

• cap3 and cap4 are pyramidal caps that fit onto block3 and block4. They appear in puzzle-redcube3b and 4b, and are put to very good use in Geometry Examples #3.

Basic Geometry

These four basic shapes don't appear anywhere except here.

• tetra is a regular tetrahedron and penta is a regular pentachoron. Both have sides of length 1. I didn't do a good job of aligning them with the coordinate grid, unfortunately.
• octa is an octahedron and hexadeca is its four-dimensional analogue, the hexadecachoron.

Although those four shapes don't appear anywhere except here, two of them do show up in different orientations.

• itetra is an inscribed tetrahedron and ihexadeca is an inscribed hexadecachoron. They appear in puzzle-redcube3c and 4c. See "Fun Geometry Facts" below for more information about how and where they're inscribed.

Here are two more basic shapes.

• dodeca is a regular dodecahedron. I only used it to make the four-dimensional trees in the geom-tree examples.
• cubocta is a cuboctahedron. It's not a regular shape because the faces aren't all the same, but it's still a very nice shape. You can construct it by truncating a cube or an octahedron. It appears in puzzle-redcube3d, and is also used as the cross section of the four-dimensional tank cars in e.g. train-cars4.

Advanced Geometry

The following shapes are all discussed at length in Geometry Examples #3.

• rdodeca is a rhombic dodecahedron. It naturally breaks into four pieces of type rdpiece.
• p5shape is hard to explain, so I'll let you go read about it on that other page. It naturally breaks into five pieces of type p5piece.
• the24cell and r24cell are both 24-cells, just in different orientations. The "r" in "r24cell" stands for "rhombic", not because the faces are rhombic in any way but because the orientation is analogous to the orientation of rdodeca. See "Fun Geometry Facts" below for more information about the 24-cell.

This last shape is discussed at length in Contributed Examples.

• the120cell is a 120-cell.

Regular Polytopes

In case it helps, here's a table of all the regular polyhedra and polychora.

shape	# faces	face shape
tetrahedron	4	triangle
cube	6	square
octahedron	8	triangle
dodecahedron	12	pentagon
icosahedron	20	triangle
pentachoron	5	tetrahedron
tesseract	8	cube
hexadecachoron	16	tetrahedron
24-cell	24	octahedron
120-cell	120	dodecahedron
600-cell	600	tetrahedron

Fun Geometry Facts

If you think of the vertices and edges of a cube as forming a graph, that graph can be colored with just two colors. If you take the vertices of one color and make a shape out of them, you get an inscribed tetrahedron. The same construction in 4D gives you an inscribed hexadecachoron. There were two colors, so there are two inscribed tetrahedra and two inscribed hexadecachora. However, if you make a shape out of the face centers of a tesseract, you get … another hexadecachoron! And if you make that hexadecachoron exactly twice as large, then it's the same size as the other two, because 2² is equal to 1²+1²+1²+1². Each hexadecachoron has eight vertices. If you make a shape out of all three sets of eight at once, you get … a 24-cell! It can't be colored with two colors, but because it's made of vertices from three hexadecachora and is self-dual, it can be colored with three colors in a nice symmetrical way.