Christopher J. Kimmer, Ph.D.
IU Southeast Informatics
iSci
the informatics of scientific computing
The Inflations of T 1
The tetrahedron T 1 may be inflated five different ways. This tile has the most possible ways to inflate it because it has the largest number of small faces of any tile. There are four short edges whose subdivision must be determined upon inflation, but the three two-fold planes of symmetry in the tile reduce the number of possible ways of inflating these edges by rendering some choices of the subdivision of different short edges equivalent after reflection through a symmetry plane. Another way to state this is that since all the small faces share edges, specifying the inflation of two small faces will specify the inflation of the tile. Consequently, choose two small faces that do not share an edge, specify their inflation, and the inflation of the tile is specified. The specification of the inflations of two faces can be written as the concatenation of the inflation of each face. For instance, LSSL suggests that one face inflates LS while the other inflates SL. The possible symmetry-inequivalent ways to inflate these two faces (along with the inflation formed by instead specifying the other two faces faces in parentheses) are LLSS(LLSS), LSSL(LSSL), SSSS(LLLL), LSSS(SLLL), and SSSL(LLLS). The ordering of the concatenated pairs is insignificant; for instance, SSLL(SSLL) is related to the LLSS(LLSS) inflation by reflection through a plane. The five entries LLSS, LSSL, SSSS, LSSSS, and SSSL will be taken as the canonical ways to specify the inflation of a T 1 . Since the LLSS and LSSL inflations are the same no matter which pairs of opposite faces are taken, these inflations create tiled regions that are invariant under reflection in all three of the two-fold planes in T 1 . The LLSS inflation is detailed in Table A.1 while the LSSL inflation is listed in Table A.3 . The SSSS inflation is listed in Table A.2 , and the LSSS inflation is included in Table A.5 . Finally, the SSSL inflation is listed in Table A.4 .
| 1 | (0, 0, 0) * | 7 | (1 + 2 τ , 1 + τ , 0) * |
| 2 | (2 τ , 0, 0) | 8 | (1 + τ , 0, 1) |
| 3 | (2 τ , 1 + τ , τ ) | 9 | (1 + τ , τ , 0) |
| 4 | ( τ , - 1, 0) | 10 | (1 + τ , - τ , 0) * |
| 5 | ( τ , τ , τ ) | 11 | (1 + τ , 1 + τ , 1 + τ ) * |
| 6 | ( τ , 1, 0) | 12 | (1 + τ , 1, τ ) |
| Tile | Vertices | Tile | Vertices |
| T 1 | 3,11,5,12 | T 1 | 9,6,5,12 |
| T 1 | 9,6,8,2 | T 1 | 9,6,8,12 |
| T 1 | 10,4,8,2 | O 1 | 1,4,8,2 |
| O 1 | 1,6,5,12 | O 1 | 1,6,8,2 |
| O 1 | 1,6,8,12 | O 1 | 7,3,12,5 |
| O 1 | 7,9,2,8 | O 1 | 7,9,12,5 |
| O 1 | 7,9,12,8 |
| 1 | (0, 0, 0) * | 7 | (1 + τ , τ , 0) |
| 2 | (2 τ , 0, 0) | 8 | (1 + τ , - τ , 0) * |
| 3 | (2 τ , 1 + τ , τ ) | 9 | (1 + τ , 1 + τ , 1 + τ ) * |
| 4 | ( τ , 1, 0) | 10 | (1 + τ , 1, τ ) |
| 5 | (1 + 2 τ , 1 + τ , 0) * | 11 | (1, 1, 1) |
| 6 | (1 + τ , 0, 1) | 12 | (1, 1 - τ , 0) |
| Tile | Vertices | Tile | Vertices |
| T 1 | 1,12,4,11 | T 1 | 6,2,7,4 |
| T 1 | 6,10,7,4 | T 1 | 11,4,6,10 |
| O 1 | 5,7,2,6 | O 1 | 5,7,10,6 |
| O 1 | 8,6,4,11 | O 1 | 8,12,4,11 |
| O 1 | 9,10,11,4 | T 2 | 4,2,6,8 |
| O 2 | 10,4,7,5 | O 2 | 10,9,3,5 |
| 1 | (0, 0, 0) * | 7 | (1 + τ , τ , 0) |
| 2 | (2 τ , 1 + τ , τ ) | 8 | (1 + τ , - τ , 0) * |
| 3 | ( τ , - 1, 0) | 9 | (1 + τ , 1 + τ , 1 + τ ) * |
| 4 | ( τ , 1, 0) | 10 | (1 + τ , 1, τ ) |
| 5 | (1 + 2 τ , 1 + τ , 0) * | 11 | (1, 1, 1) |
| 6 | (1 + τ , 0, 1) | 12 | (2 + τ , 1, 0) |
| Tile | Vertices | Tile | Vertices |
| T 1 | 4,7,10,6 | T 1 | 6,10,7,12 |
| T 1 | 11,4,6,10 | O 1 | 8,6,12,7 |
| O 1 | 9,10,11,4 | T 2 | 1,4,11,6 |
| T 2 | 10,12,7,5 | O 2 | 6,7,4,1 |
| O 2 | 6,8,3,1 | O 2 | 10,4,7,5 |
| O 2 | 10,9,2,5 |
| 1 | (0, 0, 0) * | 7 | (1 + τ , - τ , 0) * |
| 2 | (2 τ , 1 + τ , τ ) | 8 | (1 + τ , 1 + τ , 1 + τ ) * |
| 3 | ( τ , 1, 0) | 9 | (1 + τ , 1, τ ) |
| 4 | (1 + 2 τ , 1 + τ , 0) * | 10 | (1, 1 - τ , 0) |
| 5 | (1 + τ , 0, 1) | 11 | (2 + τ , 1, 0) |
| 6 | (1 + τ , τ , 0) | 12 | (2 - τ , 2 - τ , 2 - τ ) |
| Tile | Vertices | Tile | Vertices |
| T 1 | 1,10,3,12 | T 1 | 3,6,9,5 |
| T 1 | 5,9,6,11 | T 1 | 12,3,5,9 |
| O 1 | 7,5,3,6 | O 1 | 7,5,3,12 |
| O 1 | 7,5,11,6 | O 1 | 7,10,3,12 |
| O 1 | 8,9,12,3 | T 2 | 9,11,6,4 |
| O 2 | 9,3,6,4 | O 2 | 9,8,2,4 |
| 1 | (0, 0, 0) * | 7 | (1 + τ , - τ , 0) * |
| 2 | (2 τ , 0, 0) | 8 | (1 + τ , 1 + τ , 1 + τ ) * |
| 3 | ( τ , 1, 0) | 9 | (1 + τ , 1, τ ) |
| 4 | (1 + 2 τ , 1 + τ , 0) * | 10 | (1, 1, 1) |
| 5 | (1 + τ , 0, 1) | 11 | (1, 1 - τ , 0) |
| 6 | (1 + τ , τ , 0) | 12 | (2 + τ , 1 + τ , 1) |
| Tile | Vertices | Tile | Vertices |
| T 1 | 1,11,3,10 | T 1 | 3,10,9,5 |
| T 1 | 6,3,5,2 | T 1 | 6,3,5,9 |
| O 1 | 4,6,2,5 | O 1 | 7,5,3,10 |
| O 1 | 7,11,3,10 | O 1 | 8,9,6,3 |
| O 1 | 8,9,10,3 | T 2 | 7,5,2,3 |
| O 2 | 6,4,12,8 | O 2 | 6,5,9,8 |
The only tile in common to all of these inflated T 1 tetrahedra is a lone T 1 tile in the middle of the inflation. Its vertices are determined by the inflation of the long 5-fold edges L → LSL, and this rule applied to the two long edges creates the 4 vertices and the two short 5-fold edges of the smallest tetrahedron. Other inflations of T 1 share common tiled regions, and these volumes and their tilings are related to each other through the basic tile rearrangements described in Chapter 3.
Another way to characterize these tetrahedra after inflation is by considering how the three two-fold symmetry planes of the tile are broken by the choice of inflation rule for the short edges. For the two symmetric tiles, LLSS and LSSL, there is complete symmetry. The other three tiles break this symmetry along a short 5-fold edge (SSSL), a short 3-fold edge (SSSS), or along both edges (LSSS), so that in this sense, the LSSS inflation is the least symmetric of the five.
These inflated tiles may be produced from a single inflated T 1 of arbitrary type by observing the inflation matrix entries in Table 3.2 . If the entries are the same, the tile conserving flip is of the kind used to create a vertex jump along either a 3-fold or 5-fold edge, as necessary. If the tile numbers change, then the flip is the basic (1,-1,3,-2) flip described in Chapter 3.
Chris Kimmer 2011-06-01