Christopher J. Kimmer, Ph.D.
IU Southeast Informatics
iSci
the informatics of scientific computing
The Inflations of O 1
The situation for the inflation of the O 1 tile is not as complicated. It has one small face, so there are four ways to inflate it. They may be written as SL, LS, LL, and SS. Table A.6 lists the SL case. The LS data are detailed in Table A.7 . Table A.8 contains data for the LL case, and the final case is listed in Table A.9 .
The
O
1
LL and LS tile are related by tile-conserving rearrangements.
LL
LS via the rearrangement occurring in the volume
congruent to the
T
2
tile scaled by
τ
discussed in Chapter 3. In
this case, there are two of these packed regions that share the inflated
3-fold edge, so there are actually two of these rearrangements occurring.
All rearrangements of the form xL
xS in
O
2
have one
of these tile-conserving 3-fold edge rearrangements, but when the 5-fold
edge has been inflated L, there are two of them. The SL tile has the same
inflation matrix entries (Table
3.2
), and it is formed
via the tile-conserving rearrangement along a 5-fold direction where an
O
1
and 2
O
2
s flip. The SL tile can be reached from the LS tile by
first a rearrangement along the 3-fold edge and then one along the 5-fold
edge.
The
O
1
SS tile has a different inflation matrix entry, so it is related by
tile flips that do not conserve tile number. The flip is the basic
(1,-1,3,-2) flip that makes these tilings conserve density. This flip by itself
connects SS
LS, and SS is related to the other
O
1
tiles
by first flipping their tiles to form the LS tile as related in the
previous paragraph.
1 | (0, 0, 0) * | 9 | (1 + 2 τ , 1 + 2 τ , 1) |
2 | (2 τ , 1 + 2 τ , 1 + τ ) | 10 | (1 + 2 τ , 1 + τ , 0) * |
3 | (2 τ , 1 + τ , τ ) | 11 | (1 + 2 τ , 2 + 3 τ , 1 + τ ) * |
4 | ( τ , τ , τ ) | 12 | (1 + 2 τ , 2 + 2 τ , τ ) |
5 | ( τ , 1 + τ , 1) | 13 | (1 + 2 τ , 2 + 2 τ , 2 + τ ) |
6 | ( τ , 1, 0) | 14 | (1 + τ , τ , 0) |
7 | (1 + 2 τ , 2 τ , τ ) | 15 | (1 + τ , 1 + 2 τ , τ ) |
8 | (1 + 2 τ , 1 + 2 τ , 1 + 2 τ ) * | 16 | (1 + τ , 1 + τ , 1 + τ ) |
Tile | Vertices | Tile | Vertices |
T 1 | 2,15,3,7 | T 1 | 2,15,3,16 |
T 1 | 2,15,9,7 | T 1 | 2,15,9,12 |
T 1 | 4,5,3,16 | T 1 | 4,5,14,6 |
O 1 | 1,6,4,5 | O 1 | 8,2,7,3 |
O 1 | 8,2,7,9 | O 1 | 8,2,16,3 |
O 1 | 10,3,5,4 | O 1 | 10,14,5,4 |
O 1 | 11,12,2,15 | T 2 | 15,7,3,10 |
T 2 | 15,7,9,10 | O 2 | 2,8,13,11 |
O 2 | 2,9,12,11 | O 2 | 5,16,3,10 |
O 2 | 15,16,3,10 |
1 | (0, 0, 0) * | 9 | (1 + 2 τ , 1 + 2 τ , 1) |
2 | (2 τ , 1 + 2 τ , 1 + τ ) | 10 | (1 + 2 τ , 1 + τ , 0) * |
3 | (2 τ , 1 + τ , τ ) | 11 | (1 + 2 τ , 2 + 3 τ , 1 + τ ) * |
4 | ( τ , τ , τ ) | 12 | (1 + 2 τ , 2 + 2 τ , τ ) |
5 | ( τ , 1 + τ , 1) | 13 | (1 + 2 τ , 2 + τ , 1 + τ ) |
6 | ( τ , 1, 0) | 14 | (1 + τ , τ , 0) |
7 | (1 + 2 τ , 1 + 3 τ , 2 τ ) | 15 | (1 + τ , 1 + 2 τ , τ ) |
8 | (1 + 2 τ , 1 + 2 τ , 1 + 2 τ ) * | 16 | (1 + τ , 1 + τ , 1 + τ ) |
Tile | Vertices | Tile | Vertices |
T 1 | 2,15,3,16 | T 1 | 2,15,9,12 |
T 1 | 4,5,3,16 | T 1 | 4,5,14,6 |
T 1 | 9,12,2,13 | T 1 | 16,3,13,2 |
O 1 | 1,6,4,5 | O 1 | 10,3,5,4 |
O 1 | 10,3,13,2 | O 1 | 10,3,15,2 |
O 1 | 10,9,13,2 | O 1 | 10,9,15,2 |
O 1 | 10,14,5,4 | T 2 | 12,13,2,8 |
T 2 | 16,13,2,8 | O 2 | 5,16,3,10 |
O 2 | 12,11,7,8 | O 2 | 12,15,2,8 |
O 2 | 15,16,3,10 |
1 | (0, 0, 0) * | 9 | (1 + 2 τ , 1 + 2 τ , 1 + 2 τ ) * |
2 | (2 τ , 1 + 2 τ , 1 + τ ) | 10 | (1 + 2 τ , 1 + 2 τ , 1) |
3 | (2 τ , 1 + τ , τ ) | 11 | (1 + 2 τ , 1 + τ , 0) * |
4 | ( τ , τ , τ ) | 12 | (1 + 2 τ , 2 + 3 τ , 1 + τ ) * |
5 | ( τ , 1 + τ , 1) | 13 | (1 + 2 τ , 2 + 2 τ , τ ) |
6 | ( τ , 1, 0) | 14 | (1 + τ , τ , 0) |
7 | (1 + 2 τ , 2 τ , τ ) | 15 | (1 + τ , 1 + 2 τ , τ ) |
8 | (1 + 2 τ , 1 + 3 τ , 2 τ ) | 16 | (1 + τ , 1 + τ , 1 + τ ) |
Tile | Vertices | Tile | Vertices |
T 1 | 2,15,3,7 | T 1 | 2,15,3,16 |
T 1 | 2,15,10,7 | T 1 | 4,5,3,16 |
T 1 | 4,5,14,6 | T 1 | 2,15,10,13 |
O 1 | 9,2,7,3 | O 1 | 9,2,7,10 |
O 1 | 9,2,13,10 | O 1 | 9,2,16,3 |
O 1 | 11,3,5,4 | O 1 | 11,14,5,4 |
O 1 | 1,6,4,5 | T 2 | 15,7,10,11 |
T 2 | 15,7,3,11 | O 2 | 5,16,3,11 |
O 2 | 13,12,8,9 | O 2 | 13,15,2,9 |
O 2 | 15,16,3,11 |
1 | (0, 0, 0) * | 9 | (1 + 2 τ , 1 + τ , 0) * |
2 | (2 τ , 1 + 2 τ , 1 + τ ) | 10 | (1 + 2 τ , 2 + 3 τ , 1 + τ ) * |
3 | (2 τ , 1 + τ , τ ) | 11 | (1 + 2 τ , 2 + 2 τ , τ ) |
4 | ( τ , τ , τ ) | 12 | (1 + 2 τ , 2 + 2 τ , 2 + τ ) |
5 | ( τ , 1 + τ , 1) | 13 | (1 + 2 τ , 2 + τ , 1 + τ ) |
6 | ( τ , 1, 0) | 14 | (1 + τ , τ , 0) |
7 | (1 + 2 τ , 1 + 2 τ , 1 + 2 τ ) * | 15 | (1 + τ , 1 + 2 τ , τ ) |
8 | (1 + 2 τ , 1 + 2 τ , 1) | 16 | (1 + τ , 1 + τ , 1 + τ ) |
Tile | Vertices | Tile | Vertices |
T 1 | 6,14,5,4 | T 1 | 11,8,13,2 |
T 1 | 11,8,15,2 | T 1 | 12,7,13,2 |
T 1 | 16,3,5,4 | T 1 | 16,3,13,2 |
T 1 | 16,3,15,2 | O 1 | 1,6,4,5 |
O 1 | 9,3,5,4 | O 1 | 9,3,13,2 |
O 1 | 9,3,15,2 | O 1 | 9,8,13,2 |
O 1 | 9,8,15,2 | O 1 | 9,14,5,4 |
O 1 | 10,11,2,3 | O 1 | 10,11,2,15 |
O 1 | 10,12,2,13 | T 2 | 7,2,13,16 |
O 2 | 5,16,3,9 | O 2 | 15,16,3,9 |
Chris Kimmer 2011-06-01