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The Inflations of T 2

The situation for the inflation of the O 2 tile is the same as for O 1 . There is a small face with four ways to inflate it, LL, SS, SL, and LS. Table  A.14 lists the SL cases. The LS cases are detailed in Table  A.15 while Table  A.16 contains data for the LL case. The final case is listed in Table  A.17 .


Table A.14: Vertices and tiles of the LL O 2 after inflation
1 (0, 0, 0) * 7 (0, 1, 1 + τ ) 13 (1 + τ , 1 + τ , 1 + τ ) *
2 (0, τ , 1 + 2 τ , 0) 8 ( τ , 1 + τ , 1 + 2 τ ) 14 (1, τ , 1 + τ )
3 (0, τ , 1) 9 ( τ , 1 + τ , 1) 15 (1, 1 + τ , 0)
4 (0, 1 + τ , τ ) 10 ( τ , 1, 0) 16 (1, 1 + τ , 2 + 2 τ )
5 (0, 1 + τ , 2 + 3 τ ) * 11 (1 + 2 τ , 1 + τ , 0) * 17 (1, 1, 1)
6 (0, 1 + τ , 2 + τ ) 12 (1 + τ , τ , 0) 18 (2 + τ , 1 + τ , 1)
Tile Vertices Tile Vertices Tile Vertices
T 1 7,14,4,3 T 1 7,14,6,2 T 1 7,14,17,3
T 1 15,9,4,3 T 1 15,9,12,10 T 1 15,9,17,3
T 1 15,9,17,10 T 1 16,8,6,2 O 1 5,16,2,6
O 1 13,8,6,2 O 1 13,9,4,3 O 1 13,9,12,10
O 1 13,9,17,3 O 1 13,9,17,10 O 1 13,14,4,3
O 1 13,14,6,2 O 1 13,14,17,3 T 2 1,3,17,7
T 2 1,3,17,15 T 2 1,10,17,15 O 2 4,7,14,13
O 2 6,7,14,13 O 2 12,11,18,13 O 2 12,15,9,13
O 2 15,4,3,1        


Table A.15: Vertices and tiles of the LS O 2 after inflation
1 (0, 0, 0) * 7 (0, 1, 1 + τ ) 13 (1 + τ , τ , 0)
2 (0, τ , 1 + 2 τ , 0) 8 ( τ , τ , τ ) 14 (1 + τ , 1 + τ , 1 + τ ) *
3 (0, τ , 1) 9 ( τ , 1 + τ , 1 + 2 τ ) 15 (1, τ , 1 + τ )
4 (0, 1 + τ , τ ) 10 ( τ , 1 + τ , 1) 16 (1, 1 + τ , 0)
5 (0, 1 + τ , 2 + 3 τ ) * 11 ( τ , 1, 0) 17 (1, 1 + τ , 2 + 2 τ )
6 (0, 1 + τ , 2 + τ ) 12 (1 + 2 τ , 1 + τ , 0) * 18 (2 + τ , 1 + τ , 1)
Tile Vertices Tile Vertices Tile Vertices
T 1 2,6,9,17 T 1 2,6,15,7 T 1 3,4,10,8
T 1 3,4,10,16 T 1 3,4,15,7 T 1 3,4,15,8
T 1 11,13,10,8 T 1 13,11,16,10 O 1 1,3,7,15
O 1 1,3,8,10 O 1 1,3,8,15 O 1 1,3,16,10
O 1 1,11,8,10 O 1 1,11,16,10 O 1 5,17,2,6
O 1 14,9,6,2 O 1 14,15,6,2 T 2 4,8,10,14
T 2 4,8,15,14 T 2 13,8,10,14 O 2 4,7,15,14
O 2 6,7,15,14 O 2 13,12,18,14 O 2 13,16,10,14
O 2 16,4,3,1        


Table A.16: Vertices and tiles of the SL O 2 after inflation
1 (0, 0, 0) * 7 (0, 1, 1 + τ ) 13 (1 + τ , τ , 0)
2 (0, τ , 1 + 2 τ , 0) 8 (2 τ , 1 + τ , τ ) 14 (1 + τ , 1 + τ , 1 + τ ) *
3 (0, τ , 1) 9 ( τ , 1 + τ , 1 + 2 τ ) 15 (1, τ , 1 + τ )
4 (0, 1 + τ , τ ) 10 ( τ , 1 + τ , 1) 16 (1, 1 + τ , 0)
5 (0, 1 + τ , 2 + 3 τ ) * 11 ( τ , 1, 0) 17 (1, 1 + τ , 2 + 2 τ )
6 (0, 1 + τ , 2 + τ ) 12 (1 + 2 τ , 1 + τ , 0) * 18 (1, 1, 1)
Tile Vertices Tile Vertices Tile Vertices
T 1 7,15,4,3 T 1 7,15,6,2 T 1 7,15,18,3
T 1 16,10,4,3 T 1 16,10,13,11 T 1 17,9,6,2
T 1 16,10,18,11 T 1 16,10,18,3 O 1 5,17,2,6
O 1 12,13,10,16 O 1 14,9,6,2 O 1 14,10,4,3
O 1 14,10,18,3 O 1 14,10,18,11 O 1 14,15,4,3
O 1 14,15,6,2 O 1 14,15,18,3 T 2 1,3,18,7
T 2 1,3,18,16 T 2 1,11,18,16 O 2 4,7,15,14
O 2 6,7,15,14 O 2 10,11,13,12 O 2 10,14,8,12
O 2 16,4,3,1        


Table A.17: Vertices and tiles of the SS O 2 after inflation
1 (0, 0, 0) * 7 (0, 1, 1 + τ ) 13 (1 + 2 τ , 1 + τ , 0) *
2 (0, τ , 1 + 2 τ , 0) 8 (2 τ , 1 + τ , τ ) 14 (1 + τ , τ , 0)
3 (0, τ , 1) 9 ( τ , τ , τ ) 15 (1 + τ , 1 + τ , 1 + τ ) *
4 (0, 1 + τ , τ ) 10 ( τ , 1 + τ , 1 + 2 τ ) 16 (1, τ , 1 + τ )
5 (0, 1 + τ , 2 + 3 τ ) * 11 ( τ , 1 + τ , 1) 17 (1, 1 + τ , 0)
6 (0, 1 + τ , 2 + τ ) 12 ( τ , 1, 0) 18 (1, 1 + τ , 2 + 2 τ )
Tile Vertices Tile Vertices Tile Vertices
T 1 2,6,10,18 T 1 2,6,16,7 T 1 3,4,11,9
T 1 3,4,11,17 T 1 3,4,16,7 T 1 3,4,16,7
T 1 3,4,16,9 T 1 12,14,11,9 T 1 12,14,11,17
T 1 15,8,11,9 O 1 1,3,7,16 O 1 1,3,9,11
O 1 1,3,9,16 O 1 1,3,17,11 O 1 1,12,9,11
O 1 1,12,17,11 O 1 5,18,2,6 O 1 13,8,11,9
O 1 13,14,11,9 O 1 13,14,11,17 O 1 15,10,6,2
O 1 15,16,6,2 T 2 4,9,11,15 T 2 4,9,16,15
O 2 4,7,16,15 O 2 6,7,16,15 O 2 17,4,3,1

The rearrangements of the inflated O 2 tiles are essentially the same as those of the O 1 . The only difference is that the larger O 2 volume leads to an additional rearrangement corresponding to the tile conserving 3-fold flip. That is, Lx $ \leftrightarrow$ Ly occurs with three 3-fold tile-conserving rearrangements. The SL → SS rearrangement is a (1,-1,3,-2) 5-fold flip, and any other tiled inflation can be connected to SS by first transforming it to SL via tile-conserving flips.

next up previous
Next: Local Configurations of the Up: Tile inflations Previous: The Inflations of T 2
Chris Kimmer 2011-06-01

cjkimmer -at- ius.edu