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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 $ \leftrightarrow$ 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 $ \leftrightarrow$ 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 $ \leftrightarrow$ 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.


Table A.6: Vertices and tiles of the SL O 1 after inflation
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    


Table A.7: Vertices and tiles of the LS O 1 after inflation
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    


Table A.8: Vertices and tiles of the LL O 1 after inflation
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    


Table A.9: Vertices and tiles of the SS O 1 after inflation
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

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Next: The Inflations of T 2 Up: Tile inflations Previous: The Inflations of T 1
Chris Kimmer 2011-06-01

cjkimmer -at- ius.edu