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Next: The Inflations of T 2 Up: Tile inflations Previous: The Inflations of O 1

The Inflations of T 2

The tetrahedron T 2 contains two faces that share a short red edge. This sharing of an edge plus the two-fold symmetry plane of this tile means that there are four symmetry-inequivalent ways to inflate this tile. The inflations of each face obviously differ in how the short 5-fold edge is inflated, so adopting the convention of writing the face that has a L inflation first, the possibilities for this tile are LxSx and LxSy. Finally, they may be written as LLSS (Table  A.10 ), LSSS (Table  A.11 ), LLSL (Table  A.12 ), and LSSL (Table  A.13 ).


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


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


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


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

The tile rearrangements for the inflated T 2 tiles take essentially the same form as for the O 1 tiles. A flip between tiles like LxSy $ \leftrightarrow$ LySy is just the tile-conserving rearrangement along 3-fold directions. The jumps along the short 5-fold edge are either tile-conserving or not. The tiling of LLSL and LSSS is invariant under a flip along the 5-fold edge, so these flips can be realized as a reflection through the two-fold plane of symmetry of the T 2 tile. The non-conserving flips can be created by the basic (1,-1,3,-2) flip.

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

cjkimmer -at- ius.edu