For any pair of diagonally oppositetiles in a family, we may tile, withoutfaults, as shown. However, once atile has been placed adjacent to itscomplement, then that pattern islikely to be continued.
Hex family A
Hex family B(A rotated30 degrees)
Consider this edge
It may be joined by edgesin a variety of ways:
A short edge meetingtop vertex of the edge
Any edge straddlngtop vertex of the edge
A short edge nestledin the middle of theedge.
A long edge meetingtop vertex of the edge
A long edge straddlingbottom of the edge
M
M
A short edge meetingbottom vertex of the edge
In either of these cases,the remaining part of thepink edge will be overshotby whataver tile abuts thatshort edge that joins pink.
The remaining part of the pink edge will have another tile joining it that either overshootsundershoots, or matches the top vertex. These cases correspond to cases B, D, or A, above.
A
B
C
D
E
F
Here’s what’s left:
This option merely postponesthe question. See another doc.
x
x
x
x
x
x
x
Y
x
A 150 degree anglerequires a 60 and a 90.
x
x
x
x
0
1
0
1
x
x
x
x
Y
x
Y
x
Y
M
x
x
x
x
x
x
1
Y
1
0
1
0
M
0
0
C
C
0
M
0
0
0
0
0
x
x
M
x
0
Y
1
0
x
1
x
0
0
x
https://ello.co/ddailey/post/_qldt1s3dn_i56olv3-kma
One of the quasi-regulartilings covered at
M
0
0
0
x
x
x
0
M
C
0
0
0
x
x
0
0
0
x
0
0
x
x
x
x
0
0
0
0
0
M
C
x
0
0
x
x
Y
0
0
0
C
C
C
1
0
C
C
C
C
C
M
x
0
M
0
M
0
M
0
0
1
C
C
C
M
C
C
M
0
x
M
x
x
0
0
M
0
x
Y
0
1
C
C
C
M
0
C
C
C
C
M
M
0
C
C
M
M
0
0
C
C
x
C
C
0
0
C
C
M
M
Y
C
x
C
0
0
x
0
C
C
x
0
0
C
C
C
C
C
C
M
C
0
0
C
C
C
C
C
M
C
M
x
0
C
C
C
C
C
x
0
C
0
C
M
C
C
C
C
M
0
C
M
1
1
1
x
0
0
M
C
C
C
C
x
0
C
M
M
x
M
Take two copies of a sixth of a hexagon.Flip one around, abut the two, and tile.
(Only two things work.)