2-uniform tilings with regular polygons and regular star polygons · 2015-05-10 · and regular...
Transcript of 2-uniform tilings with regular polygons and regular star polygons · 2015-05-10 · and regular...
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2-uniform tilings with regular polygons and
regular star polygons
Joseph Myers
August 2009
This document lists edge-to-edge 2-uniform tilings with regular polygonsand regular star polygons in the sense of Tilings and Patterns section 2.5.Although the hand enumeration resulting in this list of 38 individual tilingsand 5 infinite families (each determined by one variable angle) is intendedto be complete, it has not been double-checked and should be treated withcaution. No attempt has been made to identify which tilings appear wherein previous literature.
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Figure 1: Tiling (3 . 4 . 4∗∗π/3; 3 . 4∗π/3
. 4 . 4 . 4∗π/3)
Figure 2: Tiling (3 . 4 . 3∗∗π/6; 3 . 4 . 3∗π/6
. 3 . 3∗π/6. 4)
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Figure 3: Tiling (3 . 4 . 3∗∗π/6; 3 . 3 . 4 . 6∗π/6
. 4 . 3∗π/6)
Figure 4: Tiling (3 . 4 . 4∗∗π/3; 3 . 4 . 3 . 4 . 4∗π/3
)
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Figure 5: Tiling (3 . 4 . 3∗∗π/6; 3 . 4 . 3∗π/6
. 3 . 6)
Figure 6: Tiling (3 . 4 . 6∗∗π/2; 3 . 3 . 3 . 4 . 6∗π/2
)
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Figure 7: Tiling (3 . 3 . 4∗∗π/6; 3 . 4∗π/6
. 4∗∗π/6. 4∗π/6
)
Figure 8: Tiling (3 . 3 . 4∗∗π/6; 3 . 4∗π/6
. 6 . 6 . 4∗π/6)
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Figure 9: Tiling (3 . 3 . 4∗∗π/6; 3 . 4∗π/6
. 3 . 4∗π/6. 3 . 4∗π/6
. 3 . 4∗π/6)
Figure 10: Tiling (3 . 3 . 6∗∗π/3; 3 . 6∗π/3
. 3 . 6∗π/3. 3 . 6∗π/3
)
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Figure 11: Tiling (3 . 3 . 4∗∗π/6; 3 . 3 . 3 . 4 . 3 . 4∗π/6
)
Figure 12: Tiling (3 . 3 . 6∗∗π/3; 3 . 3 . 3 . 3 . 3 . 6∗π/3
)
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Figure 13: Tiling (3 . 3 . 12∗π/6. 4∗∗π/3
; 3 . 4∗π/3. 3 . 4∗π/3
. 3 . 4∗π/3)
Figure 14: Tiling (3 . 12∗π/6. 12∗∗π/3
; 3 . 3 . 12∗π/6. 3 . 12∗π/3
. 3 . 12∗π/6)
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Figure 15: Tiling (3 . 8∗π/12. 3 . 4∗∗π/4
; 3 . 4∗π/4. 3 . 8∗π/12
. 3 . 4∗π/4. 3 . 8∗π/12
)
Figure 16: Tiling (3 . 8∗π/12. 3 . 4∗∗π/4
; 3 . 4∗π/4. 8 . 4∗π/4
. 3 . 8∗π/12)
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Figure 17: Tiling (3 . 12∗π/6. 3 . 3∗∗π/6
; 3 . 3∗π/6. 3 . 12∗π/6
. 3 . 3∗π/6. 3 . 12∗π/6
)
Figure 18: Tiling (3 . 9∗π/9. 3 . 6∗∗
4π/9; 3 . 3 . 9∗π/9
. 3 . 6∗4π/9
. 3 . 9∗π/9)
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Figure 19: Tiling (3 . 9∗π/9. 3 . 6∗∗
4π/9; 3 . 9 . 6∗
4π/9. 3 . 9∗π/9
)
Figure 20: Tiling (3 . 12∗π/6. 3 . 4∗∗π/3
; 3 . 3 . 12∗π/6. 3 . 4∗π/3
. 3 . 12∗π/6)
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Figure 21: Tiling (3 . 12∗π/6. 3 . 4∗∗π/3
; 3 . 12 . 4∗π/3. 3 . 12∗π/6
)
Figure 22: Tiling (3 . 18∗2π/9
. 3 . 3∗∗2π/9
; 3 . 3 . 18∗2π/9
. 3 . 3∗2π/9
. 3 . 18∗2π/9
)
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Figure 23: Tiling (3 . 9∗π/9. 3 . 3∗∗π/9
; 3 . 3 . 3 . 9∗π/9. 3 . 3∗π/9
. 3 . 9∗π/9)
Figure 24: Tiling (3 . 3 . 8∗π/12. 4∗∗π/3
. 8∗π/12; 3 . 3 . 8∗π/12
. 3 . 4 . 3 . 8∗π/12)
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Figure 25: Tiling family (3 . 3 . 3∗α . 3∗∗α ; 3 . 3 . 3 . 3 . 3 . 3)
Figure 26: Tiling family (3 . 3 . 4∗α . 4∗∗α+π/6; 3 . 4∗α+π/6
. 4∗∗α )
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Figure 27: Tiling family (3 . 3 . 6∗α . 3∗∗α ; 3 . 3∗α . 6∗∗α )
Figure 28: Tiling family (3 . 3 . 3∗α . 6∗∗α+π/3; 3 . 6∗α+π/3
. 3∗∗α )
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Figure 29: Tiling (4 . 12∗π/3. 6∗∗π/2
; 4 . 6∗π/2. 4 . 6∗π/2
)
Figure 30: Tiling (4 . 8∗π/4. 8∗∗π/2
; 4 . 8∗π/2. 4 . 8∗π/2
)
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Figure 31: Tiling (4 . 6∗π/6. 12∗∗π/2
; 4 . 12∗π/2. 4 . 12∗π/2
)
Figure 32: Tiling family (4 . 6∗α+π/6. 3∗∗α ; 4 . 3∗α . 6∗∗α+π/6
)
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Figure 33: Tiling (4 . 6∗π/6. 6∗∗π/2
. 6∗π/6; 4 . 6∗π/6
. 4 . 6∗π/6. 4 . 6∗π/6
)
Figure 34: Tiling (3 . 3 . 3 . 12∗π/6. 3 . 3 . 12∗π/6
; 3 . 3 . 4 . 3 . 3 . 12∗π/6)
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Figure 35: Tiling (3 . 3 . 3 . 12∗π/6. 3 . 3 . 12∗π/6
; 3 . 3 . 12 . 3 . 12∗π/6)
Figure 36: Tiling (3 . 3 . 3 . 3 . 3 . 3; 3 . 3 . 3 . 4 . 3 . 12∗π/6)
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Figure 37: Tiling (3 . 3 . 8∗π/12. 3 . 4 . 3 . 8∗π/12
; 3 . 4 . 8 . 3 . 8∗π/12)
Figure 38: Tiling (3 . 3 . 12∗π/6. 3 . 12; 3 . 12 . 12)
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Figure 39: Tiling (3 . 4 . 6 . 3 . 12∗π/6; 3 . 4 . 6 . 4)
Figure 40: Tiling (3 . 4 . 8 . 3 . 8∗π/12; 4 . 8 . 8)
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Figure 41: Tiling (3 . 4 . 4 . 6∗π/6. 4; 3 . 3 . 3 . 4 . 4)
Figure 42: Tiling (4 . 4 . 6∗π/6. 4 . 12∗π/3
; 4 . 4 . 4 . 4)
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Figure 43: Tiling (4 . 4 . 8∗π/4. 4 . 8∗π/4
; 4 . 4 . 4 . 4)
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