Properties
Canonical Substitution Tiling
Tilings that can be constructed using the canonical projection method, where the window is the projection of the unit hypercube to the window space.
This method was first used by N.G. deBruijn [de 81] to construct the Penrose Rhomb substitution rule. His method was used by F. Beenker to construct the Ammann-Beenker rule [Bee82].
In fact the construction used by N.G. deBruijn and F. Beenker was not the canonical projection method, but the multigrid method. The multigrid method takes several sets of parallel lines distributed over a one dimensional lattice and puts a tile at every intersection of two lines. This method was shown to be equivalent to the canonical projection method in [GR86] and [KGR88].
The method was used to find many other examples and sets of examples were found for example [Soc89],[BJK91],[BKSZ90],[BJS91],[SW92],[SMS89],[PHK00],[Ing92],[Ing93],[Zob92].
The classification of all canonical projection tilings that have a substitution rule were given by E. Harriss [HL04],[Har03].
The cohomology of the tiling spaces for canonical projection tilings is given in [FHK02].
References
[de 81]
de Bruijn, N G
Algebraic theory of Penrose's nonperiodic tilings of the plane. I, II
Nederl. Akad. Wetensch. Indag. Math., 1981, 43, 1, pp. 39--52, 53--66, 82e:05055
Algebraic theory of Penrose's nonperiodic tilings of the plane. I, II
Nederl. Akad. Wetensch. Indag. Math., 1981, 43, 1, pp. 39--52, 53--66, 82e:05055
[Bee82]
Beenker, F P M
Algebraic theory of non-periodic tilings of the plane by two simple building blocks: a square and a rhombus
Eindhoven University of Technology, 1982, TH-Report, 82-WSK04,
Algebraic theory of non-periodic tilings of the plane by two simple building blocks: a square and a rhombus
Eindhoven University of Technology, 1982, TH-Report, 82-WSK04,
[GR86]
Gähler, F and Rhyner, J
Equivalence of the generalised grid and projection methods for the construction of quasiperiodic tilings
J. Phys. A, 1986, 19, 2, pp. 267--277
Equivalence of the generalised grid and projection methods for the construction of quasiperiodic tilings
J. Phys. A, 1986, 19, 2, pp. 267--277
[KGR88]
Korepin, V E and Gähler, F and Rhyner, J
Quasiperiodic tilings: a generalized grid-projection method
Acta Cryst., 1988, A44, pp. 667-672
Quasiperiodic tilings: a generalized grid-projection method
Acta Cryst., 1988, A44, pp. 667-672
[Soc89]
Socolar, J E S
Simple octagonal and dodecagonal quasicrystals
Phys. Rev. B, 1989, 39, pp. 10519-10551, MR0998533
Simple octagonal and dodecagonal quasicrystals
Phys. Rev. B, 1989, 39, pp. 10519-10551, MR0998533
[BJK91]
Baake, M and Joseph, D and Kramer, P
The Schur rotation as a simple approach to the transition between quasiperiodic and periodic phases
J. Phys. A, 1991, 24, 17, pp. L961--L967, 92k:82071
The Schur rotation as a simple approach to the transition between quasiperiodic and periodic phases
J. Phys. A, 1991, 24, 17, pp. L961--L967, 92k:82071
[BKSZ90]
Baake, M and Kramer, P and Schlottmann, M and Zeidler, D
Planar patterns with fivefold symmetry as sections of periodic structures in $4$-space
Internat. J. Modern Phys. B, 1990, 4, 15-16, pp. 2217--2268, 92b:52041
Planar patterns with fivefold symmetry as sections of periodic structures in $4$-space
Internat. J. Modern Phys. B, 1990, 4, 15-16, pp. 2217--2268, 92b:52041
[BJS91]
Baake, M and Joseph, D and Schlottmann, M
The root lattice $D\sb 4$ and planar quasilattices with octagonal and dodecagonal symmetry
Internat. J. Modern Phys. B, 1991, 5, 11, pp. 1927--1953, 92m:52035
The root lattice $D\sb 4$ and planar quasilattices with octagonal and dodecagonal symmetry
Internat. J. Modern Phys. B, 1991, 5, 11, pp. 1927--1953, 92m:52035
[SW92]
Soma, T and Watanabe, Y
A class of patterns generated by modification of beenkers pattern
Acta Crystallogr., 1992, A48, pp. 470--475
A class of patterns generated by modification of beenkers pattern
Acta Crystallogr., 1992, A48, pp. 470--475
[SMS89]
Sire, C and Mosseri, R and Sadoc, J-F
Geometric study of a $2$D tiling related to the octagonal quasiperiodic tiling
J. Physique, 1989, 50, 24, pp. 3463--3476, 91c:52026
Geometric study of a $2$D tiling related to the octagonal quasiperiodic tiling
J. Physique, 1989, 50, 24, pp. 3463--3476, 91c:52026
[PHK00]
Papadopolos, Zorka and Hohneker, Christoph and Kramer, Peter
Tiles-inflation rules for the class of canonical tilings $\scr T\sp *(2F)$ derived by the projection method
Discrete Math., 2000, 221, 1-3, pp. 101--112, 2001e:52040
Tiles-inflation rules for the class of canonical tilings $\scr T\sp *(2F)$ derived by the projection method
Discrete Math., 2000, 221, 1-3, pp. 101--112, 2001e:52040
[Ing92]
Ingalls, R
Decagonal quasi-crystal tilings
Acta Crystallogr. Sect. A, 1992, A48, pp. 533--541
Decagonal quasi-crystal tilings
Acta Crystallogr. Sect. A, 1992, A48, pp. 533--541
[Ing93]
Ingalls, R
Octagonal quasi-crystal tilings
J. Non-Cryst. Solids, 1993, 153, pp. 177--180
Octagonal quasi-crystal tilings
J. Non-Cryst. Solids, 1993, 153, pp. 177--180
[Zob92]
Zobetz, E
A pentagonal quasi-periodic tiling with fractal acceptance domain
Acta Crystallogr., 1992, A48, pp. 328--335
A pentagonal quasi-periodic tiling with fractal acceptance domain
Acta Crystallogr., 1992, A48, pp. 328--335
[HL04]
Harriss, E O and Lamb, J S W
Canonical Substitution Tilings of Ammann-Beenker Type
Th. Comp. Sci., 2004, 319, pp. 241--279, 2074956
Canonical Substitution Tilings of Ammann-Beenker Type
Th. Comp. Sci., 2004, 319, pp. 241--279, 2074956
[Har03]
Harriss, E O
On Canonical Substitution Tilings
Imperial College London, 2003,
On Canonical Substitution Tilings
Imperial College London, 2003,
[FHK02]
Forrest, A H and Hunton, J R and Kellendonk, J
Cohomology of canonical projection tilings
Comm. Math. Phys., 2002, 226, 2, pp. 289--322, 1892456
Cohomology of canonical projection tilings
Comm. Math. Phys., 2002, 226, 2, pp. 289--322, 1892456
