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Some comments ona hexagonal monotile
Uwe Grimm
Department of Mathematics and Statistics
The Open University, Milton Keynes, UK
(in collaboration with Michael Baake, Dirk Frettloh and Franz Gahler)
Seoul, September 2010 – p.1
QuestionIs there a single shape that tiles space without gaps oroverlaps, but does not admit any periodic tiling?
Seoul, September 2010 – p.2
QuestionIs there a single shape that tiles space without gaps oroverlaps, but does not admit any periodic tiling?3D: Schmitt-Conway Danzer ‘einstein’
Seoul, September 2010 – p.2
QuestionIs there a single shape that tiles space without gaps oroverlaps, but does not admit any periodic tiling?3D: Schmitt-Conway Danzer ‘einstein’2D: Penrose tiling (two tiles):
Seoul, September 2010 – p.2
QuestionIs there a single shape that tiles space without gaps oroverlaps, but does not admit any periodic tiling?3D: Schmitt-Conway Danzer ‘einstein’2D: Penrose tiling (two tiles):
No monotile known — but Penrose’s 1 + ε + ε2 tiling
Seoul, September 2010 – p.2
Story19 Feb 2010: Email from Joshua Socolar announcing
An aperiodic hexagonal tile(joint preprint with Joan M. Taylor)
Seoul, September 2010 – p.3
Story19 Feb 2010: Email from Joshua Socolar announcing
An aperiodic hexagonal tile(joint preprint with Joan M. Taylor)
Seoul, September 2010 – p.3
Story19 Feb 2010: Email from Joshua Socolar announcing
An aperiodic hexagonal tile(joint preprint with Joan M. Taylor)
28 Feb 2010: Visit Joan Taylor in Burnie, Tasmania
Seoul, September 2010 – p.3
Story19 Feb 2010: Email from Joshua Socolar announcing
An aperiodic hexagonal tile(joint preprint with Joan M. Taylor)
based on Joan’s unpublished manuscriptAperiodicity of a functional monotile
which is available (with hand-drawn diagrammes) fromhttp://www.math.uni-bielefeld.de/sfb701/
preprints/view/420
(slight difference in definition of matching rules)
Seoul, September 2010 – p.3
Joan Taylor
Seoul, September 2010 – p.4
Joan Taylor
Seoul, September 2010 – p.4
Joan Taylor
Seoul, September 2010 – p.4
Joan Taylor
Seoul, September 2010 – p.4
Joan Taylor
Seoul, September 2010 – p.4
Robinson’s tiling
Seoul, September 2010 – p.5
Robinson’s tiling
Seoul, September 2010 – p.5
Half-hex tiling
Seoul, September 2010 – p.6
Half-hex tiling
Seoul, September 2010 – p.6
Half-hex tiling
Seoul, September 2010 – p.6
Half-hex tiling
Seoul, September 2010 – p.6
Half-hex tiling
Seoul, September 2010 – p.6
Half-hex tiling
Seoul, September 2010 – p.6
Half-hex tilinghexagonal tile
still admits periodic tilings of the plane
Seoul, September 2010 – p.6
Half-hex tiling
Seoul, September 2010 – p.6
Penrose’s1 + ε + ε2 tiling
3 tiles: 1 + ε + ε2
comparison: Ammann-Beenker tiling: 2 + 2ε2
‘key tiles’ encode matching rule information
proof of aperiodicity (Penrose)
the ε tile transmits information along edge
Seoul, September 2010 – p.7
The monotile(figures from Socolar & Taylor An aperiodic hexagonal monotile arXiv:1003.4279v2)
Seoul, September 2010 – p.8
The monotile(figures from Socolar & Taylor An aperiodic hexagonal monotile arXiv:1003.4279v2)
Seoul, September 2010 – p.8
Forced patterns(figures from Socolar & Taylor An aperiodic hexagonal monotile arXiv:1003.4279v2)
Seoul, September 2010 – p.9
Filling the gaps(figures from Socolar & Taylor An aperiodic hexagonal monotile arXiv:1003.4279v2)
Seoul, September 2010 – p.10
Filling the gaps(figures from Socolar & Taylor An aperiodic hexagonal monotile arXiv:1003.4279v2)
Seoul, September 2010 – p.10
Filling the gaps(figures from Socolar & Taylor An aperiodic hexagonal monotile arXiv:1003.4279v2)
Seoul, September 2010 – p.10
A non-repetitive Socolar-Taylor tiling(figures from Socolar & Taylor An aperiodic hexagonal monotile arXiv:1003.4279v2)
Seoul, September 2010 – p.11
Taylor’s substitution(figures from Taylor’s manuscript Aperiodicity of a functional monotile)
Seoul, September 2010 – p.12
Taylor’s substitution(figures from Taylor’s manuscript Aperiodicity of a functional monotile)
Seoul, September 2010 – p.12
Taylor’s substitution(figures from Taylor’s manuscript Aperiodicity of a functional monotile)
Seoul, September 2010 – p.12
Taylor’s substitution(figures from Taylor’s manuscript Aperiodicity of a functional monotile)
Seoul, September 2010 – p.12
Taylor’s substitution(figures from Taylor’s manuscript Aperiodicity of a functional monotile)
Seoul, September 2010 – p.12
Taylor’s substitution(figures from Taylor’s manuscript Aperiodicity of a functional monotile)
Seoul, September 2010 – p.12
Taylor’s substitution(figures from Taylor’s manuscript Aperiodicity of a functional monotile)
Seoul, September 2010 – p.12
Taylor’s substitution(figures from Taylor’s manuscript Aperiodicity of a functional monotile)
Seoul, September 2010 – p.12
Taylor’s substitution(figures from Taylor’s manuscript Aperiodicity of a functional monotile)
Seoul, September 2010 – p.12
Hexagon pseudo-inflation tiling(figures by Dirk Frettlöh)
Seoul, September 2010 – p.13
Hexagon pseudo-inflation tiling(figures by Dirk Frettlöh)
Seoul, September 2010 – p.13
Relation to Penrose’s tiling I(figures by Dirk Frettlöh)
Seoul, September 2010 – p.14
Relation to Penrose’s tiling I(figures by Dirk Frettlöh)
Seoul, September 2010 – p.14
Relation to Penrose’s tiling I(figures by Dirk Frettlöh)
Seoul, September 2010 – p.14
Relation to Penrose’s tiling II(figures from Socolar & Taylor An aperiodic hexagonal monotile arXiv:1003.4279v2)
Seoul, September 2010 – p.15
A ‘thickened’ monotile(figure from Socolar & Taylor Forcing nonperiodicity with a single tile arXiv:1009.1419)
Seoul, September 2010 – p.16
References
F. Gähler, Matching rules for quasicrystals: The composition-decomposition method,J. Non-Cryst. Solids 153& 154 (1993) 160–164.
B. Grünbaum and G.C. Shephard, Tilings and Patterns (Freeman, New York, 1987).
R. Penrose, Remarks on tiling: Details of a (1 + ε + ε2)-aperiodic set,
in: The Mathematics of Long-Range Aperiodic Order, ed. R.V. Moody(Kluwer, Dordrecht, 1997) pp. 467–497.
J.E.S. Socolar and J.M. Taylor, An aperiodic hexagonal tile,preprint arXiv:1003.4279 (2010).
J.E.S. Socolar and J.M. Taylor, Forcing nonperiodicity with a single tile,preprint arXiv:1009.1419 (2010).
J.M. Taylor, Aperiodicity of a functional monotile,preprint Bielefeld CRC 701: 0-015 (2010), available viahttp://www.math.uni-bielefeld.de/sfb701/preprints/view/420
Seoul, September 2010 – p.17
