Research

Tilings, Convex Polytopes

Symmetry Groups, Finitely Presented Groups

Hyperbolic Geometry

Graph Theory, Combinatorics

Selfsimilarity, Fractals

Model Sets, Diffraction

• Tilings Encyclopedia. An internet project in cooperation with Franz Gähler (and originally Edmund Harriss), a large collection of aperiodic substitution tilings.

• More Inflation Tilings, in: "Aperiodic Order" Vol 2, eds.: M. Baake, U. Grimm, Cambridge University Press (2017)

1. Perfect colourings of simplices and hypercubes in dimension four and five with few colours, Bulletin Mathematique de la Société des Sciences Mathématiques de Roumanie 67 (2024) 191-201.
2. With Alexey Garber, Neil Mañibo: Substitution tilings with transcendental inflation factor, Discrete Analysis, in press.
   • The slides of the talk in Milton Keynes, July 2022.
3. With Alexey Garber, Neil Mañibo: Catalan numbers as discrepancies for a family of substitutions on infinite alphabets , Indagationes Mathematicae, in press.
   • The slides of the talk in Heidelberg, Oct. 2023
4. With Christian Richter: Fair partitions of the plane into incongruent pentagons, Contributions to Discrete Mathematics, in press.
5. With Alexey Garber, Lorenzo Sadun: Number of bounded distance equivalence classes in hulls of repetitive Delone sets, Discrete and Continuous Dynamical Systems 42 (2022) 1403-1414.
6. With Alexey Glazyrin, Zsolt Langi: Hexagon tilings of the plane that are not edge-to-edge, Acta Mathematica Hungarica 164 (2021) 341-349.
• The slides of the talk in Bukarest, Sept. 2021 (part 2, slides 26-40)
7. With Christian Richter: Incongruent equipartitions of the plane into quadrangles of equal perimeters, Journal of Combinatorial Theory A 182 (2021) Article ID 105461.
   • The slides of the talk in Bukarest, Sept. 2021 (part 1, slides 1-25)
8. With Yotam Smilansky, Yaar Solomon: Pairwise non-BD sets arising from substitution tilings, Journal of Combinatorial Theory A 177 (2021) Article ID 105326.
9. With Joseph Damasco: Perfect colourings of regular graphs, AMS Contemporary Mathematics 764 (2021)
   • The slides of the talk on this topic at Kolloquium Kombinatorik, Paderborn Nov 2018
10. With Christian Richter: Incongruent equipartitions of the plane, European Journal of Combinatorics 87 (2020) 103129 (13 pages)
11. With Joseph Damasco, Manuel Loquias: Fundamental domains for rhombic lattices with dihedral symmetry of order 8, Electronic Journal of Combinatorics 26 (2019) Paper 3.5
12. With Alexey Garber: Weighted 1x1 cut-and-project sets in bounded distance to a lattice Discrete and Computational Geometry 62 (2019) 649-661 arXiv:1706.01089
13. With Alexey Garber: Pisot substitution sequences, one dimensional cut-and-project sets and bounded remainder sets with fractal boundary Indagationes Mathematicae 29 (2018) 1114-1130 arXiv:1711.01498
    • The slides of the talk on this topic at Fractal Geometry and Stochastics 6, Bad Herrenalb Sep 2018
14. With April Say-awen, Ma. Louise Antonette N. De Las Peñas: Primitive substitution tilings with rotational symmetries, Acta Crystallogr. Sect. A 74 (2018) 388-398
15. Noncongruent equidissections of the plane, in: M.D.E. Conder, A. Deza, A.I. Weiss (Eds.): Discrete Geometry and Symmetry, Veszprém, Hungary, June 2015, Springer Proceedings in Mathematics & Statistics, Vol. 234 (2018)
16. With April Say-awen, M. Louise A.N. de las Penas: Substitution tilings with dense tile orientations and n-fold rotational symmetry, Indagationes Mathematicae 28 (2017) 120-131
17. With Alexey Garber: Symmetries of monocoronal tilings, Discrete Mathematics and Theoretical Computer Science 17 (2015) 203-234
    • The slides of the talk on this topic at "Geometry and Symmetry", Veszprem July 2015 (4 MB)
18. With Hofstetter Kurt: Inductive rotation tilings, Proceedings Steklov Institute 288 (2015) 269-280
19. With Benjamin Whitehead, Michael Whittaker: Cohomology of the Pinwheel Tiling, Journal of the Australian Math Society 97 (2014) 162-179
20. With Reinhard Lück: Hyperbolic icosahedral tilings by Buckyballs, Acta Physica Polonica A 126 (2014) 524-526
21. With Christoph Richard: Dynamical properties of almost repetitive Delone sets, Discrete and Continuous Dynamical Systems 34 (2014) 531-556
22. With Edmund Harriss: Parallelogram tilings, worms and finite orientations, Discrete and Computational Geometry 49 (2013) 531-539
    • The slides of the talk 'Tilings with tiles in finitely many and infinitely many orientations' at CIRM, Luminy, Marseille, Jan 2013 (721 KB)
23. With Paulo Bugarin, Louise de las Penas: Perfect colourings of cyclotomic integers, Geometriae Dedicata 162 (2013) 271-282
    • The slides of the talk 'On colour symmetries of hyperbolic regular tilings and cyclotomic integers' in Dortmund, July 2009 (1.1 MB)
24. With Valérie Berthé, Victor Sirvent: Self-dual substitutions in dimension one, European Journal of Combinatorics 33 (2012) 981-1000
    • The slides of the talk 'About duality of cut-and-project tilings' in Davidson, NC, March 2007 (1.2 MB)
25. A fractal fundamental domain with 12-fold symmetry, Symmetry: Culture and Science 22 (2011) 237-246 (363 KB)
26. With Nikolai Dolbilin: Properties of Böröczky tilings in high dimensional hyperbolic spaces , European Journal of Combinatorics 31 (2010) 1181-1195
27. With Paulo Bugarin, Louise de las Penas: Colourings of cyclotomic integers with class number one, Phil. Mag. 90 (2010) DOI: 10.1080/14786435.2010.525544
28. With Alexey Glazyrin: The lonely vertex problem, Beiträge Algebra und Geometrie 50 (2009) 71-79
29. Substitution tilings with statistical circular symmetry, European Journal of Combinatorics 29 (2008) 1881-1893
    • The slides of the corresponding talk 'About substitution tilings with statistical circular symmetry' in Leicester, Sept 2009 (1.4 MB)
30. Counting perfect colourings of plane regular tilings, Zeitschrift für Kristallographie 223 (2008) 773-776
    • The slides of the corresponding talk in London, July/August 2008 (745 KB)
31. With Reinhard Lück: Ten colours in quasiperiodic and regular hyperbolic tilings, Z. Krist. 223 (2008) 782-784 (2 MB)
32. With M.L.A.N. de las Penas, E. Bugarin, I. Evidente, R.,P. Felix: On color groups of Bravais colorings of planar modules with quasicrystallographic symmetry, Z. Krist. 223 (2008) 785-790
33. About substitution tilings with statistical circular symmetry, Phil. Mag. 88 (2008) 2033-2039
34. Self-dual tilings with respect to star-duality , Theoretical Computer Science 391 (2008) 39-50 (595 KB)
    • The slides of the talk 'Self-duality and *-dual tilings' in Liège, May 2006 (866 KB)
35. With Michael Baake, Uwe Grimm: A radial analogue of Poisson's summation formula with application to powder diffraction and pinwheel patterns, Journal of Geometry and Physics 57 (2007) 1331-1343 (780 KB)
36. With Michael Baake, Uwe Grimm: Pinwheel patterns and powder diffraction, Phil. Mag. 87 (2007) 2831-2838
37. With Bernd Sing: Computing Modular Coincidences for substitution tilings and point sets, Discrete Computational Geometry 37 (2007) 381-407 (456 KB)
38. Duality of Model Sets Generated by Substitutions, (634 KB), Romanian Journal of Pure and Applied Mathematics 50 (2005) 619-639
    • The slides of the corresponding talk at the University of Victoria, Aug 2005: Full version, 1.6 MB , or smaller version, 546 KB (four pictures removed).
39. With Michael Baake: SCD Patterns Have Singular Diffraction, Journal of Mathematical Physics 46 (2005)
40. Selfsimilar Tilings with Integer Factor, Symmetry: Art and Science 2 (2002) 379-384

• Bounded distance and bilipschitz equivalence of Delone sets, preprint
• Highly symmetric fundamental domains for lattices in R^2 and R^3, preprint
  • The slides of the BMS-talk 'Highly symmetric fundamental cells for lattices in R^2 and R^3' at TU Berlin, 12. Dec 2012 (2.1 MB)

1. With Daniel Frohn: Wallpaper Groups, mathematik lehren 204/2017
2. With Mathias Hattermann: Frettlöh D., Hattermann M. (2016) Konzeption eines Mathematik-Förderprogramms für Informatikstudierende der Universität Bielefeld. In: Hoppenbrock A., Biehler R., Hochmuth R., Rück HG. (eds) Lehren und Lernen von Mathematik in der Studieneingangsphase. Konzepte und Studien zur Hochschuldidaktik und Lehrerbildung Mathematik. Springer Spektrum, Wiesbaden.
3. With Mathias Hattermann: Projekt richtig einsteigen. Ein Förderprogramm für Studienanfänger der Universität Bielefeld unter Einbezug erster Erfahrungen der Technischen Fakultät, khdm-Report, Online-Schriftenreihe khdm

• Habilitationsschrift: Symmetries and Spectral Properties of Aperiodic Structures (Bielefeld 2009). This thesis gives a short introduction and overview of my work until 2009.
• Ph.D. thesis (in german, 2.2MB): Nichtperiodische Pflasterungen mit ganzzahligem Inflationsfaktor ('Nonperiodic tilings with integer substitution factor', Dortmund 2002)
• Diploma thesis: Inflationäre Pflasterungen der Ebene mit D_2m+1-Symmetrie und minimaler Musterfamilie (Dortmund 1998) Also german, obviously; and regrettably without pictures

• With G. Czichowski: Die Perron-Frobenius-Fußballbundesligatabelle (Preprint 2003, update each year, last update 2018, German. It means: The Perron-Frobenius-german-football-league-table)
• With M. Baake: Is there a crystal clear definition of a crystal?, contribution to a discussion, Z. Krist.222 (2007)
• Mathematik - was, wie, wozu? A text which tries to explain aims and methods of mathematics, the spirit of mathematics, to a very broad (German speaking) audience.
• A draft text on icosahedral tilings