My research interests are diverse, but essentially they are covered by Discrete Mathematics, Algebra and Geometry. Main interests are:
  • Tilings, Convex Polytopes
  • Symmetry Groups, Finitely Presented Groups
  • Hyperbolic Geometry
  • Graph Theory, Combinatorics
  • Selfsimilarity, Fractals
  • Model Sets, Diffraction
  • ...as well as Recreational and Entertaining Mathematics. Some material about this below (less than 333 KB, if not stated otherwise). For some cool pictures related to my work, see the gallery.

    Internet Project: Book Chapter: Research Papers:
    1. With Alexey Garber, Neil Mañibo: Substitution tilings with transcendental inflation factor, submitted.
    2. With Alexey Garber, Neil Mañibo: Catalan numbers as discrepancies for a family of substitutions on infinite alphabets , submitted.
    3. With Christian Richter: Fair partitions of the plane into incongruent pentagons, submitted.
    4. 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.
    5. With Alexey Glazyrin, Zsolt Langi: Hexagon tilings of the plane that are not edge-to-edge, Acta Mathematica Hungarica 164 (2021) 341-349.
    6. With Christian Richter: Incongruent equipartitions of the plane into quadrangles of equal perimeters, Journal of Combinatorial Theory A 182 (2021) Article ID 105461.
    7. With Yotam Smilansky, Yaar Solomon: Pairwise non-BD sets arising from substitution tilings, Journal of Combinatorial Theory A 177 (2021) Article ID 105326.
    8. With Joseph Damasco: Perfect colourings of regular graphs, AMS Contemporary Mathematics 764 (2021)
    9. With Christian Richter: Incongruent equipartitions of the plane, European Journal of Combinatorics 87 (2020) 103129 (13 pages)
    10. 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
    11. 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
    12. 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
    13. 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
    14. 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)
    15. 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
    16. With Alexey Garber: Symmetries of monocoronal tilings, Discrete Mathematics and Theoretical Computer Science 17 (2015) 203-234
    17. With Hofstetter Kurt: Inductive rotation tilings, Proceedings Steklov Institute 288 (2015) 269-280
    18. With Benjamin Whitehead, Michael Whittaker: Cohomology of the Pinwheel Tiling, Journal of the Australian Math Society 97 (2014) 162-179
    19. With Reinhard Lück: Hyperbolic icosahedral tilings by Buckyballs, Acta Physica Polonica A 126 (2014) 524-526
    20. With Christoph Richard: Dynamical properties of almost repetitive Delone sets, Discrete and Continuous Dynamical Systems 34 (2014) 531-556
    21. With Edmund Harriss: Parallelogram tilings, worms and finite orientations, Discrete and Computational Geometry 49 (2013) 531-539
    22. With Paulo Bugarin, Louise de las Penas: Perfect colourings of cyclotomic integers, Geometriae Dedicata 162 (2013) 271-282
    23. With Valérie Berthé, Victor Sirvent: Self-dual substitutions in dimension one, European Journal of Combinatorics 33 (2012) 981-1000
    24. A fractal fundamental domain with 12-fold symmetry, Symmetry: Culture and Science 22 (2011) 237-246 (363 KB)
    25. With Nikolai Dolbilin: Properties of Böröczky tilings in high dimensional hyperbolic spaces , European Journal of Combinatorics 31 (2010) 1181-1195
    26. 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
    27. With Alexey Glazyrin: The lonely vertex problem, Beiträge Algebra und Geometrie 50 (2009) 71-79
    28. Substitution tilings with statistical circular symmetry, European Journal of Combinatorics 29 (2008) 1881-1893
    29. Counting perfect colourings of plane regular tilings, Zeitschrift für Kristallographie 223 (2008) 773-776
    30. With Reinhard Lück: Ten colours in quasiperiodic and regular hyperbolic tilings, Z. Krist. 223 (2008) 782-784 (2 MB)
    31. 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
    32. About substitution tilings with statistical circular symmetry, Phil. Mag. 88 (2008) 2033-2039
    33. Self-dual tilings with respect to star-duality , Theoretical Computer Science 391 (2008) 39-50 (595 KB)
    34. 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)
    35. With Michael Baake, Uwe Grimm: Pinwheel patterns and powder diffraction, Phil. Mag. 87 (2007) 2831-2838
    36. With Bernd Sing: Computing Modular Coincidences for substitution tilings and point sets, Discrete Computational Geometry 37 (2007) 381-407 (456 KB)
    37. Duality of Model Sets Generated by Substitutions, (634 KB), Romanian Journal of Pure and Applied Mathematics 50 (2005) 619-639
    38. With Michael Baake: SCD Patterns Have Singular Diffraction, Journal of Mathematical Physics 46 (2005)
    39. Selfsimilar Tilings with Integer Factor, Symmetry: Art and Science 2 (2002) 379-384

    Eternal Preprints Didactics:
    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
    Theses: Others: