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Research

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, Lorenzo Sadun: Number of bounded distance equivalence classes in hulls of repetitive Delone sets, submitted.
    2. With Alexey Glazyrin, Zsolt Langi: Hexagon tilings of the plane that are not edge-to-edge, Acta Math. Hung. 164 (2021) 341-349.
    3. With Christian Richter: Incongruent equipartitions of the plane into quadrangles of equal perimieters, J. Comb. Theory A 182 (2021) Article ID 105461.
    4. With Yotam Smilansky, Yaar Solomon: Pairwise non-BD sets arising from substitution tilings, J. Comb. Theory A 177 (2021) Article ID 105326.
    5. With Joseph Damasco: Perfect colourings of regular graphs, AMS Contemporary Mathematics 764 (2021)
    6. With Christian Richter: Incongruent equipartitions of the plane, European J. Combin. 87 (2020) 103129 (13 pages)
    7. With Joseph Damasco, Manuel Loquias: Fundamental domains for rhombic lattices with dihedral symmetry of order 8, Electron. J. Combin. 26 (2019) Paper 3.5
    8. 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
    9. 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
    10. 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
    11. 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)
    12. 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
    13. With Alexey Garber: Symmetries of monocoronal tilings, Discrete Mathematics and Theoretical Computer Science 17 (2015) 203-234
    14. With Hofstetter Kurt: Inductive rotation tilings, Proc. Steklov Inst. 288 (2015) 269-280
    15. With Benjamin Whitehead, Michael Whittaker: Cohomology of the Pinwheel Tiling, Journal of the Australian Math Society 97 (2014) 162-179
    16. With Reinhard Lück: Hyperbolic icosahedral tilings by Buckyballs, Acta Physica Polonica A 126 (2014) 524-526
    17. With Christoph Richard: Dynamical properties of almost repetitive Delone sets, Discrete and Continuous Dynamical Systems 34 (2014) 531-556
    18. With Edmund Harriss: Parallelogram tilings, worms and finite orientations, Discrete and Computational Geometry 49 (2013) 531-539
    19. With Paulo Bugarin, Louise de las Penas: Perfect colourings of cyclotomic integers, Geometriae Dedicata 162 (2013) 271-282
    20. With Valérie Berthé, Victor Sirvent: Self-dual substitutions in dimension one, European Journal of Combinatorics 33 (2012) 981-1000
    21. A fractal fundamental domain with 12-fold symmetry, Symmetry: Culture and Science 22 (2011) 237-246 (363 KB)
    22. With Nikolai Dolbilin: Properties of Böröczky tilings in high dimensional hyperbolic spaces , European J. Combin. 31 (2010) 1181-1195
    23. 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
    24. With Alexey Glazyrin: The lonely vertex problem, Beiträge Algebra Geom. 50 (2009) 71-79
    25. Substitution tilings with statistical circular symmetry, European J. Combin. 29 (2008) 1881-1893
    26. Counting perfect colourings of plane regular tilings, Z. Krist. 223 (2008) 773-776
    27. With Reinhard Lück: Ten colours in quasiperiodic and regular hyperbolic tilings, Z. Krist. 223 (2008) 782-784 (2 MB)
    28. 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
    29. About substitution tilings with statistical circular symmetry, Phil. Mag. 88 (2008) 2033-2039
    30. Self-dual tilings with respect to star-duality , Theoret. Comput. Sci. 391 (2008) 39-50 (595 KB)
    31. With Michael Baake, Uwe Grimm: A radial analogue of Poisson's summation formula with application to powder diffraction and pinwheel patterns, J. Geom. Phys. 57 (2007) 1331-1343 (780 KB)
    32. With Michael Baake, Uwe Grimm: Pinwheel patterns and powder diffraction, Phil. Mag. 87 (2007) 2831-2838
    33. With Bernd Sing: Computing Modular Coincidences for substitution tilings and point sets, Discrete Comput. Geom. 37 (2007) 381-407 (456 KB)
    34. Duality of Model Sets Generated by Substitutions, (634 KB), Romanian J. of Pure and Applied Math. 50 (2005) 619-639
    35. With Michael Baake: SCD Patterns Have Singular Diffraction, Journal of Math. Physics 46 (2005)
    36. 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: