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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
  • Selfsimilarity, Fractals
  • Model Sets, Diffraction
  • Discrete Optimisation
  • 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.

    Projects:
  • Tilings Encyclopedia. An internet project in cooperation with Franz Gähler and Edmund Harriss, a large collection of aperiodic substitution tilings.
  • Research Papers:
    1. Bounded distance and bilipschitz equivalence of Delone sets, preprint
    2. Highly symmetric fundamental domains for lattices in R^2 and R^3, preprint
    3. Noncongruent equidissections of the plane, in press
    4. 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
    5. With Alexey Garber: Symmetries of monocoronal tilings, Discrete Mathematics and Theoretical Computer Science 17 (2015) 203-234
    6. With Hofstetter Kurt: Inductive rotation tilings, Proc. Steklov Inst. 288 (2015) 269-280
    7. With Benjamin Whitehead, Michael Whittaker: Cohomology of the Pinwheel Tiling, Journal of the Australian Math Society 97 (2014) 162-179
    8. With Reinhard Lück: Hyperbolic icosahedral tilings by Buckyballs, Acta Physica Polonica A 126 (2014) 524-526
    9. With Christoph Richard: Dynamical properties of almost repetitive Delone sets, Discrete and Continuous Dynamical Systems 34 (2014) 531-556
    10. With Edmund Harriss: Parallelogram tilings, worms and finite orientations, Discrete and Computational Geometry 49 (2013) 531-539
    11. With Paulo Bugarin, Louise de las Penas: Perfect colourings of cyclotomic integers, Geometriae Dedicata 162 (2013) 271-282
    12. With Valérie Berthé, Victor Sirvent: Self-dual substitutions in dimension one, European Journal of Combinatorics 33 (2012) 981-1000
    13. A fractal fundamental domain with 12-fold symmetry, Symmetry: Culture and Science 22 (2011) 237-246 (363 KB)
    14. With Nikolai Dolbilin: Properties of Böröczky tilings in high dimensional hyperbolic spaces , European J. Combin. 31 (2010) 1181-1195
    15. 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
    16. With Alexey Glazyrin: The lonely vertex problem, Beiträge Algebra Geom. 50 (2009) 71-79
    17. Substitution tilings with statistical circular symmetry, European J. Combin. 29 (2008) 1881-1893
    18. Counting perfect colourings of plane regular tilings, Z. Krist. 223 (2008) 773-776
    19. With Reinhard Lück: Ten colours in quasiperiodic and regular hyperbolic tilings, Z. Krist. 223 (2008) 782-784 (2 MB)
    20. 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
    21. About substitution tilings with statistical circular symmetry, Phil. Mag. 88 (2008) 2033-2039
    22. Self-dual tilings with respect to star-duality , Theoret. Comput. Sci. 391 (2008) 39-50 (595 KB)
    23. 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)
    24. With Michael Baake, Uwe Grimm: Pinwheel patterns and powder diffraction, Phil. Mag. 87 (2007) 2831-2838
    25. With Bernd Sing: Computing Modular Coincidences for substitution tilings and point sets, Discrete Comput. Geom. 37 (2007) 381-407 (456 KB)
    26. Duality of Model Sets Generated by Substitutions, (634 KB), Romanian J. of Pure and Applied Math. 50 (2005) 619-639
    27. With Michael Baake: SCD Patterns Have Singular Diffraction, Journal of Math. Physics 46 (2005)
    28. Selfsimilar Tilings with Integer Factor, Symmetry: Art and Science 2 (2002) 379-384

    Didactics:
  • With Mathias Hattermann: "Konzeption eines Mathematik-Förderprogramms für Informatikstudierende der Universität Bielefeld", submitted for "Mathematik im Übergang von Schule zur Hochschule und im ersten Studienjahr", Reihe "Konzepte und Studien zur Hochschuldidaktik und Lehrerbildung Mathematik", Springer
  • 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:
  • 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 D2m+1-Symmetrie und minimaler Musterfamilie (Dortmund 1998) Also german, obviously; and regrettably without pictures
  • Others:
  • With G. Czichowski: Die Perron-Frobenius-Fußballbundesligatabelle (Preprint 2003, update each year, last update 2015, 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