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Research

My research interests are diverse, but essentially they are covered by Discrete Mathematics, Algebra and Geometry. Main interests are:
  • Convex Polytopes, Tilings
  • Symmetry Groups, Finitely Presented Groups
  • Selfsimilarity, Fractals
  • Model Sets, Diffraction
  • Discrete Optimisation
  • 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 Edmund Harriss, aiming to present everything about mathematical tilings (tessellations) one day.
  • Papers:
    1. With Christoph Richard: Dynamical properties of almost repetitive Delone sets, submitted
    2. With Edmund Harriss: Parallelogram tilings, worms and finite orientations, to appear in Discrete and Computational Geometry
    3. With Paulo Bugarin, Louise de las Penas: Perfect colourings of cyclotomic integers, Geometriae Dedicata (2012) DOI 10.1007/s10711-012-9726-0
    4. With Valérie Berthé, Victor Sirvent: Self-dual substitutions in dimension one, European Journal of Combinatorics 33 (2012) 981-1000
    5. A fractal fundamental domain with 12-fold symmetry, Symmetry: Culture and Science 22 (2011) 237-246 (363 KB)
    6. With Nikolai Dolbilin: Properties of Böröczky tilings in high dimensional hyperbolic spaces , European J. Combin. 31 (2010) 1181-1195
    7. 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
    8. With Alexey Glazyrin: The lonely vertex problem, Beiträge Algebra Geom. 50 (2009) 71-79
    9. Substitution tilings with statistical circular symmetry, European J. Combin. 29 (2008) 1881-1893
    10. Counting perfect colourings of plane regular tilings, Z. Krist. 223 (2008) 773-776
    11. With Reinhard Lück: Ten colours in quasiperiodic and regular hyperbolic tilings, Z. Krist. 223 (2008) 782-784 (2 MB)
    12. 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
    13. About substitution tilings with statistical circular symmetry, Phil. Mag. 88 (2008) 2033-2039
    14. Self-dual tilings with respect to star-duality , Theoret. Comput. Sci. 391 (2008) 39-50 (595 KB)
    15. 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)
    16. With Michael Baake, Uwe Grimm: Pinwheel patterns and powder diffraction, Phil. Mag. 87 (2007) 2831-2838.
    17. With Bernd Sing: Computing Modular Coincidences for substitution tilings and point sets, Discrete Comput. Geom. 37 (2007) 381-407 (456 KB)
    18. Duality of Model Sets Generated by Substitutions, (634 KB), Romanian J. of Pure and Applied Math. 50 (2005) 619-639
    19. With Michael Baake: SCD Patterns Have Singular Diffraction, Journal of Math. Physics 46 (2005)
    20. Selfsimilar Tilings with Integer Factor, Symmetry: Art and Science 2 (2002) 379-384

    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, updated each year, 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.
  • The slides of an introductory talk on quasiperiodic (mainly Penrose) tilings at the Physical Institute in Hamburg.
  • The slides of the talk at the annual meeting of the DMV in München 2010 (also shown at the Kolloquium Kombinatorik, Magdeburg 2008 and at the conference Fractal Geometry and Stochastics 4, Greifswald 2008).
  • A lot of images on tesselations, shown at Schülerakademie in Bielefeld on 8. April 2008 (also 8. Octobre 2010).
  • A draft text on icosahedral tilings