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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
  • 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 Joseph Damasco, Manuel Loquias: Fundamental domains for rhombic lattices with dihedral symmetry of order 8, submitted
    2. With Joseph Damasco: Perfect colourings of regular graphs, submitted
    3. With Alexey Garber: Pisot substitution sequences, one dimensional cut-and-project sets and bounded remainder sets with fractal boundary Indagationes Mathematicae, accepted arXiv:1711.01498
    4. With Alexey Garber: Weighted 1x1 cut-and-project sets in bounded distance to a lattice Discrete and Computational Geometry, accepted arXiv:1706.01089
    5. Noncongruent equidissections of the plane, to appear 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)
    6. 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
    7. With Alexey Garber: Symmetries of monocoronal tilings, Discrete Mathematics and Theoretical Computer Science 17 (2015) 203-234
    8. With Hofstetter Kurt: Inductive rotation tilings, Proc. Steklov Inst. 288 (2015) 269-280
    9. With Benjamin Whitehead, Michael Whittaker: Cohomology of the Pinwheel Tiling, Journal of the Australian Math Society 97 (2014) 162-179
    10. With Reinhard Lück: Hyperbolic icosahedral tilings by Buckyballs, Acta Physica Polonica A 126 (2014) 524-526
    11. With Christoph Richard: Dynamical properties of almost repetitive Delone sets, Discrete and Continuous Dynamical Systems 34 (2014) 531-556
    12. With Edmund Harriss: Parallelogram tilings, worms and finite orientations, Discrete and Computational Geometry 49 (2013) 531-539
    13. With Paulo Bugarin, Louise de las Penas: Perfect colourings of cyclotomic integers, Geometriae Dedicata 162 (2013) 271-282
    14. With Valérie Berthé, Victor Sirvent: Self-dual substitutions in dimension one, European Journal of Combinatorics 33 (2012) 981-1000
    15. A fractal fundamental domain with 12-fold symmetry, Symmetry: Culture and Science 22 (2011) 237-246 (363 KB)
    16. With Nikolai Dolbilin: Properties of Böröczky tilings in high dimensional hyperbolic spaces , European J. Combin. 31 (2010) 1181-1195
    17. 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
    18. With Alexey Glazyrin: The lonely vertex problem, Beiträge Algebra Geom. 50 (2009) 71-79
    19. Substitution tilings with statistical circular symmetry, European J. Combin. 29 (2008) 1881-1893
    20. Counting perfect colourings of plane regular tilings, Z. Krist. 223 (2008) 773-776
    21. With Reinhard Lück: Ten colours in quasiperiodic and regular hyperbolic tilings, Z. Krist. 223 (2008) 782-784 (2 MB)
    22. 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
    23. About substitution tilings with statistical circular symmetry, Phil. Mag. 88 (2008) 2033-2039
    24. Self-dual tilings with respect to star-duality , Theoret. Comput. Sci. 391 (2008) 39-50 (595 KB)
    25. 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)
    26. With Michael Baake, Uwe Grimm: Pinwheel patterns and powder diffraction, Phil. Mag. 87 (2007) 2831-2838
    27. With Bernd Sing: Computing Modular Coincidences for substitution tilings and point sets, Discrete Comput. Geom. 37 (2007) 381-407 (456 KB)
    28. Duality of Model Sets Generated by Substitutions, (634 KB), Romanian J. of Pure and Applied Math. 50 (2005) 619-639
    29. With Michael Baake: SCD Patterns Have Singular Diffraction, Journal of Math. Physics 46 (2005)
    30. Selfsimilar Tilings with Integer Factor, Symmetry: Art and Science 2 (2002) 379-384

    Eternal Preprints Didactics:
    1. With Daniel Frohn: Wallpaper Groups, mathematik lernen 204/2017
    2. 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
    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: