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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 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, 29 (2018) 1114-1130 arXiv:1711.01498
    4. With Alexey Garber: Weighted 1x1 cut-and-project sets in bounded distance to a lattice Discrete and Computational Geometry (2018) arXiv:1706.01089
    5. 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
    6. 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)
    7. 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
    8. With Alexey Garber: Symmetries of monocoronal tilings, Discrete Mathematics and Theoretical Computer Science 17 (2015) 203-234
    9. With Hofstetter Kurt: Inductive rotation tilings, Proc. Steklov Inst. 288 (2015) 269-280
    10. With Benjamin Whitehead, Michael Whittaker: Cohomology of the Pinwheel Tiling, Journal of the Australian Math Society 97 (2014) 162-179
    11. With Reinhard Lück: Hyperbolic icosahedral tilings by Buckyballs, Acta Physica Polonica A 126 (2014) 524-526
    12. With Christoph Richard: Dynamical properties of almost repetitive Delone sets, Discrete and Continuous Dynamical Systems 34 (2014) 531-556
    13. With Edmund Harriss: Parallelogram tilings, worms and finite orientations, Discrete and Computational Geometry 49 (2013) 531-539
    14. With Paulo Bugarin, Louise de las Penas: Perfect colourings of cyclotomic integers, Geometriae Dedicata 162 (2013) 271-282
    15. With Valérie Berthé, Victor Sirvent: Self-dual substitutions in dimension one, European Journal of Combinatorics 33 (2012) 981-1000
    16. A fractal fundamental domain with 12-fold symmetry, Symmetry: Culture and Science 22 (2011) 237-246 (363 KB)
    17. With Nikolai Dolbilin: Properties of Böröczky tilings in high dimensional hyperbolic spaces , European J. Combin. 31 (2010) 1181-1195
    18. 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
    19. With Alexey Glazyrin: The lonely vertex problem, Beiträge Algebra Geom. 50 (2009) 71-79
    20. Substitution tilings with statistical circular symmetry, European J. Combin. 29 (2008) 1881-1893
    21. Counting perfect colourings of plane regular tilings, Z. Krist. 223 (2008) 773-776
    22. With Reinhard Lück: Ten colours in quasiperiodic and regular hyperbolic tilings, Z. Krist. 223 (2008) 782-784 (2 MB)
    23. 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
    24. About substitution tilings with statistical circular symmetry, Phil. Mag. 88 (2008) 2033-2039
    25. Self-dual tilings with respect to star-duality , Theoret. Comput. Sci. 391 (2008) 39-50 (595 KB)
    26. 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)
    27. With Michael Baake, Uwe Grimm: Pinwheel patterns and powder diffraction, Phil. Mag. 87 (2007) 2831-2838
    28. With Bernd Sing: Computing Modular Coincidences for substitution tilings and point sets, Discrete Comput. Geom. 37 (2007) 381-407 (456 KB)
    29. Duality of Model Sets Generated by Substitutions, (634 KB), Romanian J. of Pure and Applied Math. 50 (2005) 619-639
    30. With Michael Baake: SCD Patterns Have Singular Diffraction, Journal of Math. Physics 46 (2005)
    31. 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: