(Homepage)
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:
- With Christoph Richard: Dynamical
properties of almost repetitive Delone sets, submitted
- With Edmund Harriss: Parallelogram
tilings, worms and finite orientations, to appear in Discrete
and Computational Geometry
- With Paulo Bugarin, Louise de las Penas:
Perfect colourings of cyclotomic
integers, Geometriae Dedicata (2012) DOI 10.1007/s10711-012-9726-0
- The slides of the talk
'On colour symmetries of hyperbolic regular tilings and cyclotomic
integers' in Dortmund, July 2009 (1.1 MB)
- With Valérie Berthé, Victor Sirvent:
Self-dual substitutions in dimension one,
European Journal of Combinatorics 33 (2012) 981-1000
- The slides of the talk
'About duality of cut-and-project tilings' in Davidson, NC, March
2007 (1.2 MB)
- A fractal fundamental domain
with 12-fold symmetry, Symmetry: Culture and Science 22 (2011)
237-246 (363 KB)
- With Nikolai Dolbilin: Properties of
Böröczky tilings in high dimensional hyperbolic spaces
, European J. Combin. 31 (2010) 1181-1195
- 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
- With Alexey Glazyrin: The lonely vertex
problem, Beiträge Algebra Geom. 50 (2009) 71-79
- Substitution tilings with statistical
circular symmetry, European J. Combin. 29 (2008) 1881-1893
- The slides of the
corresponding talk 'About substitution tilings with statistical
circular symmetry' in Leicester, Sept 2009 (1.4 MB)
- Counting perfect colourings of plane
regular tilings, Z. Krist. 223 (2008) 773-776
- The slides of the corresponding
talk in London, July/August 2008 (745 KB)
- With Reinhard Lück: Ten colours in
quasiperiodic and regular hyperbolic tilings, Z. Krist. 223 (2008)
782-784 (2 MB)
- 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
- About substitution tilings with statistical
circular symmetry, Phil. Mag. 88 (2008) 2033-2039
- Self-dual tilings with respect to
star-duality , Theoret. Comput. Sci. 391 (2008) 39-50 (595 KB)
- The slides of the corresponding
talk 'Self-duality of Galois-dual tilings' in Marseille, March 2006
(850 KB)
- The slides of the talk
'Self-duality and *-dual tilings' in Liège, May 2006 (866 KB)
- 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)
- With Michael Baake, Uwe Grimm:
Pinwheel patterns and
powder diffraction, Phil. Mag. 87 (2007) 2831-2838.
- With Bernd Sing: Computing Modular Coincidences for
substitution tilings and point sets, Discrete Comput. Geom. 37
(2007) 381-407 (456 KB)
- Duality of Model Sets Generated by
Substitutions, (634 KB), Romanian J. of Pure and
Applied Math. 50 (2005) 619-639
- With Michael Baake:
SCD Patterns Have Singular Diffraction, Journal of Math. Physics
46 (2005)
- 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