Self-affine convex discs are polygons
Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry - Tập 53 - Trang 219-224 - 2011
Tóm tắt
A disc in
$${{\mathbb R}^2}$$
is called self-affine if it can be dissected into m ≥ 2 affine images of itself. We show that every self-affine convex disc D is a polygon. As a corollary, it turns out that D must be a triangle, a quadrangle or a pentagon.
Tài liệu tham khảo
Bernheim B., Motzkin T.: A criterion for divisibility of n-gons into k-gons. Comment. Math. Helv. 22, 93–102 (1949)
Croft H.T., Falconer K.J., Guy R.K.: Unsolved problems in geometry, Problem Books in Mathematics. Unsolved Problems in Intuitive Mathematics, II. Springer-Verlag, New York (1991)
Falconer K.: Fractal geometry. Mathematical foundations and applications. John Wiley & Sons, Chichester (1997)
Hertel, E.: Zur Affingeometrie konvexer Polygone. Jenaer Schriften zur Mathematik und Informatik, Math/Inf/00/22 (2000) (preprint)
Tölke, J. Wills, J.M. (eds): Contributions to geometry. Proceedings of the Geometry-Symposium held in Siegen, June 28 to July 1, 1978. Birkhäuser Verlag, Basel (1979)