Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Thursday, June 09, 2011, 12:15 pm

Duration: This information is not available in the database

Location: OAT S15/S16/S17

Speaker: Zuzana Safernová (Charles Univ.)

On the nonexistence of k-reptile simplices in R^3 and R^4

A d-dimensional simplex S is called a k-reptile if it can be tiled without overlaps by simplices S_1,S_2,...,S_k that are all mutually congruent and similar to S. For d=2, k-reptile simplices (triangles) exist for many values of k and they have been completely characterized by Snover, Waiveris, and Williams. On the other hand, for d >= 3, only one construction of k-reptile simplices is known, the Hill simplices, and it provides only k of the form m^d, m=2,3,...

We prove that for d=3, k-reptile simplices (tetrahedra) exist only for k=m^3. This partially confirms a conjecture of Hertel, asserting that the only k-reptile tetrahedra are the Hill tetrahedra.

We also prove that for d=4, k-reptile simplices can exist only for k=m^2. For d=4 it remains to find out whether there really exist m^2-reptile simplices for m non-square and whether all m^4-reptile simplices are indeed Hill simplices.

First part is joint work with Jiri Matousek, the second one with Honza Kyncl.

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