# Noon lecture

list of noon lectures ( 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | future lectures)

On 01.04.2010 at 12:20 in S6, there is the following noon lecture:

# On the nonexistence of k-reptile tetrahedra

## Zuzka Safernová

## Abstract

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 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.

Our research has been motivated by the problem of probabilistic packet marking in theoretical computer science, introduced by Adler in 2002.

list of noon lectures ( 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | future lectures)

Webmaster: kamweb.mff.cuni.cz Modified: 25. 02. 2019