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Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Vol. 41, No. 2, pp. 589-595 (2000)
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Self-similar Simplices

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Eike Hertel

Mathematisches Institut, Friedrich-Schiller-Universität Jena, Ernst-Abbe-Platz 1-4, D-07743 Jena, Germany, e-mail: hertel@minet.uni-jena.de

**Abstract:** A Euclidean $d$-simplex ${\cal S}$ is called $k$-self-similar if ${\cal S}$ can be dissected into $k\ge2$ simplices, each similar to ${\cal S}$. Each triangle $(d=2)$ is $k$-self-similar for $k=4$ and $k\ge6$ whereas for $d>2$ most $d$-simplices are not self-similar. A first class of 3-simplices which are $m^3$-self-similar for all positive integers $m$ is characterized.

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