Minimal regular polygons serving as universal covers in \({\mathbb{R}}^ 2\) (Q1105200)
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scientific article; zbMATH DE number 4058319
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Minimal regular polygons serving as universal covers in \({\mathbb{R}}^ 2\) |
scientific article; zbMATH DE number 4058319 |
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Minimal regular polygons serving as universal covers in \({\mathbb{R}}^ 2\) (English)
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1988
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A set K in the Euclidean plane is said to be a universal cover if each set of diameter at most 1 can be covered by a congruent copy of K. The authors ask for the smallest (in diameter) regular n-gons serving as universal covers for fixed n. Using a special result of \textit{J. Pál} [Bull. Acad. Dan. 3, No.2, 35 S. (1920), JFM 47.0684.01] for \(n=6\) they prove the following nice theorem: For all \(n>3\) the minimal n-gon that serves as a universal cover is the smallest that contains a Reuleaux triangle of width 1.
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regular polygons
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universal cover
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minimal n-gon
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