$n^2 + 1$ unit equilateral triangles cannot cover an equilateral triangle of side $> n$ if all triangles have parallel sides
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arXiv
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| Natura: | Preprint |
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2023
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| _version_ | 1866909214214979584 |
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| author | Baek, Jineon Lee, Seewoo |
| author_facet | Baek, Jineon Lee, Seewoo |
| contents | Conway and Soifer showed that an equilateral triangle $T$ of side $n + \varepsilon$ with sufficiently small $\varepsilon > 0$ can be covered by $n^2 + 2$ unit equilateral triangles. They conjectured that it is impossible to cover $T$ with $n^2 + 1$ unit equilateral triangles no matter how small $\varepsilon$ is.
We show that if we require all sides of the unit equilateral triangles to be parallel to the sides of $T$ (e.g. $\bigtriangleup$ and $\bigtriangledown$), then it is impossible to cover $T$ of side $n + \varepsilon$ with $n^2 + 1$ unit equilateral triangles for any $\varepsilon > 0$. As the coverings of $T$ by Conway and Soifer only involve triangles with sides parallel to $T$, our result determines the exact minimum number $n^2+2$ of unit equilateral triangles with all sides parallel to $T$ that cover $T$. We also determine the largest value $\varepsilon = 1/(n + 1)$ (resp. $\varepsilon = 1 / n$) of $\varepsilon$ such that the equilateral triangle $T$ of side $n + \varepsilon$ can be covered by $n^2+2$ (resp. $n^2 + 3$) unit equilateral triangles with sides parallel to $T$, where the first case is achieved by the construction of Conway and Soifer. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_09533 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | $n^2 + 1$ unit equilateral triangles cannot cover an equilateral triangle of side $> n$ if all triangles have parallel sides Baek, Jineon Lee, Seewoo Combinatorics Metric Geometry 52C15 (Primary) 05B40 (Secondary) Conway and Soifer showed that an equilateral triangle $T$ of side $n + \varepsilon$ with sufficiently small $\varepsilon > 0$ can be covered by $n^2 + 2$ unit equilateral triangles. They conjectured that it is impossible to cover $T$ with $n^2 + 1$ unit equilateral triangles no matter how small $\varepsilon$ is. We show that if we require all sides of the unit equilateral triangles to be parallel to the sides of $T$ (e.g. $\bigtriangleup$ and $\bigtriangledown$), then it is impossible to cover $T$ of side $n + \varepsilon$ with $n^2 + 1$ unit equilateral triangles for any $\varepsilon > 0$. As the coverings of $T$ by Conway and Soifer only involve triangles with sides parallel to $T$, our result determines the exact minimum number $n^2+2$ of unit equilateral triangles with all sides parallel to $T$ that cover $T$. We also determine the largest value $\varepsilon = 1/(n + 1)$ (resp. $\varepsilon = 1 / n$) of $\varepsilon$ such that the equilateral triangle $T$ of side $n + \varepsilon$ can be covered by $n^2+2$ (resp. $n^2 + 3$) unit equilateral triangles with sides parallel to $T$, where the first case is achieved by the construction of Conway and Soifer. |
| title | $n^2 + 1$ unit equilateral triangles cannot cover an equilateral triangle of side $> n$ if all triangles have parallel sides |
| topic | Combinatorics Metric Geometry 52C15 (Primary) 05B40 (Secondary) |
| url | https://arxiv.org/abs/2306.09533 |