This paper arose from the observation that several (families of) near polygons <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {S}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">S</mi> </math></EquationSource> </InlineEquation>, including the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$G_2(4)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>G</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$L_3(4)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>L</mi> <mn>3</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> near octagons, share similar properties. They all have a line spread S and a set <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {Q}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">Q</mi> </math></EquationSource> </InlineEquation> of quads that behave very nicely. In particular, S and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {Q}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">Q</mi> </math></EquationSource> </InlineEquation> define a near polygon <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal {S}'$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="script">S</mi> </mrow> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation> whose diameter is one less than the one of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathcal {S}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">S</mi> </math></EquationSource> </InlineEquation>. In this paper, we derive several properties of such “polygonal triples” <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$(\mathcal {S},S,\mathcal {Q})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">S</mi> <mo>,</mo> <mi>S</mi> <mo>,</mo> <mi mathvariant="script">Q</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and obtain some classification results.

Polygonal triples

期刊论文
发表日期：

2026年3月14日

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Bart De Bruyn

摘要

This paper arose from the observation that several (families of) near polygons $\mathcal {S}$ $S$ , including the $G_2(4)$ $G_{2} (4)$ and $L_3(4)$ $L_{3} (4)$ near octagons, share similar properties. They all have a line spread S and a set $\mathcal {Q}$ $Q$ of quads that behave very nicely. In particular, S and $\mathcal {Q}$ $Q$ define a near polygon $\mathcal {S}'$ $S^{'}$ whose diameter is one less than the one of $\mathcal {S}$ $S$ . In this paper, we derive several properties of such “polygonal triples” $(\mathcal {S},S,\mathcal {Q})$ $(S, S, Q)$ and obtain some classification results.