generalized polygon
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10.37236/8476 ◽  
2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Bart De Bruyn

A triple $(\mathcal{S},S,\mathcal{Q})$ consisting of a near polygon $\mathcal{S}$, a line spread $S$ of $\mathcal{S}$ and a set $\mathcal{Q}$ of quads of $\mathcal{S}$ is called a polygonal triple if certain nice properties are satisfied, among which there is the requirement that the point-line geometry $\mathcal{S}'$ formed by the lines of $S$ and the quads of $\mathcal{Q}$ is itself also a near polygon. This paper addresses the problem of classifying all near polygons $\mathcal{S}$ that admit a polygonal triple $(\mathcal{S},S,\mathcal{Q})$ for which a given generalized polygon $\mathcal{S}'$ is the associated near polygon. We obtain several nonexistence results and show that the $G_2(4)$ and $L_3(4)$ near octagons are the unique near octagons that admit polygonal triples whose quads are isomorphic to the generalized quadrangle $W(2)$ and whose associated near polygons are respectively isomorphic to the dual split Cayley hexagon $H^D(4)$ and the unique generalized hexagon of order $(4,1)$.





2016 ◽  
Vol 24 (5) ◽  
pp. 5300 ◽  
Author(s):  
Yuntian Zhang ◽  
Fengliang Dong ◽  
Kemao Qian ◽  
Qingchuan Zhang ◽  
Weiguo Chu ◽  
...  
Keyword(s):  


2015 ◽  
Vol 32 (1) ◽  
pp. 014205 ◽  
Author(s):  
Yun-Tian Zhang ◽  
Zhi-Gang Zhang ◽  
Teng Cheng ◽  
Qing-Chuan Zhang ◽  
Xiao-Ping Wu
Keyword(s):  




2003 ◽  
Vol 271 (1-3) ◽  
pp. 223-234 ◽  
Author(s):  
Behnaz Omoomi ◽  
Yee-Hock Peng


1999 ◽  
Vol 127 (4) ◽  
pp. 343-347
Author(s):  
Bernhild Stroppel ◽  
Markus Stroppel


1997 ◽  
Vol 172 (1-3) ◽  
pp. 103-114 ◽  
Author(s):  
Yee-Hock Peng ◽  
C.H.C. Little ◽  
K.L. Teo ◽  
H. Wang


1997 ◽  
Vol 17 (4) ◽  
pp. 849-867 ◽  
Author(s):  
EUGENE GUTKIN ◽  
NICOLAI HAYDN

We study the topological entropy of a class of transformations with mild singularities: the generalized polygon exchanges. This class contains, in particular, polygonal billiards. Our main result is a geometric estimate, from above, on the topological entropy of generalized polygon exchanges. One of the applications of our estimate is that the topological entropy of polygonal billiards is zero. This implies the subexponential growth of various geometric quantities associated with a polygon. Other applications are to the piecewise isometries in two dimensions, and to billiards in rational polyhedra.



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