# Báo cáo toán học: "Constructing 5-conﬁgurations with chiral symmetry"

## Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài:Constructing 5-conﬁgurations with chiral symmetry. | Constructing 5-configurations with chiral symmetry Leah Wrenn Berman University of Alaska Fairbanks Fairbanks Alaska USA lwberman@ Laura Ng Phoenixville Pennsylvania USA Submitted Mar 16 2009 Accepted Dec 11 2009 Published Jan 5 2010 Mathematics Subject Classification 05B30 51E30 Abstract A 5-configuration is a collection of points and straight lines in the Euclidean plane so that each point lies on five lines and each line passes through five points. We describe how to construct the first known family of 5-configurations with chiral that is only rotational symmetry and prove that the construction works in addition the construction technique produces the smallest known geometric 5-configuration. In recent years there has been a resurgence in the study of k-configurations with high degrees of geometric symmetry that is in the study of collections of points and straight lines in the Euclidean plane where each point lies on k lines and each line passes through k points with a small number of symmetry classes of points and lines under Euclidean isometries that map the configuration to itself. 3-configurations have been studied since the late 1800s see . 15 Ch. 3 and more recently 9 12 13 and there has been a great deal of recent investigation into 4-configurations . see 1 2 5 8 14 . However there has been little investigation into k-configurations for k 4. Following 10 we say that a geometric k-configuration is polycyclic if a rotation by angle 2m for some integers i and m is a symmetry operation that partitions the points and lines of the configuration into equal-sized symmetry classes orbits where each orbit contains m points. If n dm then there are d orbits of points and lines under the rotational symmetry. The group of symmetries of such a configuration is at least cyclic. In many cases the full symmetry group is dihedral this is the case for most known polycyclic 4-configurations. A k-configuration is astral if it has k 1 J .

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