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Journal for Geometry and Graphics 24 (2020), No. 2, 175--191
Copyright Heldermann Verlag 2020

The Pentagrammum Mysticum, Twelve Special Conics and the Twisted Icosahedron

Nguyen Le
Dept. of Mathematics, San Francisco State University, U.S.A.

Norman J. Wildberger
School of Mathematics and Statistics, University of New South Wales, Sydney, Australia

We study an abstract pentagon in planar projective geometry via its unique circumscribing conic, associated to 15 diagonal lines and exagonal points, and describe 12 beautifully inter-related conics. These are closely connected to a distance-regular graph X which is a sister of the icosahedral graph, arising from dihedral orderings of five objects.

Keywords: Strongly regular graph, icosahedron, Pascal theorem, conic, pentagon.

MSC: 14Nxx; 51A20, 05Exx

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