Illumination of Pascal's Hexagrammum and Octagrammum Mysticum

TL;DR

This paper generalizes classical geometric configurations involving Pascal's lines and extends results to complex conic arrangements in the Mystic Octagon, using combinatorial methods and computational experiments.

Contribution

It introduces new general results on Pascal's and Mystic Octagon configurations, combining classical geometry with combinatorial and computational approaches.

Findings

Proved general results encompassing classical Pascal's lines.

Established analogous results for 2520 conics in Mystic Octagon.

Connected geometric configurations with $k$-nets of algebraic curves.

Abstract

We prove general results which include classical facts about 60 Pascal's lines as special cases. Along similar lines we establish analogous results about configurations of 2520 conics arising from Mystic Octagon. We offer a more combinatorial outlook on these results and their dual statements. Bezout's theorem is the main tool, however its application is guided by the empirical evidence and computer experiments with program Cinderella. We also emphasize a connection with $k$-nets of algebraic curves.