The Empty Hexagon Theorem

The Trisected Triangle and the Hexagon Within: Marion Walter’s Theorem

Armed only with a triangle and a dream, Marion Walter, a professor of mathematics at the University of Oregon, embarked on a quest for mathematical enlightenment. She battled rough angles and contemplated numerous strategies to find answers. In the end, Walter discovered characteristics of a triangle when each of its sides is trisected. Okay, Walter’s “quest” was not that exciting or epic, but ...

The Classical Aristotelian Hexagon Versus the Modern Duality Hexagon

1 Aristotelian versus Duality Square (1) Piaget (1949) inversion réciprocation corrélation Gottschalk (1953) complement contradual dual Löbner (1990) negation (NEG) subnegation (SNEG) dual (DUAL) Westerstahl (2010) outer negation inner negation dual Quantifier (domain/restrictor, predicate/nuclear scope) 2-place second-order predicates All (children , be asleep) <,,t> (3) All children a...

The Largest Small Hexagon

Given any n-gon of unit diameter in the plane, it is natural to inquire how large an area it can have. In 1922, Reinhardt [8] showed that for odd II, the obvious configuration is optimal, namely, among all n-gons of diameter 1, the regular n-gon has the maximum area. It was noted, however, that this is not the case for n even.l In particular, H. Lenz [7] raised the question of finding the large...

The Turán Number for the Hexagon

It is shown that the maximal number of edges in an n-vertex graph containing no cycle of length six has order of magnitude 12n 4/3.

Hexagon biquadrangle systems