Among all bodies of constant width in the Euclidean plane, the Reuleaux triangle of the same width has minimal area. But Reuleaux triangles are also minimal in another sense: if a convex body can be covered by a translate of a Reuleaux triangle, then it can be covered by a translate of any convex body of the same constant width. The first result is known as the Blaschke-Lebesgue theorem, and it...

This paper is a collection of thoughts and observations, being partly a review and partly a report of current research, on recent work in various aspects of Grünbaum colorings, their existence and usage. In particular, one of the most striking significances of Grünbaum’s Conjecture in the 2-dimensional case is its equivalence to the 4-Color Theorem. The notion of Grünbaum coloring is extended f...

After first determining criteria for wild ramification of L/K (which can only happen at primes above 2), the above result is obtained for n = 2 (e ≥ 3) by computing TL/K(OL) explicitly, and is then extended to the general case. This approach does not rely on Leopoldt’s Theorem, in contrast to the techniques used in [6]. The explicit nature of the calculations used to compute I(L/K) leads to the...

Bilattices and d-frames are two different kinds of structures with a four-valued interpretation. Whereas d-frames were introduced with their topological semantics in mind, the theory of bilattices has a closer connection with logic. We consider a common generalisation of both structures and show that this not only still has a clear bitopological semantics, but that it also preserves most of the...