Narrow Results

It is well-known that the focal set (i.e. the image of the caustic) of a given convex closed curve γ admits singular points. In this paper, we classify the diffeomorphic type of focal sets of convex curves which admit at most four cusps.

Let F be a field of characteristic p > 0, let 𝔖_n denote the symmetric group of degree n < ∞, and let M be a simple F 𝔖_n-module with vertex V belonging to a block B of F 𝔖_n with defect group Δ. We prove that |Δ| ≤ |V|!, and we also investigate the embedding of V into 𝔖_n.

We study Specht modules S^(n-2,2) and simple modules D^(n-2,2) for symmetric groups 𝔖_n of degree n over a field of characteristic 2. In particular, we determine the vertices of these modules, and also provide some information on their sources.

This chapter initiates the study of the basic details of the Mereon Matrix. Starting with Euler, links are made to the connections between unique and diverse forms. The Mereon Matrix lifts planar geometry into harmonious and elegant three-dimensional geospheres and reveals that a cluster of complex forms function with clockwork precision. It is shown how the regular Platonic and Kepler solids dynamically fit together in a vertex-to-vertex configuration; similarly, vertex-to-edge connections and ‘dual’ relationship between one form and another are studied for the solids appearing in the Mereon Matrix. In conclusion, a simple link to personal transition and social transformation is revealed.

This chapter reveals the depth of the transformational nature inherent in the Mereon Matrix showing how clusters and singles of polyhedra, 33 unified forms, scale to infinity, internally and externally. This chapter suggests significant physical, psychological and sociological implications inherent in the structure and dynamics of the Matrix’s outermost polyhedron. Referred to as the Context, this geosphere iterates between 120 and 180 faces as it opens and closes, expands and contracts. Image intense, it is revealed how localised environmental energy is required, received and processed to eventually produce new systems and/or product that expands the global environment.