If SO is a Riemann surface with a complete metric of finite area and constant curvature −1, let SC denote the conformal compactification of SO. We show that, under the assumption that the cusps of SO are large, there is a close relationship between the hyperbolic metrics on SO and SC . We use this relationship to show that lim infk→∞ λ1(Pk) ≥ 5/36, where the Platonic surface Pk is the conformal...

Foundational philosophical beliefs about mathematics in the mathematical community may have an unappreciated yet profound impact on ethics practice and education, which also affects practice. A historical basis of dominant platonic formalist views are described evaluated, after alternative evidence-based foundation for thought is outlined. The dualistic nature view based intuition then compared...

Is every edge unfolding of every Platonic solid overlapfree? The answer is yes. In other words, if we develop a Platonic solid by cutting along its edges, we always obtain a flat nonoverlapping simple polygon. We also give self-overlapping general unfoldings of Platonic solids other than the tetrahedron (i.e., a cube, an octahedron, a dodecahedron, and an icosahedron), and edge unfoldings of so...

We develop a rigorous theory of observation for dynamical systems in Hilbert spaces. This work generalizes the results of Ott and Yorke [8] for dynamical systems in Rn.

We present a construction of highly entangled states defined on the topology platonic solid using tensor networks based ancillary Absolute Maximally Entangled (AME) states. illustrate idea example quantum state AME(5,2) over dodecahedron. analyze entropy such many different partitions, and observe that they come integer numbers are almost maximal. also all solids accept AME Reed-Solomon codes s...

This paper considers Platonic solids/polytopes in the real Euclidean space R(n) of dimension 3 ≤ n < ∞. The Platonic solids/polytopes are described together with their faces of dimensions 0 ≤ d ≤ n - 1. Dual pairs of Platonic polytopes are considered in parallel. The underlying finite Coxeter groups are those of simple Lie algebras of types A(n), B(n), C(n), F4, also called the Weyl groups or, ...