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...

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.

Distance regular graphs fall into three families: primitive, antipodal, and bipar-tite. Each antipodal distance regular graph is a covering graph of a smaller (usually primitive) distance regular graph; the antipodal distance graphs of diameter three are covers of the complete graph, and are the first non-trivial case. Many of the known examples are connected with geometric objects, such as pro...

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, ...