Finite tight frames have many applications and some interesting physical interpretations. One of the important subjects in this area is the ways for constructing such frames. In this article we give a concrete method for constructing finite normalized frames using Platonic solids.

In this appendix we shall find all the irreducible representations of the symmetry groups of the Platonic solids, by a mixture of geometric methods and algebraic methods similar to those used in Chapters 5 for representations of the classical groups. We shall also see how these representations occur naturally in the harmonic analysis of functions on the Platonic solids. This is a discrete analo...

Philosophers have long debated about two ways of conceiving of universals: as Platonic universals and as Aristotelian universals. Roughly, Aristotelian universals are inherent in the particulars that instantiate them; they can be multiply located (located just where the instances are located), and they exist only if they have at least one instance. Platonic universals, by contrast, are truly ab...

Edward Feser defends the ‘Neo-Platonic proof’ for existence of God classical theism. After articulating argument and a number preliminaries, I first argue that premise three Feser’s – causal principle every composite object requires sustaining efficient cause to combine its parts is both unjustified dialectically ill-situated. then Neo-Platonic proof fails deliver mindedness absolutely simple b...

In this paper, I show that we can absolutely eliminate notions of apriority (axioms, theorems) in symbolic logic and mathematics and build up formal theories using only notions of aposteriority. The new kind of logic and mathematics is said to be anti-Platonic. Both satisfy basic presuppositions of unconventional computing. 1. Platonism in Logic and Mathematics According to Platonism, mathemati...

Plato’s geometrical theory of what we now call chemistry, set out in the Timaeus, uses triangles, his stoicheia, as the fundamental units with which he constructs his four elements. A paper claiming that these triangles can be divided indefinitely is criticized; the claim of an error here in the commentary by F.M. Cornford is unfounded. Plato’s constructions of the elements are analyzed using s...

We develop a combinatorial method to show that the dodecahedron graph has, up to rotation and reflection, a unique Hamiltonian cycle. Platonic graphs with this property are called topologically uniquely Hamiltonian. The same method is used to demonstrate topologically distinct Hamiltonian cycles on the icosahedron graph and to show that a regular graph embeddable on the 2-holed torus is topolog...