The two-dimensional sphere can’t be approximated by finite homogeneous spaces. We describe the optimal approximation and its distance from sphere. compare this to achieved all Platonic Archimedean solids.

All sciences need and many arts apply mathematics whereas mathematics seems to be independent of all of them, but only based upon logic. This conservative concept, however, needs to be revised because, contrary to Platonic idealism (frequently called "realism" by mathematicians), mathematical ideas, notions, and, in particular, numbers are not at all independent of physical laws and prerequisites.

We consider rotations A, B of finite order in SO(3), about axes separated by an angle of restricted type, and attempt to classify the possible group relations between A and B. We show that the relations responsible for the symmetries of Platonic solids have consequences far beyond that simple geometric setting.

A number of perennial doubts plague genre theory. Are genres really 'out there' in the world, or are they merely the constructions of analysts? Is there a finite taxonomy of genres or are they in principle infinite? Are genres timeless Platonic essences or ephemeral, time-bound entities? Are genres culturebound or transcultural?... Should genre analysis be descriptive or proscriptive? (Stam 200...

theory of ideas (muthul) in fact delineates the infrastructure of plato’s philosophy and metaphysics. it can be stated with certainty that by presenting this theory, plato intended to justify the generation of the mundus sensibili in view of its transitoriness and not being assuredly recognizable. encountering this theory, aristotle believes that professing to it would yield illogical results. ...

This paper is an artist statement describing a manipulation of the Mandelbrot Set equation as the basis for creating semi-figurative images, using a genetic algorithm. Modernist painting is referenced in terms of the play between representation and abstraction. The Platonic implications of the Mandelbrot Set are considered as a point of departure for manipulation of the complex function.

The notion of a spiral unfolding of a convex polyhedron, resulting by flattening a special type of Hamiltonian cut-path, is explored. The Platonic and Archimedian solids all have nonoverlapping spiral unfoldings, although among generic polyhedra, overlap is more the rule than the exception. The structure of spiral unfoldings is investigated, primarily by analyzing one particular class, the poly...

In this note we will determine the symmetry groups of the Platonic solids. We will use Maple to help us do this. The five Platonic solids are the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron. We will view the symmetry groups of these solids in two ways. By definition, the symmetry group of a solid is the set of isometries of R that stabilize the set. In each of t...

Consider the following situation: Alice is a hand-held device, such as a PDA. Bob is a device providing a service, such as an ATM, an automatic door, or an anti-aircraft gun pointing at the gyro-copter in which Alice is travelling. Bob and Alice have never directly met before, but share a key as a result of secure hand-offs. Alice has used this key to request a service from Bob (dispense cash, ...

Mathematicians study shapes, structures and patterns. However, there are shapes, structures and patterns within the body and practice of mathematics that are not the direct objects of mathematical study. Rather, they are part of the explanation of how mathematical study is possible, and thus demand the attention of epistemologists and phenomenologists as well as mathematicians. Partial philosop...