The Pythagorean Theorem and variants of it are studied. The variations evolve to a formulation in terms of noncommutative, conditional expectations on von Neumann algebras that displays the theorem as the basic result of noncommutative, metric, Euclidean Geometry. The emphasis in the present article is finite dimensionality, both "discrete" and "continuous."

The Pythagorean tree is a fractal figure with a branching structure based upon the Pythagorean theorem. This paper describes the use of a greedy algorithm to encourage a homogenous spread of detail for asymmetrical trees and avoid localized regions of high detail. A model of prioritized branch growth based on this idea is used to optimize tree shape, allowing more visually interesting trees to ...

The Pythagorean tree is a fractal figure with a branching structure based upon the Pythagorean theorem. This paper describes the use of a greedy algorithm to encourage a homogenous spread of detail for asymmetrical trees and avoid localized regions of high detail. A model of prioritized branch growth based on this idea is used to optimize tree shape, allowing more visually interesting trees to ...

The Pythagorean theorem can be proven geometrically through the use of dissections of squares and triangles. Four of these decompositions are paired into two compound dissections, which are used to create novel images. Pythagorean Theorem Many proofs of the Pythagorean theorem are given in the literature, for example, at the MathWorld web site [1]. Of particular interest to this work are dissec...

Recently, Gao introduced some moduli from Pythagorean theorem. In terms of these moduli, he got some sufficient conditions for a Banach space X to have uniform normal structure, which plays an import role in fixed-point theory. In this paper, we mainly discuss the moduli E (X) and f (X). Let X be a Banach space. By SX and BX we will denote the unit sphere and unit ball of X , respectively. For ...

A famous theorem in Euclidean geometry often attributed to the Greek thinker Pythagoras of Samos (6th century, B.C.) states that if one of the angles of a planar triangle is a right angle, then the square of the length of the side opposite the right angle equals the sum of the squares of the lengths of the sides which form the right angle. There are less commonly known higherdimensional version...