6/26/2023 0 Comments Hyperspaces geometry![]() ![]() For a cube which had sides a, b, and c, the diagonal, d, inside the cube, crossing from one corner to another was given by a 2 + b 2 + c 2 = d 2. The Pythagorean Theorem from the Greeks had shown that in a two-dimensional world where a and b are the short sides of a triangle and c is the hypotenuse, then a 2 + b 2 = c 2. Riemann had discovered what we now called "field theory," which connects forces in the universe with the geometry of space. In a brilliant lecture on Jthis shy, mentally unstable young man toppled the Euclidean world order once and for all. But a sickly, brilliant mathematician-the second of six children born to a poor Lutheran pastor-Bernhard Riemann (1826-1866) blew the world apart when he proved mathematically that more than three dimensions were not only possible but also highly likely. Until the middle of the last century there was not much talk of a possible Fourth Dimension. Medieval art even accommodated this orthodoxy-paintings were flattened and two dimensional so the viewer could (sort of) see the world as God sees it. God would surely live there, thus He could watch everything that was happening in our 3-D world. Aristotle and Ptolemy added their weight to Euclid by "proving" that any more than three dimensions was "impossible." 1īut of course for those who believed in God, there must be a fourth dimension. Beyond that it was, for centuries, thought impossible for more dimensions to exist. ![]() A line had one dimension, a square had two, and a cube, three. The Greek philosopher Euclid (330-275 B.C.) put this down in a mathematical format we now call plane geometry-which ruled the world for the next 2000 years-almost like a religion! In grade school we all learned that the angles of a triangle must add up to 180 degrees, and that a straight line is the shortest distance between two points. We do not have to be told that the world we live in has three dimensions: length, width, and height. The analogous results for self-conformal sets that satisfy the Open Set Condition are developed in Chapter 4.Born into this world with two eyes, two ears, two arms, two legs and a wonderful data-crunching computer system in both halves of our brains, we humans develop a perception of space as small infants. In Chapter 3, a one-parameter family of gauge functions is constructed which computes the dimensions of the hyperspaces of graph-self-similar sets that satisfy the Strong Separation Condition, after which the approximations of Chapter 2 are applied to extend the result to graph-self-similar sets which satisfy the Open Set Condition. In Chapter 2 it is shown that the dimensions of the underlying fractals may be approximated by the dimensions of sets invariant under particularly constructed subiterated function systems that satisfy the Strong Separation Condition. This dissertation further generalizes these results to include graph-self-similar and self-conformal fractals satisfying the Open Set Condition in Rd. Hyperspaces have been extensively studied by topologists since the 1970's, but the measure theoretical study of hyperspaces has lagged, Boardman and Goodey concurrently provided a characterization of a one-parameter family of Hausdorff gauge functions that determine the dimension of H(), and this result was extended by McClure to H(X) where X is a self-similar fractal satisfying the Open Set Condition. H(K) is itself a metric space under the Hausdorff metric dH. Given a metric space (K, d), the hyperspace of K is defined by H(K) =.
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