Karl Menger, the Menger Sponge, and the Institute for Figuring
Intro Courtesy from the Institute For Figuring
"In the 1920’s a young Austrian named Karl Menger extended the work begun by his mathematical predecessor Siepinski. Menger attended a course of lectures by Professor Hans Hahn at the University of Vienna entitled What’s New Concerning the Concept of the Curve; under Hahn’s encouragement he embarked on an exploration of the concept of dimension that him to an expanded definition of this seemingly obvious term. Several years later Menger reported his discovery of a three-dimensional version of Sierpinski’s Carpet, which came to be known as the Menger Sponge. Where the Carpet is poised between a line and a plane, the Sponge hovers of the boundary of the plane and the solid - its fractional dimension is 2.73.
Though it manifestly occupies a volumetric space, the Menger sponge is essentially a linear object – it possesses a topological dimension of 1. Menger proved that it is indeed the universal curve - that is, any possible one-dimensional curve is mathematically identical to some part of its infinitely complex internal morphology. Though the classical Menger sponge is constructed in three-dimensions, it can be embodied in any number of higher dimensions; consequently any geometry of loop quantum gravity can be embedded in a Menger Sponge. Interestingly then, the structure of spacetime may be allied with this foam-like form."
The Menger Sponge is the 20th century adaptation to the 17th century idea of "Spissitude". Henry More described Spissitude as the fourth spacial dimension. In More's view length, breadth, height, and then spissitude would be the unit of measurement for any object.
Spissitude would find a sound scientific basis in the Minkowski Space theory. Hermann Minkowski, using Einstein's theory of special relativity, acknowledged a single dimension of time, which engulfs the the three known dimensions of space; the effect is a "four-dimensional manifold for representing spacetime. Charles H. Hinton coined the word "tesseract". The physical representation of the fourth dimension, the tesseract was linguistically created in addition to kata/ana. Greek in origin, kata means down from and ana means up toward, kata/ana are similar to the theory of vectors and matrices in their nature.
Intro Courtesy from the Institute For Figuring
"In the 1920’s a young Austrian named Karl Menger extended the work begun by his mathematical predecessor Siepinski. Menger attended a course of lectures by Professor Hans Hahn at the University of Vienna entitled What’s New Concerning the Concept of the Curve; under Hahn’s encouragement he embarked on an exploration of the concept of dimension that him to an expanded definition of this seemingly obvious term. Several years later Menger reported his discovery of a three-dimensional version of Sierpinski’s Carpet, which came to be known as the Menger Sponge. Where the Carpet is poised between a line and a plane, the Sponge hovers of the boundary of the plane and the solid - its fractional dimension is 2.73.
Though it manifestly occupies a volumetric space, the Menger sponge is essentially a linear object – it possesses a topological dimension of 1. Menger proved that it is indeed the universal curve - that is, any possible one-dimensional curve is mathematically identical to some part of its infinitely complex internal morphology. Though the classical Menger sponge is constructed in three-dimensions, it can be embodied in any number of higher dimensions; consequently any geometry of loop quantum gravity can be embedded in a Menger Sponge. Interestingly then, the structure of spacetime may be allied with this foam-like form."
The Menger Sponge is the 20th century adaptation to the 17th century idea of "Spissitude". Henry More described Spissitude as the fourth spacial dimension. In More's view length, breadth, height, and then spissitude would be the unit of measurement for any object.
Spissitude would find a sound scientific basis in the Minkowski Space theory. Hermann Minkowski, using Einstein's theory of special relativity, acknowledged a single dimension of time, which engulfs the the three known dimensions of space; the effect is a "four-dimensional manifold for representing spacetime. Charles H. Hinton coined the word "tesseract". The physical representation of the fourth dimension, the tesseract was linguistically created in addition to kata/ana. Greek in origin, kata means down from and ana means up toward, kata/ana are similar to the theory of vectors and matrices in their nature.
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