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Even in a universe that displays quantum structure, we cannot avoid "continuum", which means: 'connections'. But quantum architecture forces 'separation'. And more importantly, our mathematical house is built on the bedrock of another more subtle separation, that found in Pythagorean trigonometry. We assume that trigonometric descriptions of wave form relationships are sufficient and complete as they stand. But what relationship does trig rely on? If a wave is the domain basis of a system, trigonometry defines that domain from a locus which is not part of the domain, it establishes the domain from a locus set apart, as the radial distance from the center of a circle. The focus of the circle is not part of the circumference, it is separate, apart. But what is the prime characteristic of topology and symmetry breaking? It is the continuity of connection, of a locus dimensionally decompressing into extraordinary dimensionality. The dimensionally participant locus gives birth to dimensional pluralities, yet always being resident within all arising domains and topologies. Trigonometry is unquestionably valid, so where are these other equally valid and important relations.
Apollonius did it by exploring an alternative way of rendering a circle.