A Review and Mathematical Treatment of Infinity on the Smith Chart, 3D Smith Chart and Hyperbolic Smith Chart

This work describes the geometry behind the Smith chart, recent 3D Smith chart tool and previously reported conceptual Hyperbolic Smith chart. We present the geometrical properties of the transformations used in creating them by means of inversive geometry and basic non-Euclidean geometry. The beauty and simplicity of this perspective are complementary to the classical way in which the Smith chart is taught in the electrical engineering community by providing a visual insight that can lead to new developments. Further we extend our previous work where we have just drawn the conceptual hyperbolic Smith chart by providing the equations for its generation and introducing additional properties.

SELF-SIMILAR SPACE-FILLING SPHERE PACKINGS IN THREE AND FOUR DIMENSIONS

Inversive geometry can be used to generate exactly self-similar space-filling sphere packings. We present a construction in two dimensions and generalize this construction to search for packings in higher dimensions. We discover 29 new three-dimensional topologies of which 10 are bearings, and 13 new four-dimensional topologies of which five are bearings. To characterize the packing topologies, we estimate numerically their fractal dimensions and we analyze their contact networks.

CUTTING SELF-SIMILAR SPACE-FILLING SPHERE PACKINGS

Any space-filling packing of spheres can be cut by a plane to obtain a space-filling packing of disks. Here, we deal with space-filling packings generated using inversive geometry leading to exactly self-similar fractal packings. First, we prove that cutting along a random hyperplane leads in general to a packing with a fractal dimension of the one of the uncut packing minus one. Second, we find special cuts which can be constructed themselves by inversive geometry. Such special cuts have specific fractal dimensions, which we demonstrate by cutting a three- and a four-dimensional packing. The increase in the number of found special cuts with respect to a cutoff parameter suggests the existence of infinitely many topologies with distinct fractal dimensions.

On a Diophantine Equation That Generates All Integral Apollonian Gaskets

A remarkably simple Diophantine quadratic equation is known to generate all, Apollonian integral gaskets (disk packings). A new derivation of this formula is presented here based on inversive geometry. Also, occurrence of Pythagorean triples in such gaskets is discussed.