In 1899, the 1st proof of Morley’s theorem was given by Frank Morley, professor of mathematics at Johns Hopkins University, Baltimore, Maryland. Subsequently many investigators provided more proofs. This theorem is here to stay. Remarkably, it took roughly 1200 years since Euclid invented Euclidean geometry for this elegantly simple theorem to become a mathematical truth. More remarkable is the fact that it was discovered at the eve of the quantum revolution initiated by Max Planck in 1900. Although global infinity is unbounded and incomprehensible, the quantum nature of reality hides the truth about infinitesimal regular infinities.
One of these is commonly called the spin. This property is shared by both real and virtual elementary particles. Without it nothing can or will exist. Although spin zero particles exist, they are simply a balance of opposite spins: the up and down, left and right, forward and backward spins of unit h/2p. Where and when normalized by 2p/h, the spin quantum numbers are multiples of ±˝. Another regular infinity is electric charge which is shared also by all elementary particles. Like spin, electric charge is also quantized with at least six irreducible rational states: ±1, ±1/3, and ±2/3. The first pair is reserved for leptons and most colorless mesons and baryons. The last two pairs are reserved for quarks. Another regular infinity is color charge. There are again six states: red, antired, green, antigreen, blue, and antiblue. These have no physical relation with the primary colors of visible light. However, they provide the eight quantum states of the gluons. Mysteriously, color charge and fractal electric charge are exclusive property of quarks responsible for the strong nuclear force while integer electric charges are responsible for the electromagnetic force. Including the weak charges of intermediate vector bosons: ±W and Z0 with chargeless photon, they completely describe the electroweak force shared by both quarks and leptons while the electromagnetic force is predominantly for leptons and colorless (color balance) particles such as protons and neutrons forming the chemical atoms and molecules. More regular infinities are the bipolarities of magnetic charge of north and south poles. Each elementary particle must exist exactly with one magnetic north pole and one magnetic south pole and for every pair the pole strengths are always equal. Efforts to separate this duality into distinct magnetic monopoles have all failed. Magnetism is a universal physical property shared by all elementary particles whether real or virtual. One lasting regular infinity is the existence of gravitational charge or inertial charge usually called gravitational mass or inertial mass. As far as experimental verifications go, they are truly equivalent. This principle of equivalence becomes the point of departure for the general theory of relativity. However, a theory of mass for its regular infinity of values inclusively between zero and infinity does not exist. Fortunately, all these regular infinities can be described together with a principle of directional invariance by the theory of space-time quantization of squares of energy using the binary operational calculus of second order Hadamard matrices embedded in a sieve of Diophantus.