[AntonioLao]

It can easily be shown using a compass and straight edge that less than half of the points of a given circle of finite radius can be completely mapped into every point of an infinitely extended straight line in both opposite directions.

This singular mathematical fact was not known to Euclid (2300 years ago) when he invented geometry. Since infinitely extended lines were never mentioned in any of the 13 books of the Elements. The Renaissance period (1400–1600) saw the emergence of projective geometry and widespread application of artistic infinitely extended lines called perspectives. One culmination of discoveries is the theorem on the hexagon by Blaise Pascal (1623–1662). In modern language, it asserts that if a hexagon is inscribed in a circle, the three points of intersection of the pairs of opposite sides lie on a line. On the other hand, if the opposite sides are parallel, the points will lie on the line at infinity. This realization prompted him to state: Nature is an infinite sphere, whose center is everywhere and whose circumference is nowhere.

The rise of differential and integral calculus heavily depended on the theory of limits as approaches to infinitesimals and infinities. Subsequent extensions to differential geometry, complex analysis, and conformal mapping allowed Riemann (1826-1866) to invent his complex z-sphere, recalling that a complex number z=x+iy is composed of a real part x and an imaginary part iy where x and y are real numbers and i is the complex unit. This gave meanings to both z=0 and z=∞, respectively, as the south pole and north pole of the corresponding Euclidean sphere which is extensively used in stereographic projection. However, one of the many other subtle implications of Riemann sphere is that the entire Euclidean universe can be mapped completely into less than one hemisphere. This shows that a circle of arbitrary radius (even Planck length) would still be less than half occupied even if the entire universe is mapped by point-set topology of one to one correspondence. Extended to many to one Riemann surfaces would still make the point at z=0 unoccupied even if x<0 and x>0 were topologically connected.

This connectedness can only imply the existence of Riemann closed curves equivalent to Hopf rings. These Hopf topologies could be described by Hadamard matrices. Adding them defines space-time charges: electric, weak, and color. Multiplying them defines physical mass. The ratio of Hadamard proton to Hadamard electron is 1832. The accepted experimental value is approximately 1836. This quarter of a percent discrepancy might be due to the fact that a proton is a composite particle of 2 up quarks and 1 down quark while the electron is truly fundamental.

Going beyond conventional wisdom, it can be constructively argued that both quarks and leptons are composites of absolutely fundamental space-time charges of H+ and H- where and when a circular set is more than enough to cover any linear set that is infinitely extended.