At the turn the 20th Century, scientists began measuring micro elements as quantum's rather than as continuous units of measure. As scientists rather than as mathematicians, they had no way of interpreting these quantum measurements as a mathematical conundrum. A mathematician could recognize that these measurements were quantum because they had tapped an unrecognized fourth dimension which was composed of quantums. A quantum is defined as a unit of distance that cannot be divided. It is a whole number unit. This is in contrast to standard units of distance which may be infinitely divided.
Further, one of the most significant measurements revealed the mathematical principles governing this unknown geometric dimension. Jann Rydberg had demonstrated that the spectrum of light emitted by hydrogen was distributed as whole number negations of the shortest wavelength in this series (91 1/7 nm). All wavelengths in this spectrum of hydrogen were negations of subdivision of the inverse of this 91 1/7 nm wavelength. The negation of subdivision it Is the only way that non-divisible quantum's may be differentiated into smaller units. The negation of subdivision for the Rydberg distribution is the following:
wavelength-1=(1-1/n2)1/(91 1/7nm) ; 'n'=whole number between '2' and '8'; subdivision=1/n2, negation of subdivision=(1-1/n2). The Rydberg distribution became the basis for the shell/sub-shell distribution of orbits in the standard model of the Atom.
The second quantum measurement which confounded early 20th century science was Max Planck's Constant. In a study of black bodies which absorb all light Energy as heat, Planck found the energy changed by whole-unit packages which he called "quanta." Using the Rydberg formula for the orbitals available to their electron, that formula also establishes direct divisions of the quantum. Instead of"(1-1/22), using "1/22" alone gives a Direct division of the quantum. This division results in a wavelength of 365 nm which is known as black light and causes some hydrocarbons to fluoresce. Similarly, the direct division of the quantum by"1/32" results in an infrared wavelength of 820nm. This direct division by "1/32" also causes objects to fluoresce. This florescence is accompanied by a loss temperature and demonstrate that molecule must expend its own energy for its electrons to reach the orbit. The divisional orbits, 365 and 820 nm, are negative radiation in that they cool rather than heat matter.
We conducted an experiment with black light on cotton and measured the dropping temperature accompanying the florescence. When the data was tested in an Energy equation the data derived Planck's constant. The loss of temperature in in a fluorescing material was a function Planck's constant and the number of hydrogen bonds in the hydrocarbon. The mathematics governing this is experiment as well as the fully developed mathematics for the quantum dimension are available in the book Four Dimensional Atomic Structure.