A good while ago Eigenstate asked more or less where one could put the mass energy of an atom within a sphere.

In 3D space, and our senses are tuned to this real world of space, the volume of a sphere is 4/3Pi*r

^{3}. Our senses are tuned to the even distribution of density and its product with volume as mass and the Archimedes "Eureka" moment.

But what if we move up a dimension to 4D or a hypersphere. The nomenclature varies somewhat since they incorporate real dimensions and complex space coordinates-

http://en.wikipedia.org/wiki/3-sphere for background

Since I'm thinking of atoms I'm more interested in Lie group nonAbelian representations of U1 SU(2) and SU(3) peripheral to Pi

^{2}/6 that EdV raised.

Relevant to EdV's interest- is the number of positive integers as the sum of four squares and the volume of a 4D sphere in both real and complex spatial coordinates.

I came across Robin Chapman's(2003, Exeter Univ Math Dept) 14th proof in a compendium of Rieman Zeta(2) proofs= Pi

^{2}/6

http://www.secamlocal.ex.ac.uk/people/staff/rjchapma/etc/zeta2.pdfIn nature we see a continuum of black body radiation in spectra, that smooth Raleigh Jeans curve, but it's those peaks and spikes of stellar and nebulae etc spectra that provide us with all the astronomically based understanding of the cosmos. The photons absorbed and/or emitted by atoms come in discrete packets of energy that are integer multiples of plancks constant and the radiation frequency. The energy of those photons is related to the reciprocal of wavelength or its frequency if you prefer. Those energies of photons, those spikes and valleys, once we deal with the origin and formation of normal matter are also related proportionally to the mass of the emitting matter( the matter that we say is represented by particles, and stable particles for astronomers) and to the fourth power of the electric charge. What we call the attributes of particles or charges, namely mass for instance is a mass charge, that force exerted by the gravitational force. The electric charge, manifests itself as part of the electromagnetic force that is in part the electroweak force. An apple although containing billions of trillions of atoms falls to the Earth. The Earth with incredibly more atoms attracts the apple by its gravitational force. An electron also has mass as an attribute and also an electric charge but it is not wholly described as wholly electric charge even though we ascribe its total mass energy as electron volt mass energy equivalents. This later charge, the electric charge is just over 10

^{42}fold greater than the gravity force that attracts electron to electron; also it can attract and repel whereas gravity is always attractive . A few spare electrons, as electrstatic charge and the electromagnetic effect overwhelms that of gravity. Although you don't notice the effects in every day lab experiments that electron or proton carries other charges. Mass charge and electric charge, infact any other attribute charge bar one is measured in the laboratory.

It is important to realise that the magnitude and position of the photon wavelength is related fundamentally to the Rydberg constant, that Pi

^{2}/n term. The actual wavelength we see is from the product of this term with what we call the mass and electric charge

^{4}th power, attributes whose magnitude has to measured in the lab, for the only matter we know yet of, that which we call normal matter and are made of from ourselves to cosmic dust. Were we to find other matter types, and we haven't nor yet in low energy collider experiments with other attributes we would have to empirically measure them in the lab to predict those wavelength and/or frequencies, but REGARDLESS of what they are measured as, that Rydberg or Riemann zeta(2) function would still be the foundation of the predicted frequency. It may appear complex but that S

^{3} Lie group representation for a 4D space is the simplest construct nature has provided us with. The physics is amenable to an understanding by many whereas were the maths more complex every future generation of humanity would be ignorant of the detail, a sparse few beings in billions capable of an understanding of how nature operates.

The attributes of matter are described by all those conservation laws you have heard about, momentum conservation, energy conservation, charge(s) conservation and others but these become expressed as inner dimensions, firstly with one complex dimension as spin charge, then with two as the complex plane as isospin conservation, then finally with three complex spatial dimensions or complex volume that incorporates colour charge conservation. All these " laws" bar one are conserved for normal matter, the stuff we know of. An electron emits a photon and way away another electron absorbs it. It could be a proton doing the same thing or a gluon exchanged by a quark, even a single quark, for although you are aware that quarks cannot be separated, held tight together in twos or threes by the strong force gluon carrier at short range it is fundementally long ranged! It's a remarkable feature of nature that after ploughing through pages and pages of wavefunction(psi) logic, that the energy of the force carrying is what is called the fudge or cheating function A(x). By whatever change to the wavefunction psi(xt) when it is moved in distance and time to a different spacetime locale, and this takes pages of maths hieroglyphs then, A(xt) is algebraically negative to psi(xt) inorder to maintain all of the conservation laws.

If you want to determine the photon energy or gluon energy or wavelength to ten decimal places for any matter matter interaction then the cheating function miraculously is

q*A(xt) or g *(xt) fold psi(xt).

We go about it in the other way, we measure q and g, the electron and gluon charge empirically. The pi

^{2}/6 number constant is implicit in the equation that gives the wavelength, the one you see in the SDSS spectral interpretion. The wavelengths of photon emissions are calculated from energy differences the product of the Rydberg and an inverse square relation of integers.

So we start from what is already proved in terms of Physics Proofs rather than Maths proof ; the state of the art is still, is the Riemann Zeta Conjecture true or false? Yet for physics we would say it is true; every time an atom emits/absorbs a photon throughout the history of time it says "true, true true.....every time an atom jiggles" That might not be a mathematical proof, but then I'm not a mathematician.

So everything physically proved condenses down to this line(s) -

Pi

^{2}/6 is ~ q*A(...)*psi(...)

~ g*A(...)*psi(...) as the correcting function to whatever the psi(xt) is locally and globally in one dimension for simplicity.

So if anyone asks you why are there 3 quarks in a proton or two in a meson, or why is colour charged conserved or any of many that would require volumes of complex text and a deep understanding that takes a even pecialist many years to acquire you can tell them-

It's because of the energies of their interactions, when you measure them , include a Pi

^{2}/6 relation most easily understood to mass energy confinement in a 4D hypervolume.

It could be a damned lot harder to understand in even higher dimensions. Every time you say that a

^{2}=b

^{2} + c

^{2}you don't have to derive Pythagoras first or prove that the diagonal across a 3D box is given from b

^{2} + c

^{2}+ d

^{2} nor that it could be a 4D box with a complex dimension or more. That's where the Lie algebra appears, not all the inner dimensions are real axes.

What about strings and branes in even higher dimensions? I don't know- it's up to them to show that the same cheating term corrects for phase charge invariance.

Could there be antimatter around in this universe? What about multiverses?

Of all the conservation laws self consistent with say the q fold cheating function (it does work, this is not a criticism, a particle physicist might also call it a fudge term or function) I was amiss in not including Parity Conservation. You know already the answer- parity or mirror reflection isn't conserved. Of all the conservation laws that provide us with that one liner

q*(A...) for a force carrier energy, partly as mass energy,

many accept the fact that parity isn't conserved is because that's the way it is, it just happens to be so; or that right handed mirror specific stuff is far off, say the same universe or even some other universe. One place locally where it isn't is here locally for energies below those to make the top quark anti quark pair. Anyone worth his/her salt considers that parity is conserved somewhere, even if it's in another multiverse. Yet it would be nice for it to be local as an unknown form of matter. How about relic particles, still normal matter in the sense parity is still accepted as not conserved ie. it's still left hand stuff. Parity violation has to be restored somewhere, either here as shadow matter, perhaps by allowing different mass for particles, and that has to be high or by changing e and that can be lower since it goes as q

^{4} for photon energies.

The q*A(...) function incorporates the effects of mass and charge. Let's stick with the electric charge for simplicity. For the g colour charge there are 8 of them; g

_{1},g

_{2}... etc fold A(.....) simple, or 3 for isospin, not too difficult to appreciate, it's all in the Lie Algebra.

Now psi(...) doesn't include the attributes; these are measured and put in the A(...) function for whatever psi(..) might be, A is exactly equal but opposite in sign. What is the force of the interaction that sends that photon or gluon on its journey? It is the measured q value fold the A(...) function. Any wavelength in spectra inherently include these attributes eg. there's an e,h and m in the energy potential expression.

The q* fudge function gives the correct answer every time for whatever that energy function is and its change with space and time , -A(..) corrects it and q gives the actual magnitude or strength of that force, electric in this case but it could be spin charge or g strong force( there are eight of these).

How can we incorporate and restore parity. Matter or normal matter is left handed in this universe. Photons don't discriminate between parity but all else does either particles or their carriers for normal matter, and these are 100% parity specific favouring the left hand "mirror image" selection is absolute or as they call it, maximal for LH. What an asymmetric universe it is! All those hurdles to clear since Archimedes and the last one is called a hidden symmetry rather than an inner symmetry. Everything is conserved bar parity but that leaves a cack handed universe. I could say fine, I'm cack handed myself , but to come all this way to find nature prefer left handers grates.The photon though is expressed as a composite , carrying a little of the B

_{0}wavefunction( that doesn't incorporate empirical measurables) to give its non specificity.for LH or RH particles. All we need to do is construct a mass, charge balace to balance a new form of matter with different m's and or charges that has RH preference. For instance if there is 4% normal matter mass in the universe and as the universe is largely H or protons essentially, nominal mass 1, how much other matter is needed to restore parity in this universe both locally and globally?and by local in is the galaxy reference; it's all in and around galaxies rather than saying how much is on the Moon or very locally sequesterd ?

How different is this other matter? Well how nice to obey ALL those conservation laws and energy packages- to say of it that it's because fundementally the energies are related to EdV's Pi

^{2}/6 law and energy sequestration in 4D hyperspace, whatever that means.What is an inner dimension that has one two or three complex dimensions as in a +b*i/ . No-one knows but it produces a self consistent bedrock of mathematical logic that works, once we recognise that charges have to be empirically input.

Normal matter mass fraction 0.04; other matter fraction x , q?, spin h/nPi, isospin? g etc? There are many apparent unknowns. There's a running coupling constant diagram produced from empirical collider results up to a couple of hundred giga electronvolts that goes up to and in the white heat of creation it's still the same photons that interact albeit with huge energies, ca. one per particle and that includes antiparticles. The photon unlike the other carriers is lonely when it comes to attributes, none other than h and no electric charge or colour charge etc. Normal matter is composed of up and down quarks but there are six of them, why? we don't know but we do know why there aren't more of them; the others of higher mass energy are unstable except in the early universe, but all carry the same electric charges of -1/3 and 2/3 q and masses and the same g values, but all these change slightly in the running coupling constant. Other matter could be of different family masses and charges. The simplest conceivable case is to keep matter mass energy essentially constant and admit one other electric charge, never ever recorded, whose magnitude other q restores the parity imbalance, so that all conservation laws are conserved, including the Pi

^{2}/n relation to actual measured energy differences and the Planck Charge I have not mentioned.

For Parity conservation the physics fudge function reduces to

q* (A 0.04 mattermassLH) + q/3* (A 0.32allomattermassRH)

Namely the other matter is ca. 8 fold more abundant than normal matter and is RH specific rather than left hand specific. In all respects it is the same as normal matter in attributes other than its electric charge.

Whereas one might say there is ca. 4% ordinary matter(LH specific) and 20% Dark matter(?) as matter mass energy in this universe one would predict that here there is now ca. 36% matter mass of which 32% is (RH) specific, as a desire to balance parity. But if one is going to predict an unforseen electric charge why introduce just one? and further, that there is empirical proof that ca. 25% is matter mass and the rest or 75% is Dark Energy in this universe. It is assumed that Dark Matter is LH specific. The Pi

^{2} /6 relation applied to photon energies and the photon is not parity specific at even high energy, appears to reflect that nature's book keeping hands out packets of energy stored in 4D hypervolumes but that hypervolume has to satisfy the SU(3) Lie group restrictions; infact the simplest ones fortunatey, otherwise my script wouldn't be here. We could sacrifice Dark Energy in that nobody likes it anyway, a facetious comment but many don't appreciate the latest evidence for an increasing expansion of the universe. Doing away with completely and going for 100% matter mass that gives the accepted critical density is done by postulating three electric charges in total instead of the only one we know of, characteristic of normal matter, that q or 1.602.. *10

^{-19}Coulomb. With two charges we have q and q/3. For three charges we postulate 3q as the third addition with a troublesome problem, we need four two specific for LH and two specific for RH; three is a trouble ; its sign or it has to LH specific like ordinary matter; if it's not specific or zero parity. Were the H atom to have a charge 3 on the proton and -charge 3 on the electron , all energy transitions rather energy differences for emitted photons come in with 3

^{4}th power greater energies, ie. energies differences precisely 81 fold greater or less. For a mass 1particle or its molecule you would be searching for a 21cm fold 81 emission or conversely 1/81 fold 21 cm emission. Thus, similary 1/3

^{4} for the lower charge candidate. The theory is identical to everything for normal matter , the Rydberg constant is the same and incorporates the Pi

^{2}/n relation.

Pi

^{2}/6 has an important and independent role to play, hopefully

. all that's missing is the capture and discovery of this novel matter