An Irrational hybrid in an Analogue-Digital world?

Nature's four great irrationals ẟ, ϕ, 𝝿 and e together seem to be encoded in much of what we describe as physical reality. An instructive (if arbitrary) way to characterise them could be to demarcate them according to their discrete or continuous ("analytic") roots.

A fifth irrational, the lesser-known Euler-Mascheroni constant,  ɣ traverses the analogue functions of the analytic (mathematical) modeller and the apparent discrete reality of the physical world. It appears in their regularisation programs, that coarsen an otherwise too finely-tuned view of the world. That is in the development of a non-divergent, non infinity-riddled quantum field theory of electrodynamics. Does ɣ deserve its place in Nature's hall of fame?

• The "Digitals" of sequencing, non-linear growth:

• ẟ-Feigenbaum's number 4.669201609..., that universal constant of chaos theory being the limiting value characterising the velocity of period-doubling, the ratio of the intervals between bifurcation points of dynamical system approaching chaotic behaviour (planetmath.org).

• ϕ- phi-golden ratio (√5-1)/2=1.618033988...that irrational limit to which ratios of successive numbers in the iterative Fibonacci sequence 1,1,2,3,5,8,13,.. tends. The ratio is characterised by the log spiral and the non-Markovian (see below) nature of the sequence growth.

• The "Analogues" of analyticity of Calculus and trigonometric linearity:

• 𝝿 - pi, 3.14159265359.. that ratio of circumference over diameter that reflects the flatness of the space in which a locus of points a fixed distance from a central point scribe. Alternatively the area that that scribed circle (x² +y²=r²) encapsulates if its radius, r is one unit.

•  e- Euler's number, 2.71828182845.. that number beyond x=1 (on the real number line) at which the definite integral (area) under the hyperbola y=1/x is one unit.That is when the domain of function is (1,a=e) as in the graph below,

To these four irrationals should we add to Nature's true natural numbers the "hybrid"(as yet to be confirmed definitively irrational):

•  γ≈0.5772156649…, Euler-Mascheroni constant?

Logarithmically growing Harmonic Series

Whereas  ln(n) is the limit of the area of the analytic hyperbola 1/x as we extend "a" in the graph above to infinity, the "digital" harmonic series:

cumulatively grows"logarithmically' slowly.

The Euler-Mascheroni constant,  ɣ  is the finite difference (the light grey below) between this "logarithmically slowly" cumulatively growing divergent Harmonic series and the divergent hyperbola: 

It is the grey area in the graph below, the difference between the two series 

being summarised as follows:

setting its range between 0.4851 and the limit of the convergent alternating harmonic series:

Rather than exploring the pinning down of the number in terms by using the Psi (digamma) and Gamma functions which is done ably here (mae.ufl.edu) we merely nod at its acquaintances both mathematical and physical.

Alternating Harmonic Series limit of ln(2)

The alternating harmonic series, unlike its corpulent brother is a convergent series,

converging to ln(2), which itself is equivalently the area under the hyperbola defined between the domain interval of (1, 2):

This ln(2)limiting value is a number familiar to those acquainted with exponential radioactive decay. The deterministic model for a decaying (aggregate) set of discrete elements, N(t), governed by quantum selection rules, says that the mean lifetime an element remains in the set relates to the decay rate, λ (characteristic) of the sample:

The time required for the decaying quantity to fall to one half of its initial value is the ratio of this limit to decay rate:

When this expression is inserted in the exponential equation above we move from base e to base 2:

The statistics of this exponential decay can be derived using a stochastic Markov chain process, that satisfies the ("memory-less") property that predictions for the future of the process can be based solely on its present state, ignoring the process's full history. Unlike the memory-full(er) growth determined by golden ratio ϕ, ɣ rather characterises that decay which retains no memory of the history that delivered the aggregate to its present state.  Rather than associate Euler's constant, e with decay, perhaps it is more appropriate to consider such spontaneous decay as governed by ɣ.

We may summarise the irrationals then according to the following:

• 𝝿 and e embody the reductionist principle of describing the scattering of light (electromagnetic waves) with homogenous solids by sets of interacting linear trigonometric functions;
• ϕ, ẟ encode the complex non-linear dynamics inherent to interacting systems involving non-homogenous condensed matter such as gases and fluids as modelled for example by Navier-Stokes equations;
• ɣ traverses the regime between these memory-less and memory-full processes. Somewhere between the analogue functions of the mathematical analytic model and the discrete reality of the physical world.  

In theories of Quantised fields, renormalization is the process of subtracting counter terms at each order of the "perturbation" (about the linearised) theory. Regularization is the modification of the theory at (shorter) distance (“cutoff") scales so that the theory becomes well-defined (UMD physics). A "regulator" is a process which renders finite a momentum integral which is superficially divergent.  The Euler-Mascheroni constant γ appears in the finite part of the integral having no apparent physical significance being merely an "artefact" of a coarse-graining subtraction scheme to counter divergences arising from deploying otherwise analogue idealised functions.

How Linear is Light?

Maxwell's field equations governing the "Classical" electromagnetic field are described as "Linear". In the following we describe this quality through its vector calculus and differential form formulations. We note some expressions of apparent non-linearity in the electromagnetic field as suggested by the quantized version of its field theory.  As such the following questions are addressed:

1. What makes intertwining Electric, E and Magnetic, B fields in some sense a linear wave yet have possess a non-linear character? 
2. How is handedness characterised in complex "Dual" formulations of Maxwell equations?
3. What does Ahranov-Bohm have to say about the linearity of intertwined E and B phenomena?
4. How is is that QED says that light-light interactions are possible? 

Linear in Maxwell's Field equations

The principle of superposition applies to electromagnetic field described by Maxwell's equations given they are linear in both the sources and the fields. So if you have two solutions to Maxwell's equations for two separate sources then the sum of those two solutions will be the same solution as if you just added together the two sets of sources.  Such linear superposition is in contrast to non-linear interacting water-soliton or gravitational waves in which as in case of water waves, surface tension and frictional forces between layers create positive feedback loops. The formation of cross wave soliton being an example of the sum of two waves being more than the sum of their parts. Such are the complexities of the fluid dynamics as described by non-linear Navier-Stokes equations. Light at least before it scatters is trite in comparison.

So for example, linearised polarisation of light arises out of non polarised light scattering off the flat interface of the air and water. Due to linearity, such linearised polarized radiation can be described as a combination of right-circularly and left-circularly polarized radiation, a form of Fourier decomposition of the wave.The equations describing the interaction of light with charged matter, their gauge freedom and linearity as captured in Maxwell's equations are briefly summarised in the following.

Magnetic monopoles unlike electric charges (as described by Gauss' law) do not exist:

Such Dipole generated fields, B are divergence free because they may be written in terms of the curl of a gauge potential, A

that in themselves are defined uniquely only up to addition of an gradient of a scalar field. Faraday's law describes how that changing magnetic field, B gives rise to a curling Electric field,

This equations is the source of Lenz's law of induction. We envisage a cascade of induced fields, E as a result of a change in magnetic flux. The Electric field is itself defined up to an equivalence set of scalar potential fields, V

In a tensor formulation we form a four-vector A=(V, A) (𝜇=0,1,2,3) for (x,y,z=1,2,3)  that is defined modulo a gauge ("phase") transformation:

This gauge freedom or invariance under a change of phase is expressed as a rotational U(1) group symmetry through Euler's formula which captures the deep relationship between the exponential and trigonometric functions in the complex (Argand) plane. Maxwell equations can be cast as an Abelian gauge theory.

A circle as the locus of points scribed by a radial vector of length, r can be described by radial (polar) co-ordinates (r, φ). Here φ is both angle defining a complex number on the unit circle and the phase of a sinusoidal function.

It is these very trigonometric (sinusoidal) functions of Ptolemy's astronomy that are the (eigen) function (complex exponentials) invariants with respect to time translation. An experiment’s results should not depend on the day they were made and the sine function is as homogenous as they come. As such they are the appropriate functions for linear systems. Indeed they are more than time translation invariant being palindromic as well. Quantum states being absolutely additive have such trigonometric functions as their eigenfunctions.

The Electric, E and Magnetic fields, B can be encapsulated as

which suggests an antisymmetric matrix form:

.

Ampere's law which through symmetry requires the introduction of time change displacement "current" field for charge and current densities can be written as

that is,

In order for this set of equations to be captured in Lorentz invariant matrix form a (Hodge) dual ★ "metric structure" needs to be added to our space. That is using the Levi-Cevita like symbol defining an orientation we have the object ★F

.

The multivector form of Maxwell's equations then takes the extremely compact form of a Bianchi identity and a source equations that makes their linearity apparent:

The differential form operator d when applied twice to a Field, F delivers zero. The main takeaway, mathematics aside is that the electromagnetic field is F=dA is the physical, observable field and that A the field potential, as merely an equivalence set is not physically realisable. 

The proceeding field theoretic formulation is called a non-chiral (i.e. Parity symmetric preserving), "vector" (as in gauge boson) theory. The familiar "vectors" of three-dimensional space are the objective-invariants -staying the same when the underlying basis (co-ordinate axis) set is rotated. This vector quality of a mathematical object renders it as a realisable "observable" in nature. 

A purely chiral formulation of Maxwell's equation can be made by forming the complex valued ant-self dual object, F-i★F.

The Relativity of Helicity

In the quantised field versions of classical field theories massless gauge vector bosons have spins that are in the same direction along their axis of motion regardless of the point of view of the observer. That their Chirality is absolute in this sense is due to both the invariance and finite speed of light. That such massless particles move at the speed of light, means that a massive observer (travelling at less than the speed of light) cannot travel in a faster reference frame in which the particle would appear to reverse its relative direction.

The photon's handedness is unambiguous in that all real observers see the same chirality. Accordingly we say that the direction of spin of the massless particles is not affected by a Lorentz boost (the relativistic equivalent of a Galilean change of reference frame) in the direction of motion of the particle. As such the sign of the projection (helicity) is fixed for all reference frames.

A photon’s “twistedness” has a sense in that helix described the rotation of its electric field (say) can be clockwise or anti-clockwise. By definition the helicity of a particle is right-handed if the direction of its spin is the same as the direction of its motion. It is left-handed if the directions of spin and motion are opposite. 

Aharanov-Bohm Double Slit experiment

In the double slit experiment we observe an interference pattern based on the superposition of waves arriving on screen in phase but travelling different path lengths be the missile a set of (or indeed single) electron(s) or (coherent set) of photon(s).

The Aharonov–Bohm effect is the shift in the interference pattern of a quantum mechanical double-slit experiment in which a magnetic flux carrying solenoid is placed between the two slits in the barrier. The phase, α, of the wave-function of the electrons going through each of the slits and onto the screen are phase shifted thus translating the interference pattern.

The effect shows the physical nature of the vector potential, A which is classically otherwise deemed the convenient mathematical ("Gauge Theoretic") artifice described previously.

The electrons move in a region, outside the solenoid where the magnetic field strength B=∇xA is zero. This means there is no Lorenz force acting on the electrons. The gradient of the field potential, A however just falls off as 1/r from solenoid. The A as discussed is gauge variant (ambiguously defined up to the addition of a gauge transformation of the derivative of scalar potential, V). The resulting phase difference however is gauge independent since it can be turned into a (gauge invariant) surface integral of the magnetic field. A variant invoking Faraday's law is the following scenario.

A cylinder, with a wire along its axis, takes the place of the solenoid.

The wire is fixed in space and the cylinder is free to rotate about it. The wire and cylinder carry equal and opposite charges, distributed uniformly; outside the cylinder their electric fields cancel, and once again the electron feels no electric force.

The cylinder, if it rotates, produces a magnetic field that vanishes outside it (like the magnetic field of a solenoid). So the electron also feels no magnetic force.

However, the electron acts upon the cylinder: the passing electron generates a magnetic field, and the changing magnetic flux through the cylinder induces an electric field curling around it (by Faraday’s law). This electric field changes the angular speed of the cylinder, but only briefly: the cylinder rotates freely with the same angular speed ω before and after the electron passes.

In all these scenarios the electric (electron) source's trajectory is guided by a gauge potential. the accumulated source of that potential thus does not dictate its motion. Linearity appears lost.

Laser-Laser Interactions

Light-by-light scattering is a quantum effect, whereby the two photons scatter from a virtual electron. Since this involves four photon-electron interactions, the process is heavily suppressed by a factor of roughly 1/137^4, making this process really rare (science mag).

Feynman rules of QED associate a factor -ie to each photon-electron-positron vertex, and there are four of those factors in the leading diagram (which is the "box diagram" with four such vertices).The fact that in QED there is no photon-photon scattering diagram with only two factors of -ie is the reason why photons do not scatter off each other as strongly as electrons.

In the classical picture an accelerating charge generates electromagnetic radiation. However, an accelerating mass generates no gravitational waves, which are only generated when the acceleration of the mass itself is changing. That is when the mass has a non zero “snap” (third order derivative of space”, to the fourth and fifth orders of “Crackle” and “pop”). 

Gravitational waves if they are second quantizable would be carried by gravitons. A graviton is the excitation in the boson-gravitational field, travelling (according to 2017 results form binary-Neutron star coalescence to 15 orders of magnitude in precision to effectively ) the speed of light. It is a spin-2 particle, the only one, which means that it somehow needs only spin half a revolution before it arrives in the same position.  Unlike a photon it is a source for the gravitational field itself. That is a graviton-graviton interaction. In this sense the graviton is non-linear (formed from a non-Abelian gauge theory) whereas the photon is not. The laser-laser interaction suggests otherwise at very high energies.

The Left-handed signature in our flesh

Our universe of apparently immutable Natural laws, appears to be well characterised by the sets of symmetries that can be viewed as generating them. In particular we observe a handedness choice of Nature more profound than that which differentiates our opposable thumbs. Notwithstanding the universe in aggregate being modelled pretty fairly at large enough scales as Isotropic with no preferred directionality to both its mass-energy content and the movement of that content, is Nature's "chiral" choice really any deeper than the necessity for an extended (non point-like "elementary") objects to have an orientation?

Symmetry and Chiral biases in Nature

An object is "chiral", possessing a handedness if it cannot be superimposed on its mirror image such that a Parity transformation would invert the object into its image.

In summarising the handed-orientability symmetry choices of Nature we will collate terminology from Chemistry, Physics and Mathematics and look at the algebra of the linear structures that underpin fundamental microphysics.

On a macroscale from Cosmology we see the signature of handedness in the polarised afterglow of gravitational waves embedded in the last scattering surface that is the Cosmic Microwave Background Radiation (CMBR). That we are built from left-handed amino acids begs a natural Anthropic question.

Lucky Linearity of Quantum Mechanics

Classically, the universe behaves in a weakly chaotic manner. All it's clockwork-like cyclicality is mostly illusory. The apparent simple periodicity observed in the universe belies a complex struggle between bifurcations of plausible orbits in the short term against the pull of long-term stability. That we have such stability at all in our rotating solar system is profoundly due to the fact that space is 3 dimensional. The multiplicity of words to describe Periodicity, cyclicality and orbits trace a tribute to some simple governing inverse square law, but some precession of the orbits point elude to this uneasy state of equilibrium.

That chaos is just a small perturbation away is due to the non-linearity of the gravitational field equations describing the systems' trajectories: a system of interacting particles given a little kick (perturbation) by the addition of another small body, will behave in a manner that will embody more than the sum of these two parts: it's behaviour will not be deterministic (independent). .

The simplicity of Linearity is present in a system comprising the addition of a static electric charge (electron) to a set of other static charges. Applying the (apparently) natural Superposition principle gives the resultant Coulomb force on each particle as

being equal to just the net linear sum of all the Coulomb forces of the other charges. See diagram aside (dmr-physicsnotes.blogspot).

In the quantum description Superposition conjures up multi-paths of a particle (-wave) appearing, as in a many-worlds manner, to coherently follow as it passes between two slits all available paths to a screen in the classical interferometer experiment.

Happily for those seeking to reduce our patterned but ultimately turbulent world to a mechanics of primitive basic building blocks these simple rules of linearity preside over the quantum realms of the very small. The mechanics of the quantum, be the fundamental root element of point-like, string-like or brane-like form, is linear in nature.

That is, the possible states of a physical object at its most basic level form a linear space that are eloquently described by Sophius Lie's (continuous) Group theory.

Thus these laws can be systematically applied to the electron cloud orbital shell systems around an atom's nucleus. The supreme reductionists have revealed the theory of linear representations of Lie algebras as underpinning fundamental symmetries of their universal primitives. Those entities composed of no other sub-entities, as Prime numbers: those non composite numbers with no simpler roots. A little discouragingly for believers in the reductionist programs (over dynamical system theorists) we know that just as the Prime numbers are infinite in number so are the number of coherent String theories that look to extend the primitive from a point to a wiggle.

Spin Sensitivity of Matter

In Quantum (Linear) Mechanics, operators give rise to the observables (vector) states that we see as the s and p shell eigenvalues of an atom. Such Linear Operators mark out the energies of the atom's states. The nuclei of such atoms themselves possess dipole moments, in which their Angular momentum, J preccess in an external magnetic field akin to a gyroscope immersed a torque inducing "gravitational" field.

The gyromagnetic ratio equal to  is the proportionality constant relating the "Larmor" frequency of this precession to that external magnetic field, B. In nuclear physics the g is a composite of the nucleon spins, their orbital angular momenta, and their couplings. As such spinning, be it intrinsically a "point" elementary particle (like an electron) or a composite nuclei marks out an orientation for aggregated matter. That most aggregated matter is charge neutral but not inertial mass neutral means that though it may not precess in any ambient magnetic fields it will exhibit De-Sitter (geodetic) and Lense-Thirring precession by virtue of being in the presence of a static (resp. rotating) central larger massive body.

Left-handed -amino acids

We describe a “chiral" molecule as one that possesses a handedness in the sense that it is not superimposable with its mirror image.

So, just as left and right hands have thumb and fingers in the same order are mirror images but (arguably functionally) not the same, chiral molecules are molecule pairs that have the same atoms attached in a complimentary configuration, but which are mirror images and thus, not deemed identical.

Although most amino acids can be synthetically produced in both left and right-handed forms, left-handed amino acids almost exclusively dominates the life forms of Earth. It has been shown that some of the amino acids that fall to earth from space are more left than right handed prompted the search for such Chirality on the recent Rosetta-RSA mission to comet 67P/C-G.

Most DNA double helices are right-handed.

That is, if you were to hold your right hand out, with your thumb pointed up and your fingers curled around your thumb, your thumb would represent the axis of  the helix and your fingers would represent the sugar-phosphate backbone. Only one type of DNA, called Z-DNA, is left-handed. Formation of this structure is generally unfavourable, although certain conditions can promote it. 

A crude evolutionary analogy serves to indicate why a dominant configuration has prevailed. Consider the choice of handedness of the spiral staircases in medieval castles that served best as a means to defend. They were (almost) always built with the spiral going in the same direction according to the right hand screw rule as one looks down the stairs from above so that the defending right- handed swordsman, who would either be coming down the stairs or backing up in reverse, could freely swing his sword.

 The attacking swordsman (ascending the stairs), unless left-handed would have his swing blocked by the wall. (cmrp blog)

As such only a army of left-handed soldiers will have parity in the battle. Of course if they happen to succeed they will be at a disadvantage in defending the castle against another horde of lefties and be afforded no advantage over subsequent right-handed foes.

Polarization of electromagnetic fields

An ordinary incandescent bulb (just like the Sun’s radiation) emits (many wave packets of) photons pointing in random directions so its radiation is not said to be Polarised in any one direction. The propagating electromagnetic wave comprises intertwining electric and magnetic fields. By definition a wave's direction of polarization is defined to be the direction in which its electric field is oscillating. We can have linearly polarised, elliptical and circularly polarised waves.

While looking at the source, if the vector of the electric field part of the light coming toward you appears to be rotating counterclockwise then the light is said to be right-circularly polarized. Using such a wave description, the electric field vector and its intertwining magnetic field vector of circularly polarized (wikipedia) radiation describe a helix.

As with all field equations those of Maxwell's varying electromagnetic fields are invariant to the reversal of time: change t to - t and they still work. (This is in contrast to what we know from his Thermodynamics where the arrow of time points only in the direction of increasing Entropy, to a universe sinking to a final equilibrium state of zero useful working energy). The helix is palindromic in time in the sense that the horizontally oriented helix is identical whether traversed into or out of the page in the diagram above: a reversal merely inverts the direction of the propagation of the light and does not modify the sign of the helicity. By convention though we define left handed helicity by the motion of the Helicity vector scribing the tip of the hands of a standard clock as it moves with the light ray with its face directed forwards.

Light-Handedness?

Does this choice of helicity reflect a real certain intrinsic handedness present in the light? Unlike the inversion of time there is a real effect of inverting the sign of spatial directions on the description of the light ray. Such an inversion operation is called a Parity transformation. It turns out that the study of such orientation dependence has had deep influence in the building of fundamental particle theories. The inversion of space transforms right-circularly polarized radiation into left-circularly polarized radiation. As such the helicity of radiation flips sign with a parity transformation and thus we can assert that circularly polarized radiation possesses a handedness called chirality. The Quantum field theory take on the distinction between helicity and chirality is a little more subtle and was discussed in from-chirality-to-helicity Whereas helicity (meaning "twisted") is defined via the direction of the momentum and angular momentum of a particle it is not a Lorentz-invariant property for massive particles. In the massless case of light photons, helicity is synonymous with chirality as you cannot attach an inertial frame to a photon to observe it moving in a relative manner.

Optical Rotation

When linearly polarised radiation (say that has scattered off the plasma at Baryogenesis) traverses a medium that is "chiral" in the sense that it is filled with handed "enantiomer" molecules, one of the circularly polarized components is propagated less rapidly than the other and the plane of the polarized light beam becomes rotated. This phenomenon is called optical rotation ( chemgapedia).

If the beam is traversing a medium that is chromophoric, one of the two circularly polarized components is absorbed more than the other resulting in the phenomenon of circular dichroism.

Uneven-handed questions

The question you might ask then is why is there an apparent handedness choice present in the chiral compounds that gave rise to the life that observes this universe. In particular, why is it that almost exclusively there are only left handed amino acids within life on earth?

Is this due to some handedness in the radiation signature that was emitted at the Big Bang? 

More precisely (harvard.edu) does the light emitted from the surface of last scattering with its circularly polarized signature (a.k.a the Cosmic Microwave Background Radiation, CMBR) reflect some very early choice that has been made in the subsequent ert life that would come to exist in it. As the fireflies drawn to the light are we not drawn to the inescapable conclusion that our chiral flesh was cast from the tumult of circular polarised waves, the primordial evolutionary filter, the progeny of left-fit enantiomers.

Categorically Confusing

A story here of the nature of numbers that descends into being a placeholder: an excuse to catalogue the Mapping types between categories of objects. By looking at the relation between the set of irrational numbers generated by the Riemann Zeta function and the pyramid ('Power Pascal triangle") series characterised by its Bernoulli coefficients we see an instance of a functorial relationship from Category theory.

The language used to describe some of our dearest ideas in mathematics seems overly onerous so I try here to traverse the foothills of category theory by sampling from the irrationals drawn from Bernoulli and Riemann Zeta "functional" series.  The conclusion couched in the pseudo-arcania of Category theory is that modulo some coefficients pi^(2n) acts as "functor" relational mapping 

between the Category of Riemann Irrationals, Xi and the Bernoulli Rationals, B.

Number types

The rules for operations with pairs of (Natural, N) numbers are designed to give the same results as

the operations with fractions (Rationals, Q) that we would have first learned without reference to pairs of numbers, (p/q). The rules for the operations with sequences, that is, with Irrational numbers R, still belong to that [same] category of rules. Defining the Integers, Z as positive and negative Natural numbers with a zero.

Technically we have the following definition of two otherwise familiar algebraic structures:

• Monoid: the integers Z under multiplication (but with no inverses)
• Abelian group: the integers or real numbers under addition

The laws to the right are the basic operations we look for in sets of objects: that they remain "closed" under their operational influence. 

Irrationals as sums of Infinite Series

An irrational (Real) number is but an infinite sequence of rational (reals). The archetype irrational number is that defined as the Riemann zeta function defined as an infinite sum of Real-(field)
valued rational, Q numbers:

So for instance pi can be recast as the square root of (6{1^2 +1/2^2+..}) as in its Basel formulation. As long as the ordering of terms in the sequence is respected the rules of addition and (multiple) multiplication can be carried over from Countable numbers to this less Natural system of numbers.

Respecting summation order in Method of Differences

As an example of this respecting of ordering when manipulating series in order to determine their (finite) sum, consider the method of differences employed to determine the sum of the first 100 terms of the series 1/r(r+1):

If we want to determine the finite sum of

1/2+1/6+1/12+1/20+..1/100(100+1) =100/101,

we use the method of differences to cancel out subsequent terms.


For our purposes it is sufficient to note that the such (irrational) reals form an Abelian group under the addition + operation.



Similarly we can use this method to unpick the i=2:




"Power Pascale" pyramid formula by taking the difference of the left hand side of the i=3 formula and its (n-1)^3 equivalent.

Indeed the i=2 series can be written in terms of the Bernoulli numbers:

where the set of Bernoulli numbers, B are themselves an infinite series of rationals.

Whether or not the series is of rational 1/q (Riemann Zeta, Z) or polynomial (Bernoulli, B) form we talk of these numbers being irrational Real (-valued) fields. Indeed these are but an infinitely small subset of the set of irrational numbers.

Class of Algebraic structures (Object) set types

The operator types be they + or x define the Algebraic structure or "object type".  Possible features of such structures are summarised  (mathsphysicsBook) below.

• Semigroup but not monoid: the positive reals less than 1 under multiplication (no identity)
• Monoid but not group: the integers under multiplication (no inverses)
• Abelian group: the integers or real numbers under addition
• Ring but not integral domain: the ring of integers mod n for n not prime (zero divisor pq=n=0)
• Integral domain but not field: the integers (no multiplicative inverses)
• Field: the real numbers; the complex numbers 

Their behaviour under addition being Abelian means that the order in which the objects are added is unimportant.  That the operation commutes under that operation.

If this ordering requirement is loosened for multiplication (which is just multiple addition after all) typically we see that we need to insist on multiplicative Associativity to maintain structural consistency. From the sets themselves to the mappings operating within them.

Morphism and Composition Mapping Types

A Functor can be thought of as a homomorphism (structure-preserving map between two algebraic structures of the same type) between two categories C and D;  a mapping that

• associates to each object  in C an object- its image,  f(X) in D,
• associates to each morphism  in C a morphism preserving identity morphism and composition of morphisms

Within the algebraic structure of groups, rings, modules, etc.  morphisms are usually (type preserving) homomorphisms. 

The notions of automorphism, endomorphism, epimorphism, homeomorphism, isomorphism, and monomorphism all find use in describing relations between algebraic structures.

An object class consists of sets with a structure:

• in which morphisms would be mappings between these objects in which elements of one set X map one-to-many or many-to-one to a set Y of other elements 
• the category would consist of both the class and the mappings.

Functor between Z and B

Now to set of sets and maps as we define a category $C$ :

1. class of objects ob(C) 
2. a collection of sets of morphisms mor(X,Y) between these, 
3. morphism composition operator -"functor". 

The set of Riemann Zeta Irrational functional objects, Z  :  map to the Bernoulli "polynomial" Rationals, B functions   through the functor mapping:

.

As such, the irrational pi^(2n) (modulo some coefficients) is the "functor" the relational mapping the Category of Riemann Irrationals to the Bernoulli Rationals.

Real valued Groups

We see below the functor mapping between Group Categorical object to Set and then to function Category the bottom most map of which defines the function in the Real Number field.  Illustrations from mathsphysicsBook as:

Three Categories each with their closed sets of objects and mappings types are thus illustrated. While the morphisms operate within each category the functors make clearer their algebraic structural relations.