Maths of Physics and Politics.

Displacements from equilibrium.

 Pascal's triangle is generated by the binomial distribution.  It displays endless mathematical relationships. It has been said that when someone finds another such a property in this array of numbers, the only person, that it comes as a surprise to, is its discoverer.

 Take any row of Pascal's triangle which is generated by an expansion of the binomial theorem.  Say, 1, 4, 6, 4, 1, and multiply this respectively by from nought to four.  This gives: infinity, 4, 12, 12, 4: that is to say another symmetrical distribution, always neglecting the first term, infinity.

 It is also possible to derive a symmetrical distribution by dividing from one to five: 1, ½, ½, 1. 5, always neglecting the last term.  Larger distributions show more clearly that this fractional distribution is somewhat exponential.

If you drew these distributions, they would look like steep-sided bowls. A vertical line down the middle would be like a mirror on one side to the other.  You could call this vertical axis, or y-axis, the mirror axis. The horizontal axis, or x-axis, would be at zero level, the bottom of the bowl. 

The vertical or y-axis intersects the x-axis at zero, the lowest point of the bowl. The x-axis to the right is positive, while the left is negative. 

 The positive and negative limits may be equal in magnitude, called the amplitude, as in the crests and trofs of a wave of equal height and depth. The zero level, equally in between, is called the equilibrium and it corresponds to the level of a calm sea without any waves.

See diagram of potential energy parabola:

Energy graph of simple harmonic motion.

Having located the essential features of a wave, in the bowl diagram, it is possible to redraw them as wave forms. But these waves wouldn't look like the simple harmonic waves. The above is a bowl diagram that shows the energy relations of harmonic waves but the curve of the bowl is described by a parabola.

A parabola is a function of a variable squared. Up till now, I have been thinking in terms of the binomial distribution, because constituency seats and voters preferences are readily modeled on this distribution. Now to turn to a different election model.

First principles differentiation of election quotas.

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Equal representation is served by proportional or elective quota counting. The Hare quota counts how many voters are represented per seat, by dividing the total votes by the number of seats in the parliament: V/S.

 The Droop quota counts the proportion of votes needed to elect each representative to a seat in a multi-member constituency. The Droop quota is the total votes in the constituency, V/C divided by one more than the number of seats in the constituency, S/C. That is: (V/C)/{(S/C)+1}.

 Assuming that there is only one constituency or C = 1, it is possible to treat these two quotas in terms of traditional differentiation from first principles.

 Let q stand for quota, so that (1):

q = V/S.

 Suppose a small change in the number of seats produces a small change (given by sign, #) in the quota, so that (2):

 q + #q = V/(S+#S).

 Therefore, (3):

 #q = V/(S+#S) – V/S = -#S.V/S(S+#S).

 Therefore (4):

 #q/#S = -V/S(S+#S).

 The value, #S, could be 1, as in the Droop quota. Going thru the traditional differentiation procedure of reducing #S to a limit of zero, and replacing the hash sign with a d-for-differentiation, (5):

 dq/dS = -V/S^2 = -q/S.

 This is the so called derivative (the older term is differential coefficient) of the quota with respect to the seats. It represents a change in the quota with respect to a (zero-limit) change in the seats. Or, the rate of change in the quota with the change in seats.

 The point is that the derivative’s independent variable is a square and so it offers the prospect of a parabolic function, which is what is required.


Election quotas as equilibrium levels.

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It may be worth speculating what is the possible meaning of differentiation from first principles that begins and ends in terms of the Hare quota, with only the working bearing a passing resemblance to other quotas like the Droop quota.

Going back to the early nineteenth century inspiration for proportional representation, children queue behind their most popular candidates. The longest queues elect their representatives to a limited number of seats on a committee.

The lengths of the queues oscillate. Some children see their queue has more support than their candidate needs to be elected, so they drift away to swell the ranks of a next prefered candidate. Likewise some candidates cannot win enough support to be elected, so their support drifts away to help a next prefered candidate, who is nearly elected.

Eventually, the queues reach equilibrium. That is to say, all the winning candidates have queues just long enough to elect them. That leaves a residue of voters, not quite big enough or agreed enough to put a runner-up in contention.

(A tie is possible and then the exclusion of one winning candidate is decided randomly. For instance, by "taking the short straw" amongst a bundle held out for the winners to chose.)

In a contest for only one seat, the winner needs over half the votes. The minority have no choice but to be represented by the winner who they did not vote for. That's not much of a choice and presumably many of them might not think much of him as a second preference.

If the election is in a multi-member constituency, of, say, five seats, then at least five-sixths of the voters may be represented. This leaves another sixth of the voters unrepresented, if they prefered other candidates than the five with most support.

Nevertheless, their situation is much rosier for choice than in a single member constituency. The losing sixth of the voters might still decide they prefered enough one or other of the five winners to join their queues, so that all the voters might choose to show that they were satisfactorily represented by the five winners.
If this were the case, then, in effect, the candidates have been elected not only by the Droop quota, satisfying the minimum votes, each winning candidate needs for election, but the winners have also been elected, in effect, by the Hare quota, for maximum representation of the voters, or adequate representation of all the voters.

Thus, differentiation that effectively winnows the Droop quota back down to terms of the Hare quota may be expressing a reality about electoral counts.
In abstract terms, the reality is that, as the number of seats grows, then the change, of, say, #S = 1, becomes relatively less and less important, till its relevance is vanishingly small: the (so-called zero limit-taking) condition of differentiation.

An increase in choice from increasing number of seats in a larger multi-member constituency may provide a a tipping point from the lower level of equilibrium, provided by the Droop quota's minimum level of representation, to the higher level of equilibrium, provide by the Hare quota's maximum level of representation.

There is an analogy from physics of phase transitions, the abrupt changes of form in matter such as from ice to water to steam. (This is the meaning of the term, phase, in chemistry. The word, phase, is also used below but in its mathematical sense.) This concept is widely used in physics, such as in stages of cosmological evolution. In these examples, matter under-goes, stage by stage, an increasing disorganisation, or "entropy", of its state.

The above electoral example has been considered in terms of increasing order. The single member election has relatively high entropy or disorder, because about half the voters, who did not first prefer the winning candidate, only have a second preference, moreover that can only go to the one winning candidate, who they may not be willing to rank as their second choice.

On the basis of chance (as determined by the binomial theorem pattern 1/4, 2/4, 1/4) for a single vacancy, one quarter the voters equally prefer both candidates; one quarter equally prefer not to vote for either candidate. The two quarters are those voters who prefer one or other of two candidates.

Thus when US Presidential elections or single member constituencies have only fifty per cent turn-outs, that may well reflect low theoretical expectations from low-choice single vacancy elections.

With an increasing choice from more seats in larger multi-member constituencies, the election may under-go a phase transition, in effect, from a Droop quota election to a Hare quota election. This would be a state of increased order, where voters preferences are sufficiently extended to the point where everyone is represented by some winning candidate.

Of course, voters might be confused by too much choice, but such arguments usually leave out the important consideration that voters can home-in on the limited number of candidates who they will recognise as most representative, before ordering their final choices.

Another possible quota, I christened "the Ross quota" is considered below. (This supersedes my previous evaluation of a so-called Ross quota. See foot-note 1.) This represents another possible equilibrium level in an electoral system, where the requirement of equal representation is perturbed by a conflicting, if lesser, requirement to make constituencies follow natural community boundaries.
Like the Droop quota, this Ross quota makes for a somewhat lower, but still stable, level of order than the Hare quota.

"The Ross quota."

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The JFS Ross book, Election and Electors has a particular affection in my memory, because it converted me, as a student, from the complacency of my compatriots, to the single transferable vote.

Later, I asked Frank Britton, as secretary of The Electoral Reform Society, if they had a spare copy and, amazingly, he posted me an old edition (not on their sales list of literature).

 JFS Ross gave the rule for cross-constituency proportional representation in a single member system, with respect to community boundaries.

It is not possible to have PR within a single member constituency and the rule for PR within multi-member constituencies is given by the Droop quota. The rule for PR between constituencies, with no relaxation within or across constituencies, is given by the Hare quota.

 The Ross rule states that a single member constituency must split equally in two, if its electorate goes so far above the required average (given by the Hare quota) such that two constituencies both half its size would be nearer to the required average in sizes of electorate.

Suppose an electorate of 36,000,000 has a Parliament of 600 seats. Thus a single member constituency must have 60,000 electors, to give equal representation to voters thru-out the nation.

The upper limit of variation, from 60,000, is 80,000.This is because a constituency of 80,001 would be more equitably divided into two constituencies of 40,000 and 40,001, since the latter is closer to the 60,000 quota, than is 81,001.

 In the 1950s, Ross complained that governments did not understand this natural rule. And it’s still true now in 2011, where the House of Lords has disputed (according to David Lipsey in a Guardian article) the Tory coalition’s strict limit on variation to a mere 5%. The Labour peer wanted it relaxing to 10%, which he said would not harm Tory representation. On the contrary.

Even a 10% variation is less flexible and more continually fragments natural boundaries and messes-up local identities than the Ross rule. And that’s bound to stir up more local inconvenience and discontent and acrimonious dispute. All because the two parties, who can control a First-Past-The-Post single member system, are jealous of any slight advantage between them, while tacitly agreed on a system that under-represents or fails to represent anyone but themselves.

 However, the concern here is how to generalise the Ross rule for multi-member constituencies with the same number of seats, S. Also, a required average of votes per constituency, V/C. This is given from the total votes in the nation divided by the number of constituencies.
In the traditional English two-member system. If total electors, V, = 36m. and votes per seat, V/S = 60,000, and seats per constituency, S/C = 2, then V/C = V/S x S/C = 120,00.

 Let the least sized constituency be x voters, and the most voters permitted be y. Let their equal difference in voters, from the average of v, be z voters.

Therefore, v – z = x

And v + z = y.

The Ross rule shows that the smallest permissible constituency in a single member system, is half the size of the largest permitted.

The quota within a single member constituency is also one-half, or fifty per cent of the voters are needed to elect a candidate.

 In other words, s/(s+1) = ½ in a single member system, applies both to within-constituency representation and representation across the single member constituency system.

 Applying this Ross rule to multi-member systems, that is two member systems or three-member or four-member systems etc, with s, seats per constituency:

 Then, x/y = s/(s+1).

It follows that:

(v – z)/(v + z) = s/(s+1).


vs + v – zs – z = vs + zs.

 Or: v = 2zs + z.

Or: z = v/(2s + 1).

 But the permitted variation in votes from the smallest to the largest constituency is 2z.

And: 2z = 2v/(2s + 1) = v/(s + ½).

This new quota between the Hare quota and the Droop quota, I call the “Ross quota.”

 Suppose the Hare quota, V/S = v/s, is 60 (which is simpler to work than 60 000) then the required vote in a two-member system, is s.v/s = 2 x 60 = 120.

The smallest or largest permitted constituency is thus minus or plus z = v/(2s+1) = 120/(4+1) = 24.

Therefore, smallest constituency allowed is: 120 – 24 = 96. Largest constituency is: 120 + 24 = 144 voters.
This is a cross-constituency proportional representation of 96/144 = 2/3, as expected for a two-member system, which also has a two-thirds PR within constituencies.

Multiplying the Ross quota by s, for: sv/(s+1/2), gives the minimum permitted constituency (e.g. 2x240/5 = 96). While multiplying by (s+1) gives the maximum permitted constituency.

Classical mechanics as a model for maths of elections.

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An analogy between mechanical systems in simple harmonic motion and an electoral system was ruled by certain considerations. The simplest case in elections of displacement from equilibrium is perhaps given in the above section on the Ross quota. This just involves some allowance made for natural variations in the size of communities, such that some constituencies have to be made larger and some smaller than strict proportional representation requires.

Hence, the displacement had to be in votes per constituency above and below the norm or equilibrium position. Note that in the above diagram, the required average votes per constituency would be re-set to zero, so that the displacement, above and below the norm, is only the number of votes per constituency above or below that norm.

Mechanics, however, consists of two distinct systems, kinematics and dynamics, which brings mass into the picture. To be true to the analogy with elections, I had to make a distinction, essentially between representatives and represented. This meant that the represented, as the voters, were likened, literally, to "the masses."

Thus "kinematic" elections consist of representatives, S, and candidates, K, as would-be representatives, just as kinematics in physics consists, respectively, of time and space, but not the third quantity in the classical physics triumvirate, mass.

My implicit definition of the candidates is as the minimum number of voters. That is to say an election, in which there are only candidates voting, is an election not swelled by any extra weight of voters, inessential to the bare working of an election.

I have departed from strict analogy between classical physics and elections, in that my definition of the equivalent term to mass, m, unlike mass itself, is not a primitive term but a ratio of two terms, votes per candidate. Instead, energy is chosen as the equivalent term to votes, This is more in keeping with modern physics, in which energy is a rather more basic concept than mass.

In the appendix, there is a table on the close relations between linear motion and rotational motion. Essentially, rotational motion allows for a return or repetition. In an election system, especially a uniform member system, there is a repetition of constituencies. Northern Ireland or Ulster has a six-member system. In other words, each constituency is a repetition of six seats per constituency, and indeed repeats a proportional representation of six-sevenths of the voters in each constituency.
Thus, the maths of rotation expressed number of turns or revolutions in terms of constituencies, C.

Over the years, I have made several attempts at a good analogy between maths of physics and politics. This pages endeavor is summarised in table 1.

Table 1: Maths of Physics and Politics.
Physics: Politics:
x = Qr = 2πnr, length. K, candidates. (K = hpS, seats times parties times a constant, h.)
y, length up to radius or (maximum) amplitude. k, candidates per constituency. (k = hps, constant times parties times seats per constituency. Letting h = 1, p = s, k = s^2.)
r, radius or (maximum) amplitude. X, maximum variation from proportionly required average candidates per constituency
t = Pn, time equals period multiplied by revolutions. S = 2πs.C/2π, seats (or representatives).
u = wr, velocity equals angular velocity times radius. K/S = k/s = hp. Candidates per seat.
Q = 2πn, angle. C, constituencies.
n, number of revolutions. C/2π, constituencies divided by 2π.
P, period or inverse frequency. 2πs, 2π times seats per constituency.
w = Q/t, angular velocity (or angular frequency). C/S = 1/s, inverse of seats per constituency.
a = -yw^2, acceleration. -k/s^2 = -hp/s, or -1, if k = s^2, minus candidates per (constituency seats) squared.
m, mass. M = V/K, voters per candidate.
p = mu, momentum. V/K(K/S) = V/S = q, vote quota.
b = mw^2, elasticity. M/s^2.
E* = (1/2)br^2, total energy equals kinetic energy plus potential energy. V, total votes.
f = ma, force V/K(-K/S^2) = -V/S^2 = -q/S.


*Note that energy can be expressed in terms of work done, which equals force needed to move a mass body thru a given distance, or: W = fx.

The distance, x, can be considered as the velocity, from a start of zero to a final velocity, u, over a given time, t. This implies, on average, a velocity of half the final velocity, or u/2.

Thus, W = fut/2.

By Newton's second law, basic principle of mechanics, force equals mass times acceleration, or, f = ma = mu/t.

Substituting this, W = fut/2 = fumu/2f = (1/2)mu^2,

which is the formula for the kinetic energy of a moving body.

The total energy, as shown in the diagram, equals the potential energy plus the kinetic energy.

That is (1/2)br^2 = (1/2)mu^2 + (1/2)by^2,

where there is an elasticity constant, b, for instance governing the degree of oscillation to a spring. The distance, y is up to the maximum amplitude, r, of the oscillation. When y = r, the whole of the energy is potential energy. That is the two points at the top of the parabola, in the diagram, marking the furthest extents of the springs oscillation up and down.

There are any number of other examples: the amplitude might be the furthest extents of the swing of a pendulum back and forth. Or, a ball at the top of a bowl, before it is roled down, and upon reaching the top of the other side of the bowl. Tho, this neglects friction impeding the swing motions.

The kinetic energy, in contrast, is greatest, and consists of all the energy, in the middle of the oscillation or the swing, marked by the zero point, at the middle base of the diagram.

Following the rules of basic calculus, the potential energy can be reverse differentiated ("anti-differentiated" or "integrated") with respect to distance, y.

This transforms (1/2)by^2 into: by.

The kinetic energy also integrates with respect to y.

Thus: (1/2)mu^2 = (1/2)m(y/t)^2 integrates to: my/t^2 = -ma = -f.

These integrations or summings of the complementary potential and kinetic energies are equal. Hence, f = -by.

That is to say that the force is proportional to the displacement, y, and opposed to it, as shown by the negative sign. The force is always acting towards the equilibrium or rest position, where displacement is zero, and therefore the force is zero, between oscillations or swings.

The so-called simple harmonic motion, of an oscillating spring or swinging pendulum or ball rolling up and down in a bowl, is given by the displacement, y as a sine equation, such as: y = r.cosQ.
Or it could be, say, y = r.cos(Q + π/2) = r.sinQ. Or indeed any so-called phase (call the phase: &), besides angle, π/2 radians, could be used to signify the starting angle (*initial phase") of the wave motion.
Sometimes the whole of the bracketed angle (Q + &) is called the phase, with angle, &, called the "initial phase."

The maximum sine value is unity, which is when y = r, the (maximum) amplitude. This is either the crest of the wave or its trof, the corresponding negative displacement, from an equilibrium such as a calm water level.

Given that y = r.sin(Q + &) = r.sin (wt + &),

the operation of differentiation with respect to time, t gives:

velocity, u = dy/dt = wr.cos(wt + &).

A further differentiation of velocity with respect to time, or in other words, a second order differentiation of distance, y, with respect to time, gives:

d²y/dt² = du/dt = a = -rw^2.sin(wt + &) = -yw^2.

From: f = -by,

m.d²y/dt² + by = 0.


d²y/dt² + yw^2 = 0.

This second order differential equation is the standard equation of simple harmonic motion, which is solved in terms of the equation for y.

Election models of wave motion.

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The relevance of this basic physics for politics is that an election system can be similarly measured.

An ethical force or moral imperative of proportionality attracts oscillations or swings in sizes of constituencies back to equilibrium of the required average.

For instance, you might have a constituency system in which there was a seasonal shift of electorates from an equilibrium of equal constituencies. This is not necessarily a single member system but could apply to multi-member systems.

Suppose migrant workers doing seasonal work deplete one constituency to augment another constituency, by an equal amount. This would represent a seasonal swing in the size of these two constituencies, in principle, subject to the same kind of (simplified) mathematics just discussed.

There would be, so to speak, a surplus transfer of voters to one constituency leaving a deficit transfer from another constituency.

This surplus transfer, across constituencies, compares to the conduct of a proportional count for an election within a multi-member constituency. Candidates elected, with more votes than the quota they need to win a seat, have the value of that surplus vote transfered to their voters next preferences.

In accord with the democratic principle of one person one vote, the transfering of surplus votes means that each voter, for a candidate elected with a surplus, effectively has a proportionate fraction of their one vote given as a "transfer value" to a next prefered candidate. The balance of that one vote is kept, by the candidate elected with the surplus, and is called the "keep value."

Akin to physics, the vote has an energy. The proportional count achieves equilibrium when the surplus votes are transfered to prefered candidates still in deficit of a quota.

Thus, the proportional count is not like a wave of uniform amplitude but a damped wave down to equilibrium. Its equation modifies the uniform sine equation with a damping coefficient in the form of an exponential function with a negative index.

Before the votes are counted, they are all potential energy. The count of the votes is how they move, or their kinetic energy. In common parlance, to pass a motion means to take a vote. The most popular candidates with big surplus votes will have the smallest keep values and the correspondingly largest transfer values. That is to say their popular vote has the smallest kinetic energy and the correspondingly largest potential energy for transfer to next prefered candidates.

Simple harmonic waves are described by sine curves. These can also be mathematicly expressed in terms of Euler formulas using an exponent with an imaginary number in the index. That is of the form for a complex number, z = r.e^iQ = x + iy = r(cos Q + i.sin Q).
I merely mention this, without attempting to explain it here, because, on other pages, I have given election models of another type of exponential function function, the normal distribution. At its simplest, this is of the form, y = e^-(x²/2).

The various types of exponential function share the property that conventional differentiation leaves the independent variable unchanged, tho the coefficients are affected according to standard differentiation rules.

The wave equation, given above, involving a second order derivative, is but one of several second order differential equations (which may be partial differential equations having more than one independent variable) not only with wave solutions but solving in terms of random or normal distributions.

My treatment of the normal distribution was rather different to convention, because I considered differential calculus on a statistical basis. This allowed me to introduce a geometric mean form of differentiation, which derived an exponential function, rather than just took it as found, under traditional differentiation.


Appendix: corresponding strait-line and rotational motions.

I havent troubled to give electoral equivalents to all the physical variables Ive mentioned. For instance, one physical concept not mentioned on either table 1 or 2, that of Action, equals energy multiplied by time, might be given an election equivalent of votes, times representatives, equals number of preferences for representatives.

Some of the more complicated variables in table 2 might be compared similarly in terms of election preferences. And there is always the fall-back of primaries or candidates for candidature. But such ingenuity, at such an early stage in a new subject, might not be particularly helpful.

Table 2: Physics of corresponding motions.
Strait-line motion: Rotational motion:
distance, x = Qr angle, Q
velocity, u = wr angular velocity, w
acceleration a = αr angular acceleration, α
mass, m moment of inertia, I = mr^2 (for all mass on wheel rim)
force, f = ma torque, T = fr = Iα
kinetic energy, (Ek) = (1/2)mu^2 kinetic energy, (1/2)Iw^2
momentum, p = mu = d(Ek)/du angular momentum, L = mur = Iw = d(Ek)/dw
impulse, change of momentum, ft angular impulse, change of angular momentum, Tt

Note: This table is based on Milton A Rothman's The Laws of Physics.

The principal text consulted for this web page was Alonso and Finn: Fundamental University Physics, volume 1. (This classic text was later revised in one volume as "Physics" under the general editorship of Paul Davies.)

Foot-note 1:

Or #S could be, say, 2, as in another quota, I formerly hazarded to call the Ross quota (a term I now use for a new quota above). I relegated the old page, on the former Ross quota, to My Archive. I don’t know whether it contains a useful germ of truth or is just plain wrong! In any case, it was not conformable to a parabolic function.
In this respect, my early work for a “Ross quota” gave an unsatisfactory lop-sided answer but here I want to give an equal amplitude variation limit to both smaller and larger than average multi-member constituencies.

Richard Lung.
24 May 2011;
minor revisions, 16 June 2011.

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