- The geometric mean in special relativity and STV with controled re-counts.
- Surplused candidates already elected as representatives.
- The Interval in Mechanics and Electics.
- Constituency dimensions, observer participants, and from majority to surplus.
- Comments on Mechanics and Electics table entries.
- Appendix 1: the Interval related to the Lorentz transformations.

Some previous web pages show that Ive persisted in trying to match physical theory, as represented by Special Relativity, with electoral method, essentially transferable voting. I suppose the best by-product of that was to show that Special Relativity is better conceived as a statistical theory.

The electoral modeling or analogising with Special Relativity remained incomplete. That is to be expected. It will still remain so, after this web page. But some basics perhaps can be met.

Proportional representation ( in the original, logical application of a transferable vote ) makes possible the kind of arithmetic accuracy in voting that is found in an advanced science. For a long time, I relied on the concept of quotas or elective portions of the vote as the link with the mechanics of motion. It seemed that applying the Fitzgerald-Lorentz contraction formula to quotas could squeeze out marginly more proportions of representation for the voter. Those were the lines I was thinking on in the early 1990s. ( I put on-line some of my notions of that time. )

Much later, I ventured a web page on an electoral model. Looking back, I am surprised by how much milage I seem to have made out of mainly a quota model. Soon after, I realised the greater flexibility of employing the concepts of keep value and transfer value for the Relativity analogy. Quotas are mainly about how close one can approximate complete representation, whether within or across constituencies. But the value concepts cover the whole possible range of support, for a candidate and next prefered candidate, from all to none. Similarly, relativistic motion considers velocities from rest up to light speed. And an analogy has to reflect that.

It is fair to say that a quota and values model still hadnt supplied a consistent analogy of electoral method with mechanics.The test is: do you have an isomorphic relation? That is scientific method's jargon for a one-to-one relationship. In other words, for every concept in mechanics, have you a matching concept in "electics?" ( "Electics" is another jargon word Ive just made up on the spur of the moment, not to have to keep repeating the phrase, the study of electoral method. ) The two groups of concepts match when every concept in either group is mathematicly related to its fellow concepts in the same way as its matching concept in the other group is related to its fellow concepts.

The basic concepts of classical mechanics are space, time and mass. Since relativity, the concept of mass has been somewhat over-shadowed by that of energy. It is possible to base physics on alternative conceptual foundations, which are just as consistent and just as valid a framework in which to study the natural world. For instance, it is possible to rest mechanics on a two-concept foundation of velocity and mass.

This is worth bearing in mind for would-be analogisers between scientific disciplines. A previous page shows my attempt to align the mathematical characteristics of space, time and mass with those of candidates, representatives and voters. That began to look more promising than naive early attempts at a correspondence, but in itself it didnt seem to help much.

As it happens, this three-three correspondence of basic concepts does seem to be the most convenient analogy I can consistently apply. But it took another distinction to be made, for a consistent comparison between the two systems of mechanics and electics. This page is the up-shot of the new distinction and a switching round to make the analogy work.

To put this new departure in context, I refer to my pages which show that Special Relativity is statistical. The Lorentz transformations and the Interval may be considered as geometric mean relations. That is to say, observers' measurements of an event, in high velocity physics, may be considered as a kind of average that suitably represents the range of observations they make in relation to each other. In particular, the interpretation of the Interval is the commonly observed geometric mean, or representative measurement of an event from observers' different point of views, as a statistical range of observations.

Also previously explained, the simplest showing of the geometric mean in Relativity is the contraction factor. So, this is a convenient starting point for a comparison with the geometric mean as found in the single transferable vote ( STV ). The logical objection of premature exclusion was made to transferable voting on the grounds of the chance excluding a candidate with least votes, just when there happen to be no more surplus votes to transfer to next prefered candidates.

To avoid this, Ive given web page examples of the transferable vote's count being re-run like a controled experiment or test of public opinion. No candidate is definitely excluded in these re-counts but the controled or systematic withdrawal of one or other candidate to allow re-distribution of votes means that candidates are given a result for each re-run. These multiple results are each expressed as keep values.

A keep value expresses the ratio of the quota to total transferable vote received by a candidate. The quota is the portion of votes any candidate needs to be elected, according to how many seats or vacancies going in the constituency. The total transferable vote is how many votes the candidate actually got, if more than the quota. Say these were a total of first preference votes to the candidate that comes to more than the quota. Then that candidate has a surplus, which is transferable to next prefered candidates. They share the surplus in proportion to their shares of next preferences out of that total transferable vote.

An elected candidate always has a keep value of one or less. A keep value of one means that the candidate's total vote equals the quota, that is exactly what he needs to be elected. The bigger the total transferable vote than the quota, the smaller the ratio of the quota to the total transferable vote. In other words, the most popular candidates are those with the smallest keep values.

Traditionally, keep values have only been considered in these terms of candidates with surpluses of votes above the quota. But I introduced the possibility of keep values over one. That is for candidates with total votes in deficit of the quota. It was necessary to do this, if you have controled re-runs of STV. The point of re-runs is to average their range of results to find the most typical. In one count, a candidate may get an elective keep value of less than one. In the re-run, he might get an unelective keep value of more than one. Roughly speaking, if the candidate's deficit of votes, below the quota, in the re-run is less than his surplus of votes, over the quota, in the first count, then on average, the candidate still has a keep value of less than one. In that case, the average keep value elects the candidate.

The suitable average, for the re-run ranges of candidates' keep values ( in Binomial STV, as I called it ) was in fact the geometric mean, the same average Ive claimed is implicit in Special Relativity. As explained before, the reason for using the geometric mean as an average for both proportional representation and special relativity is because that is the suitable average for their kinds of ranges, subject to diminishing returns, as both increasingly high velocity and increasingly proportional representation are.

My reformed STV might have just two controled counts. Each candidate may receive two unequal results, where one will be more or less than the other, as such constituting a range. To find their geometric mean, multiply the end-values of the range. There are only two here and, in that case, their square root is taken for their geometric mean keep value.

The transfer value, already mentioned on other pages, is the complement of the keep value. The two values always add up to one. This is the egalitarian one person one vote value. ( It is in fact a realisation of the electoral reformers' slogan: one person one vote, one vote one value. ) For instance, if a candidate has just enough votes to be elected, his keep value is one but his transfer value is zero, because there are no surplus votes to transfer to next prefered candidates.

In short, keep value, k, plus transfer value, t, equals one, or, k + t =
1. Weve also said that the keep value equals the quota, q, divided by the
total transferable vote, T, or, k = q/T.

So, t = 1 - q/T.

Compare this with the famous contraction factor from special relativity. This is the square root of all the following: one minus the ratio of the square of velocity, v, divided by the square of the light speed, c. The velocity is the relative velocity between observers and the factor corrects for observers' respective high velocities, to or from each other, given a limiting speed of light.

In short, the contraction factor is:

( 1 - v²/c² ).

The geometric mean is calculated as an average of, say, two end-values of a range of values, by multiplying those end-values and taking their square root. Treating the contraction factor as a geometric mean, to find its end-values, you square it, which gets rid of the square root. Then factorise the remaining ( 1 - v²/c² ).

This gives: ( 1 - v/c )( 1 + v/c ). The factor with the plus value is the upper end of the range. The factor with the minus value is the lower end of the range. Compare the lower value with the transfer value as 1 - q/T. But I found out, thru trial and error, that the electoral analog of relativity wasnt quite amenable to consistent treatment in quite these terms.

Consider an STV election. Several candidates compete for a lesser number of seats for representatives, in a multi-member constituency. Suppose that some of these candidates have totals of first preferences in excess of the quota. It is an arithmetic certainty that there cannot be more candidates with surpluses than there are vacancies to give them.The quota count ensures that if some candidates have surpluses, those surpluses will be needed to help other candidates fill the remaining seats.

The number of representatives ( call that number r ) required in the multi-member constituency may be distinguished from the number of candidates with surpluses ( call that number s ) who are, therefore, already elected as representatives. At the first stage of the count, s means number of candidates with first preference surpluses. We may sum up the previous paragraph by saying that in general the number of representatives is greater than the number of surplus-voted candidates. Or, r > s.

The pattern of counting the transferable vote is that a few large surplus candidates give way to small surplus candidates till all the surplus candidates have lost their surpluses and are equal to the required number of representatives.

This distinction, not made in my previous pages, of surplus voted candidates, s, from representatives, r, is analgous to special relativity's distinction between maximum velocity, being light speed, c, and lesser velocities, v. Velocity is a ratio of distance to time. And s and r belong to ratios. Let V mean the total votes in a multi-member constituency. Then V/r means voters per representative in the constituency. This may be called the quota, q = V/r. It is used for deciding the number of voters per constituency in a constituency system. ( We bear in mind the refinement that the actual quota of votes any candidate needs to be elected within a constituency is V/(r+1). )

Our electoral analogy makes the quota equivalent to light speed, c. Light speed is the ratio of distance to time that no body's velocity may exceed. Similarly, the quota is the ratio of votes to representatives that no candidate may exceed. The value of the surplus votes is always transferable to next prefered candidates.

We already have given the letter "s" to number of surplus-voted candidates. We give the number of surplus votes its own letter "b" for balance of votes over the quota. The term, balance, is also used because it suggests the meaning "more or less". And one bears in mind that a candidate may equally well win as many less votes than a quota as he might win more votes than a quota. This is the idea that transfer values may be deficit as well as surplus.

Statistics is in terms of a range of values more or less about an average. If we add up all the surplus-voted candidates' balances of votes above the quota and divide by that number, then b/s is the total surplus vote per surplus-voted candidate. That is the average surplus vote.

Here we come up against a problem for a strict electoral analogy with special relativity. Suppose there is only one surplus-voted candidate, s = 1. That candidate might be very popular and have a surplus vote, b, that is bigger than the quota. That means that as well as winning a quota to elect himself, he has a big enough surplus transferable to elect another candidate, with still some votes to transfer to subsequent preferences.

That possibility is analgous to a body with velocity greater than light speed, which is not allowed. There is a theoretical escape clause in quantum electro-dynamics. This is the sum over many-paths, which allows extreme short range variations in the speeds of photons, which average out over macro-scopic distances to the constant speed of light.

However, in the course of an STV count, any surplus vote, initially itself more than an extra quota of votes, will rapidly cease to be so, in the successive stages of transfering the vote.

Moreover, our electoral analogy isnt allowed a single comparison, because mechanics is double-parted. Dynamics treats of masses in motion, while kinematics doesnt consider masses. Kinematics treats of velocities but dynamics treats of such things as momentum or mass times velocity.

Perhaps, the simplest way to show how the mechanics-electics correspondence works is to go to the key formula in special relativity, the Minkowski Interval for both kinematics and for dynamics. In the latter form, mass effectively replaces time in the former. ( For the Interval relation to the Lorentz transformations, the other well-known formulas in special relativity, see appendix 1. )

The basic kinematic Interval ( actually the Interval squared ) is (1):

t²(c² - u² ) = t'²(c² - u'² ).

The two sides of the equation show how two observers correspond their respective space-time measurements for their respective velocities, u and u' significantly approaching light velocity, c. Space or distance equals time multiplied by velocity, or s = tu, and s' = t'u'. The Interval is usually given with two more dimensions of space but this is not crucial for our purposes. Even one dimension can be considered as a vector resulting from a three-dimensional tug of war, as it were. But a tug implies force, involving mass, which is the province of dynamics.

The basic dynamic Interval squared replaces the time outside the brackets with mass (2):

m²(c² - u² ) = m'²(c² - u'² ).

The text books usually express this in energy terms after E = mc² as (3):

E²/c² - p² = E'²/c² - p'².

where the momentum is p = mu. ( Again there will be two more dimensions, this time in terms of x, y z, co-ordinate directions of momentum. )

As explained on previous pages, the Interval is a logical extension of Pythogoras' theorem for 4-dimensional space-time. In effect, the Interval is the hypotenuse of this extended geometry. And you take the square of the hypotenuse to equal the sum of the squares of the other two sides at right angles in a triangle. Of these other two sides, say one is a vertical side and the other is a horizontal side. Say, the latter side is itself the hypotenuse to two base dimensions of a cube. This effects Pythagoras' theorem for three dimensions.

Physics texts usually leave the Interval as its square, really a hypotenuse squared, rather than the hypotenuse itself. This matters when the Interval is considered as a geometric mean, which is the square root of the ends, of a geometric range of values, multiplied together. When the Interval is left squared, we dont need to take the square root, to trace the end-values. To get them, we factorise the respective sides of the Interval squared.

But current physics doesnt recognise the statistics of special relativity and it is easier to show the Interval, as a geometric mean, in electoral terms. For this page, I arrived at the result, as follows, omitting some trial and error.

The key idea was to express the contraction factor, alias geometric mean, in terms of light speed, c, equivalent to quota, V/r. Velocity, v, of a body, approaching but not exceeding light speed, equivalent to average surplus, which equals total surplus of votes divided by surplus-voted candidates, or, b/s.

We noted above that the keep value, k = q/T, or the quota divided by the total transferable vote. But this assumes only one surplus-voted candidate, or s = 1. This is used when returning officers count the total transferable vote of surplus-voted candidates, candidate by candidate. But here we are concerned, for study, with the average surplus equals total surplus vote per surplus-voted candidates. This means adding the total transferable votes of every such surplused candidate and dividing by the number of them: T/s.

Thus the keep value becomes, k = q/(T/s) = (V/r)/(T/s).

And T/s, the total transferable vote per surplused candidates, is the same as the quota plus the total surplus vote per surplused candidates: T/s = V/r + b/s.

Therefore (4): k = (V/r)/{ V/r + b/s}.

This is an elective keep value, because the ratio is less than unity. But we also discussed how a reformed STV could give average keep value results for each candidate. And such results might equally well show deficits as surpluses of votes about a quota for a candidate.

Hence the equal possibility for a candidate, in a re-run of a controled STV count, to have an unelective keep value of greater than one, like so (5):

k = (V/r)/{ V/r - b/s}.

To find the geometric mean keep value, multiply the two keep values, the elective and unelective ends of a value range, and take the square root. In fact, this result averages slightly less than one. Therefore that candidate would be elected on the basis of her geometric mean keep value, given a capital letter, K. Hence (6):

K = ((V/r)/{ V/r + b/s}.(V/r)/{ V/r - b/s}).

We square this equation, then invert it for, 1/K² as (7):

1/K² = { (V/r)² - (b/s)² }/(V/r)².

Then multiply thru by K² and (V/r)². Finally, multiply thru by z², so the Interval value, on the left side, is not left equal to the equivalent of light speed, which the Interval generally is not. Likewise, the variable outside the curly brackets, on the left side, cannot be left as just a factor K. Hence, equation (8):

(zV/r)² = (zK)²{ (V/r)² - (b/s)² }.

We can repeat this procedure for another geometric mean K' and another average surplus equals total surplus vote per surplused candidates, b'/s'. It equates to the above equation, as (9):

(zV/r)² = (zK)²{ (V/r)² - (b/s)² } = (zK')²{ (V/r)² - (b'/s')² }.

This is the special relativity Interval in electoral form, with the Interval squared on the left side. To be equivalent to the Interval in physics, zK = s, because the time, t, occupies both those positions in the Interval in physics. Likewise, the other physical observer's time, t' must be equivalent to both zK' and s'. The two must be equal for the electoral analogy with the Interval to hold.

It is sufficient to test one of the required equalities: zK = s. Take (10):

(z/s)² = 1/K² = { (V/r)² - (b/s)² }/(V/r)² = 1 - (b/s)²/(V/r)².

Therefore, z² = s² - b²/(V/r)².

And (11): (V/r)² = b²/( s² - z² ).

This equation tells that the quota squared ( equivalent to light speed squared ) on the left side, is larger than b²/s², which is equivalent to sub-light speed velocity squared. This means that the requirement, of making zK = s, seems to have left the analogy consistent.

Thus we can confirm the electoral analogy to the Interval as (zV/r)².
Actually, the Interval here compared is the kinematic Interval in terms of
time. The dynamic Interval, that replaces time with mass, requires a slight
modification of the formula. In physics, mass divided by time is proportional
for the two relativistic observers: m/t = m'/t'.

So, to correspondingly modify the electoral analogy, take some further
coefficient, y, then multiply thru by the square of y/z.

This gives the dynamic Interval an electoral equivalent, (yV/r)².

The question is what could the coefficients "z" and "y" stand for electorally, since they are needed to make the comparison with special relativity's formulas?

The simple equality zK = s shows that z is just a modification of s by factor K.

Coefficient, y, can be absorbed in the numerators inside the curly
brackets of equation (9). That means that we could let V = yC and b = yc and
b' = yc'. All we have done is make coefficient, y, into a ratio of voters, V,
to candidates, C. In an election where all the voters are candidates then
coefficient, y, reduces to one, or y = 1. Then total votes in the
constituency is replaced by total candidates as pan-candidate voters. The
surplus votes over a quota are the surplus of pan-candidate votes, c, or c'
in the case of the other side of the Interval.

It's difficult to see how we would accommodate the concept of candidates in
the analogy without the help of coefficient, y. Now we have to remember that
light speed,c ( not to be confused with the aforementioned electoral use of c
) is not equivalent to V/r but C/r.

There is still some tidying-up to be done. We have treated the Interval in one spatial dimension instead of three. Admittedly a single vector can be the balance of three dimensions. But still, what electoral interpretation can we offer for this? Well, the first page I put up on the subject ( tho I subsequently put up an earlier effort ), Electoral model of Special Relativity was mainly about permitted variations in proportional representation across a uniform member constituency system, allowing for geographical variations in the size of constituencies.

I couldnt think of how to express permitted variations in proportional representation within a constituency. Later, I thought of an STV system of controled re-counts ( Binomial STV ) producing a range of keep values, for each candidate, which are averaged for the final result. Permitted variations in the keep value, measured in statistical deviations, from the electively required keep value of one, could assess whether a candidate was probably elected or unelected. And this might determine whether a marginally elected candidate might be unelected thru unpopularity in an exclusion count. Or, a marginally unelected candidate might be elected thru a significant lack of unpopularity in the exclusion count.

Thus, there are statistical variations of PR within a constituency and of PR across a uniform member constituency system. That leaves the possibility for statisticly permitted PR variations between constituency systems. So, we could think in terms of three dimensions of constituency PR comparable to mechanics' three dimensions of space. This page's discussion has been implicity just in terms of PR within a constituency.

This leaves us, right at the end, with perhaps the most basic question as to the meaning of the electoral analogy. Almost the whole point of Special Relativity is to ensure all observers observe the same natural laws, especially under the extreme conditions of high energy physics fairly approaching light speeds. ( Essentially, proportional representation by a transferable vote, even in its traditional form, consistently observes a moral law to high standards of equality in freedom, as mentioned on my page The Laws of Motion and Election. )

An electoral meaning of two ( or more ) observers might be that there are two ( or more ) groups or parties of candidates. These each have one or more surplused candidates. Then b/s and b'/s' are the respective parties' totaled surplus votes, b and b', divided by each party's number of surplused candidates ( meaning already elected candidates ) s and s'. It is also possible that an Independent ( or more than one ) might have a surplus over the quota. In the Independent case, b stands for her one surplus divided by one surplused candidate, s = 1.

Of course, this implies a large number of seats in the constituency. The more such "observers" or participants, the more seats there have to be for them to possibly gain a number of surplused candidates in each party.

We have been concerned to establish a comparison with special relativity. However, electics should have an intelligible cross-over to classical mechanics. ( My early paper, The Laws of Motion and Election, discussed this. ) The present page is new in matching velocity to surplus votes, in the respective mechanic and electic systems.

In classical mechanics, light speed is not a significant issue in measuring relative speeds. Similarly, one might speak of a classical electics, which has not conceived the quota. Indeed, this is the case. Proportional representation is a product of the nineteenth century.

Yet it still makes sense to think of the surplus vote, the "more-than vote" in pre-quota terms, as a majority. Majority voting is akin to the Galilean relativity principle of motion. Both involve simple additions and subtractions, whether of observers' relative motions or candidates' relative majorities.

Children had those line drawings with little numbers, inside the shapes, telling you which color to use for each shape. The electoral analogy, with the Interval of Special Relativity, was not like crayoning by numbers, I can assure you. But we can make it look like that in the end. Our electorally color-coded Interval for kinematics and dynamics, can be summed up in table 1 of Mechanics and Electics.

Galileo or Lorentz relative velocity are both symbolised by v, because they both refer to a relative velocity, tho the former is just for relatively low velocities. Ive also let v mean its equivalent electic term, relative average surplus.

The equivalent term to light speed, c, is q, the candidates per representative, C/r. The letter q stands for quota and may, for instance, be thought a quota of parties. The reason is that a number of representatives stand to represent a distinct group of people or their opinions. But each of these groups or parties will contest who is the best candidate to represent them. Ideally, as far as each party is concerned, the party candidates would win all the seats, representing not only themselves but all other groups. Thus, the number of candidates, C is determined by the number of representatives times the number of parties or groups, g.

So, C = rg. Or, g = q = C/r. In that case, it can be said that maybe C = r². Then the electic equivalent for energy, E = mc², which is V(C/r)² becomes VC. This is the votes multiplied by the candidates. Suppose an election in which all the voters, V, express a preference, p, for every candidate, C. Then p = C and Vp = P, where P is the total preferences expressed by the voters. Then P might be taken as the electic equivalent of energy.

The equivalent for momentum follows from that. Admittedly, what we really wanted was for the vote quota, V/r, to have an obvious equivalent term from mechanics. It is actually the ratio m/t, without a well-known symbol of its own. Nevertheless, m/t switches the Interval ( and the Lorentz transformations ) between kinematic and dynamic versions. So, it cannot be said to be unimportant. In any case, we would not necessarily expect that all the most important terms in one system were matched by all the most important terms in the other system. It is sufficient that there is a well-defined correspondence.

Voting preferences are a mathematicly useful concept. For instance, one could draw a distribution graph. Number of preferences per voter could be the x-axis. Number of voters, for each number of preferences, could be the y-axis. The number of preferences could go from zero for abstention or one for the statement of only one preference up to the voter stating an order of choice for every candidate. That is maximum preferences per voter equals the number of candidates.

On a graph or bar chart, the number of voters, who expressed zero, one, two, three, etc ranked choices, are represented by the successive lengths of vertical lines. The shape of the mass of lines might be a binomial distribution or perhaps a Poisson distribution. The area of this distribution, which is the sum of voters for each number of preferences, gives the total preferences.

By the way, some think-tank's policy ultimatum, this month, that the public should be forced to vote, mis-represents zero preference, which is as much an option as one preference, all our mis-begotten one-party government, by one-in-five, as usual offers.

Mechanics | Electics |

t, t' times | s, s' ( representatives who are ) surplused candidates |

s, s' ( space ) distances | C, C' candidates |

u, u' velocities | c/s, c'/s' average surpluses |

v, Galileo or Lorentz relative velocity | v, relative majority or relative average surplus |

c, light velocity | q = C/r candidates per representative. |

m, mass | V, voters |

mv = p, momentum | V.C/r, total preferences per representative |

E = mc², energy | V(C/r)², total preferences, P ( if C = r² ). |

Starting with equation (9):

(zV/r)² = (zK)²{ (V/r)² - (b/s)² } = (zK')²{ (V/r)² - (b'/s')² }.

Using the equivalences, zK = t and zK' = t'. And V/r = c; b/s = u; b'/s' = u'.

That is equation (1):

t²(c² - u² ) = t'²(c² - u'² ).

Or, t²/t'² = (c² - u'² )/(c² - u² ).

But t²/t'² can also be expressed as a ratio of the respective observers' Lorentz transformations for time, which cancels out their respective contraction factors.

Observer O records time, t = t'( 1 - u'v/c²)/( 1 - v²/c² ).

Observer O' records time, t' = t( 1 + u'v/c²)/( 1 - v²/c² ).

Therefore, t²/t'² = ( 1 - u'v/c²)/( 1 + u'v/c²).

This equals the derivation from equation (1). So,

( 1 - u'v/c²)/( 1 + u'v/c²) = (c² - u'² )/(c² - u² ).

Cross-multiplying:

1 + uv/c² - u'²/c² - uvu'²/c² = 1 - u'v/c² -u²/c² + u'vu²/c²c².

Cancel 1 from either side of the equation and multiply thru by c²:

uv - u'² - uvu'²/c² = -u'v - u² + u'vu²/c².

Therefore, v(u + u') + (u - u')(u + u') = vuu'(u' + u)/c².

And, v + u - u' = vuu'/c².

So, u + v = u'(1 + uv/c²)

Or, u' = ( u + v )/(1 + uv/c²).

This is the Lorentz transformation of velocities going from observer O to observer O'.

*Richard Lung.
13 May 2006.*

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