Arrested For Living Too Far Up The Road

Credit where credit is due on this one.  This is based on a dream a friend had a few years ago about someone being arrested for living too far up the road.  More recently, I spent too much time considering how this could conceivably happen.  I say "conceivably" because the way my thoughts went, it isn't actually possible.  Unfortunately it's now occurred to me that there are circumstances in which it really could happen.  For instance, in Apartheid South Africa it's possible that people of a certain ethnicity could be arrested for living in the "wrong" area and in some places this amounts to "living too far up the road", but that isn't really how things worked there because people wouldn't've been able to move into such an area in the first place.  Nonetheless, such a situation is, unfortunately, possible.

After some cogitation, I realised there is a pre-existing fictional setting into which this would fit quite well:  the sitcom 'Ever-Decreasing Circles'.  In case you don't know, this is an '80s sitcom based around one Martin Bryce, whose obsessiveness has led to him fixating on community activities in his local cul de sac in the Home Counties, stressing out his long-suffering wife Anne.  Meanwhile he is completely unable to compete with his neighbour Paul Ryman, who effortlessly succeeds in outpacing him at every turn.  Hilda and Howard Hughes are his main allies and acolytes.  His job at Mole Valley Valves, where he is a fairly low-status drone although this being the '80s he does have his own office, occasionally impinges on the action.

I'm not going to turn this into a review of the series, but it's notable that Martin is played by Richard Briers, who also played Tom Good in 'The Good Life', about a suburban couple who decide, or rather he decides, to become self-sufficient, and watching the two sitcoms together lead to a completely different take on what's going on in the earlier one.  It brings out Tom's pig-headedness much more, and consequently also his wife Barbara's dedication and also put-uponness.

When I mentioned my story idea about the brainwashing helmet, the problem was that there could easily be a sudden jerk into implausibility when that plot device is introduced.  This isn't necessarily a bad thing, but it also applies to what I'm doing with this.  The main focus of this post is on the device I use to weld the two thoughts together:  of someone being arrested for living too far up the road and 'Ever-Decreasing Circles'.  This is how I see this particular fanfic proceeding:

The opening scene is based on the episode 'Half An Office', where Martin, at Mole Valley Valves, is due to have his office halved in size owing to the expanding role of his neighbour.  I get the impression that his situation at work helps explain his attitude to the close he lives in, and is a realistic pattern for many people.  Because he's unable to achieve the success he might at work, he attempts to do so in other areas and he sees himself as the king of The Close.  Now, although it hardly seems to enter the picture at all in the sitcom, Mole Valley Valves is of course a firm which makes valves.  This is the jumping-off point for my idea.  The company develops a new valve which seems to work like a pinch valve.  A pinch valve reduces fluid flow by using a pinch effect.  Apparently, among other things of course, they're used in water pistols.  The word "fluid" colloquially tends to refer only to liquids but in fact gases are also fluids, as is plasma.  A major difference between gas and liquid is that the latter is hardly compressible at all.  This can be seen in the contrast between the atmosphere and the ocean.  The pressure in the atmosphere reduces logarithmically:  over a certain height, the pressure is half what it is at sea level, and at twice that height it would be a quarter, and so on.  Underwater, the situation is very different.  The pressure is twice what it is at sea level only ten metres down, then thrice twenty metres down, and so on.  Although I think there's some compression, I do know it's very small.  Pressure in a liquid is linear.  This means, for example, that filling a balloon with water leads to an easily calculated volume and diameter, but filling it with air is very much non-linear.

Mole Valley valves develops what they consider to be a pinch valve of unusual design.  The flow of water out of the valve is smaller than the flow into it, but the excess liquid seems to disappear completely.  This is very hard for the developers to explain, and they realise that their own knowledge is no better than anyone else's in this respect, so they decide to engage in the '80s equivalent of crowdsourcing the research by allowing everyone at the firm to investigate the valve.  Martin receives a valve and accidentally spills coffee into it.  He is surprised to find that although coffee emerges from the other end, the volume is much smaller than expected.  After some investigation, he discovers that the valve reduces the size of anything which passes through it in one direction while magnifying anything passing through it in another.  More specifically, the dimensions of any object passing through at the inlet are halved at the outlet.  This effect also continues outside the valve, so that objects begin to shrink in proximity to the outlet before they enter it and continue to shrink on the other side.  This effect only exists at a distance equal to the depth of the valve.  It means that objects have one eighth of their volume immediately after passage, but the total shrinkage is greater, and I'm going to make this up on the hoof:  total shrinkage is sixteenfold, with half occurring on each side.  Also, objects have proportionately less mass:  they do not become denser or less dense.  Finally, and this is crucial to the story line, all dimensions are shrunk, including time, so objects on the far side of the valve are twice as "fast":  a watch passing through the valve runs fast and covers twelve hours in only six, or in fact three if the complete distance is taken into consideration.

Martin finds that the valve's properties don't depend on its material.  Instead, it's to do with the geometry of the valve.  Valves made out of polystyrene, wood or papier maché have exactly the same properties, and they can be any size.  He eventually discovers that the valve induces space parallel to the axis to be non-Euclidean and anisotropic.  I'll go into more detail about this later.  He also realises that he can get his office back to its previous size by constructing a giant version of the valve against the wall and sliding it across the floor halfway across the remaining available space.  This in fact creates a subjectively larger office than he had before, because the effect continues on either side, and the floor space is in fact up to eight times larger.  It also means, however, that anything passing through the valve and then around the side in one direction is shrunk compared to the other side of the office, and more disturbingly grows in size circulating the other way.  This leads to a housefly circling the ceiling lamp growing steadily in size until it's as the size of a heron, at which point it collapses and suffocates, and also means his secretary, previously only half-available to him, shrinks but is able to do the same work in half the time provided she goes around the edge of the valve.

He then realises that he can also use the valve on the Close.  If he constructs a valve which passes through the drains into the back gardens of the houses, it will reduce the size of the Close on one side.  If he then devises a way to pass the end of the Close repeatedly through the valve, it will continue to shrink the Close, and each time he does this, the mechanism whereby the action is performed will at least double in speed. If he can also duplicate the item passing through the aperture, he can create an infinitely long cul de sac.  Contradiction intentional.  This is where I have to depart even more from reality.  Martin canonically owns a spirit duplicator.  If he is able to use this to duplicate the items passing through the valve, and rig the duplicator up so that it can crank its own handle with 100% efficiency, he can do this in only twice the time it takes him to create the first copy of the end of the Close.  Moreover, at the very end of the Close, time will pass infinitely fast.

This situation has its limits given the Planck Length.  This is the minimum possible distance, and is 1.616255(18)×10^−35 m.  The filming location was in Dell Lane, which is a T-shaped close:

The actual "Close" itself is the bottom right branch, and is around seventy metres long including the gardens.  This is the bit I envisage being duplicated.  In order for the Planck Length to become equivalent to those seventy metres with the distance halving each time, it would have to be duplicated 119 times, which is an equivalent distance of one and a half teraparsecs.  This is very long but still not infinite.  Therefore I've chosen to ignore the Planck Length and assume classical physics, with infinitely divisible space.

What, then, am I on about?  Well, over two thousand years ago there was a Greek geometry guy called Euclid who wrote a textbook called 'Euclid's Elements', which was reprinted for so long that there's actually a copy of it in this house.  He came up with a number of postulates, which were supposed to be self-evident axioms on which the whole of geometry could be based.  Most of these seemed fine, such as "a straight line can be drawn between any two points", but the fifth was a bit weird.  It reads (in translation):  

If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles

This really doesn't seem basic or self-evident at all, and there's a reason for this:  it isn't.  In fact, strictly speaking it isn't actually true either, although in most of the circumstances we find ourselves in it may as well be.  Geometry is a branch of mathematics, and maths is not meant to be based on observation but logic.  When it was realised that Euclid's fifth postulate wasn't necessarily true, mathematicians were able to invent different forms of geometry.  The fifth postulate is also known as the parallel postulate, and amounts to the idea that parallel lines never meet.  Rather surprisingly, this is not in fact true.  Real space does not obey this law at all, but it fails on such a vast scale and in such extreme situations that we rarely get to observe it.

There are three big geometries which are well-known.  One is of course Euclidean geometry, which we assume applies to the world as we know it.  The ways in which it's wrong are sufficiently small on the surface of this planet that we can pretend they don't exist.  On a large scale, and in extreme settings such as near a black hole, it becomes clear that Euclidean geometry is wrong, and in fact Riemannian geometry is a better way of describing space.  Parallel lines vary in such a geometry, but on a large scale they do meet.  Riemannian geometry is a superset of various types of geometry, including Euclidean geometry itself.  The crucial thing to remember here is that this kind of geometry depends on observation, not just axioms.  The axioms are there but which axiom applies depends on observation.  Hence geometry becomes an empirical undertaking like physics.

Hyperbolic geometry can be understood using the following metaphor.  Suppose you are in the centre of a sphere five metres in radius.  If you move two and a half metres, you shrink to half your size and the surface of the sphere is now still effectively five metres away.  So you move again, half the distance remaining, and find yourself still five metres from the surface.  This continues forever.  In fact you're always at the centre of the sphere, and its surface is infinitely far away.  Distances shrink in every direction.

This is similar to the situation inside the valve.  Parallel lines get closer as they approach the mouth of the valve.  Within the valve they get steadily closer, and they continue to get closer after they leave the valve, for a while.  All dimensions therefore shrink, and because time is a dimension, so does that.  The line graph describing the convergence starts flat at a distance from the apertures, curves into a diagonal line as the parallel lines are tracked through the bore, then becomes curved once again as it levels out on the other side.  However, this is not quite the same as hyperbolic geometry since the situation is anisotropic:  it has different properties in different directions.  Relative to the centre of the valve, parallels converge in one direction and diverge in the other.  It's also important to note that all parallel lines undergo this as well as the ones passing through the bore.  The ones at right angles to the axis also do, for example.

This anomalous geometry has the merits of being mathematically possible and established by observation.  This is not the same thing as being physically possible, but it means that Martin can carry out experiments which confirm or refute his observation that the valve can shrink or enlarge the objects passing through it.  It does, however, violate the laws of thermodynamics in the divergent direction, since matter appears to be created out of nothing.  It must also change the size and properties of subatomic particles passing through it, and it apparently also changes the speed of light.  There is rather a lot about this object that violates the laws of physics.  However, there's another aspect to the way space works near and within it which may not be as contradictory as it seems.

There are, counterintuitively, objects with only one side and edge, and objects whose inside is also their outside, namely the Möbius Strip and the Klein Bottle.  Although these are potentially regarded as silly, these are their topological properties.  There really can be an object with only one side, although the physical realities of the situation are that the strip is a three-dimensional object with an "edge" which is in fact a surface, so it's more like one of Eschers trefoil knots in some ways:

A topologist once said that there was technically no reason why there couldn't be an arrangement which led to a shape with no edge and no side at all, which would presumably mean the object in question would disappear the moment it was completely in that configuration.  This definitely does not seem possible.  

There is, though, another approach to this in the form of "topological space".  This is space in which closeness is defined but can't be measured.  To clarify this, it's been noted that to a topologist a tyre is the same as a discus with a hole through it.  Alternatively, a ring is the same as a teacup - the only thing that matters is the handle and the actual cup bit is a continuous surface.  Space considered topologically would have distances but the actual idea of something being a metre or a light year away is disregarded.  Instead, its a set of points with an additional structure called a topology, which is a set of neighbourhoods for which closeness is axiomatically defined.  At this point I'm going to have to abandon this explanation in detail because I don't understand it, but there seems to be a way to relate topological space to different kinds of geometry.  

Now to proceed with an outline of the entire story:

Martin Bryce arrives one morning at Mole Valley valves to find that his office is to be halved in size.  He thinks to protest against this to his boss, but can't pluck up the courage to do so.  At the same time, he has to share his PA Mrs Ripper with another employee.  Neither of these things make him particularly happy.  That afternoon, his boss delivers the anomalous valve, explaining that its peculiar properties mean that no employee is particularly qualified or experienced to investigate it.  It acts like a pinch valve but the pressure of the fluid leaving it is decreased and it doesn't have blowback.  His boss asks him if there's anything else he wants to discuss, and Martin is too timid to say he's unhappy with the new arrangements.  His boss leaves.

Alone, Martin peers inside the valve and mutters to himself that the bore seems to comprise a series of ever decreasing circles.  Baffled, he asks Mrs Ripper to make him a cup of coffee, but she can't because she's working for the other guy and he has to make it himself.  He brings it in and starts drinking it, placing the valve on the desk.  He ends up spilling the entire cup into the valve, and notices that most of the coffee seems to have disappeared instead of emerging at the other end.  The coffee remaining is less viscous and flows much faster, and also cools down more quickly than expected.  This intrigues him, so he takes off his wristwatch and drops it through the hole.  This shrinks the wristwatch and causes it to speed up, and he realises he has an anomaly on his hands.

Taking the valve home with him, he goes to his shed and measures it carefully, making a second model of the valve using the lost wax process.  This helps him to measure it more precisely by using the mould as a template.  By this time he's spent so long in the shed that Anne comes in and remonstrates with him.  They spend a relatively normal evening, then after bedtime he sneaks out again, makes a smaller model out of wood and finds it still works.

The next day, rather tired, he returns to work and proceeds to pass the smaller valve through the original, magnifying it in size beyond that of the original.  He then uses that valve to pass the original through it.  The valve was initially about the size of a coffee cup, 8 cm wide, so passing it through in contact (he can't do it from a distance because it grows until it won't fit through) doubles its size.  Doing this five times results in a valve with an aperture 2.56 metres in diameter, which is big enough to walk through and to pass the office through it.  He achieves this by making a polystyrene valve, and also discovers that compressing the whole design horizontally in the axial direction while maintaining its structure produces an equally effective valve.  His work in the afternoon is to make a large polystyrene arch which acts as a valve against one wall and slide it across the office floor, thereby increasing the size of his office.  The incident with the fly then occurs and he has a giant fly corpse to dispose of, which he throws out of the window just as Mrs Ripper comes in.  He then walks around the outside of the arch and asks her to join him.  She is surprised at the size of the office and shrinks to the size of a small child while speeding up, allowing her to fit in more work.  However, she is not actually willing to do more work for him and finds it difficult to perform her duties because the office equipment is now too large for her to operate.  She also doesn't realise she can return to her original size, which causes her to panic.

Martin dismantles the polystyrene arch and takes it home.  Once there, he asks Paul and the Hugheses if he can do an experiment with their drains.  Mystified, they agree.  He assembles the arch through the drains and over the roofs so that the entire width of the close is encircled, and drags the arch across the drains, causing the length of the close to increase slightly relative to anyone walking down it while decreasing its length as measured from the outside.  He then hires a crane and moves the arch from the end of the close to half way up it, increasing its length to 140 metres.  This requires dragging it through everyone's gardens.  After doing this eight times, he has managed to reduce the external length of the end of the close to about the size of a sheet of A4 paper.

Then he gets his spirit duplicator and rigs up the valve to it, fits the end of the Close into it and turns the handle once.  This copies the valve, the end of the Close and the duplicator, and since the handle of the duplicator is connected to a smaller model within it to turn it every time he turns the full-size one, this creates an infinitely long Close within a few seconds.  He has also created an infinite number of Harolds, Hildas, Annes and the various other residents of the Close, such as Tommy Cooper.  Time is also passing at infinite speed at the end of the close.  This means that the Harolds and Hildas beyond a certain point wish to enter old people's homes but can't reach the start of the Close, and end up setting up their own old people's homes within the Close.

Paul returns, having been busy at the hair salon.  Martin tells him what he's done and Paul casually mentions that he has a mate in higher dimensions who has been helping him set up an infinite hair salon and pet grooming service in hyperspace which can serve all beings throughout space and time.  Infuriated at Paul upstaging him, Martin takes up residence at the house at the end of the infinite cul de sac and declares himself emperor of the Close.  He then finds that Paul has set up an infinite number of hair salons all along the Close. The interdimensional police then turn up and arrest him for living too far up the road and owing an infinite amount of council tax.  Paul pays his fine with his infinite income from the salon chain and bails him out.  Martin comes to an arrangement with the council to brick up the end of the Close and live in normal space, but there is a gate to let through the ten thousand Harolds and Hildas who want to enter the old people's homes on their subjective dates who occasionally arrive to do so.  Hence Paul still has the upper hand, particularly as he is declared a god by the residents of the Close.

This is a seven paragraph explanation which misses out various details, such as the fact that the duplicator was imperfect and creates peculiar copies of the houses and individuals, the passage of time leading to evolution of organisms, including humans and Martin channelling his spiritual predecessor Tom to encourage the Close to become self-sufficient due to the lack of supplies from the outside world.  The possibilities of the infinitely long Close are also ripe for exploration:  perhaps there is sufficient variation within it for events similar to 'To The Manor Born' and 'Yes Minister' to transpire as well, though in a different environment.  Each of these paragraphs would be less than five minutes of action, which is really cutting things fine.  However, there is a way around this in keeping with the theme.  Suppose the whole episode is ninety minutes long.  The first forty-five minutes can run at normal speed, the next 22½ minutes at double speed and so on.  That way, the seven sections can fit into a total of almost eighty-eight minutes and eighteen seconds, with the last section lasting forty-two seconds, and there is naturally room for an infinite number of further sections after that.  Expanding this to a sensible length would provide enough for an entire series.

I'm not sure what to do with this.  I think it has possibilities but it needs to be compressed somehow.