Empire of Fashion

[Pleater]

We prepared to wait on the top of the row of offices, on the rough wooden beams suporting the ceilings. We took boxes, flattened them, and made several interlocking layers over the area we would be occupying, we put several stacks of boxes around the space, to make it look as though it was used for storage, and then built ourselves a hide.

We prepared for a long stay, just in case, and at the appointed time we closed our office, signed out and then the two of us slipped back inside when one of the senior Blue Meanies came to chivvy his minions.

At noon we heard the announcement that the new colour clothes were to be worn, and we were rather glad, as it was very hot up on the roof and we were grimy and dripping with perspiration. Luckily we had brought large amounts of water for drinking and even made provision for washing, or at least wiping ourselves, but crawling around, or any movement had to be kept to a minimum.

About one o'clock, to judge by the shadows - we had no watches and the office clock was below us and so invisible, there was movement in the hall, and a group of the Hall's Golden came in. They came to the locked store, which was behind the clock, and began to bring out various items. We could hear them being put down on the walkway outside. They were talking - one of the younger ones was quibbling about not being allowed to operate the equipment, and the older ones began to shout questions at him to try to trip him up.

After a few minutes I realised that they were questioning him about the cross ratio of points in the complex plane, Apollinius circle formulae, conic sections and similar mathematical concepts.

They did not catch him out, but their chief employed the argument used for generations in similar circumstances. 'If you think I'm trusting my pension to you, you've got the wrong idea about who is in charge here, son.'

They set up the equipment, a tripod, a rod, a thing like an astrolabe combined with what looked like handlebars, just far enough from the front of the row of offices for us to see.

They took another rod and carried it to the end of the hall near the large doors, and replaced an identical one which lay in a compartment in the floor, which they brought back to the store. Before putting it away they unscrewed one end and carefully removed a series of disks, which they carefully packeged - except the young man who had been refused the chance to show off his abilities took a couple of them and clashed them together so they sparked. They were huge coin cell batteries.

The young man was sent away, and the remaining Golden got ready for an arrival. First more Golden came in to witness the event, then the vertical ring was created amd moved to the door end of the hall, and finally there was the sound of engines, the ring moved back up the hall to reveal a hovercraft, painted in gold and turquoise, which settled on the ground and the front was opened up.

From the ring there emerged a ramp, and there was shouting and the sound of marching feet. The entrance into this world was not as impressive as it might have been, for as the men passed through the ring bringing heavy trolleys on thick pneumatic tyres, they were flung forward so they staggered and stumbled into the hall, and one trolley got away from its minders and ran over a couple of men before crashing into the trolley in front. The flow of men stopped, and those already here began to try to sort themseves out and tend the injured.

The newly arrived Golden left the hovercraft and swept up the hall - she wore high boots of glossy turquoise, and over a glittery figure hugging little dress, a long golden cloak which fluttered wonderfully. I tensed as I recognised her - Ivana Rulitall.

I glanced sideways at the Captain, and from her expression she also knew the woman. If looks could have killed we would have seen Ivana vaporise with a hideous shriek, but she advanced steadily, with almost all eyes upon her.

Ivana tore into the equipment operators, demanding that they correct the problem. One of them dared to try to explain, but she was not listening, she turned and stalked off towards the higher ranking Golden, who quailed before her.

I wouldn't have held out much hope for his pension.

Eventually she returned to the hovercraft, the top was closed over the cab and the engine started, and she left the building.

I listened carefully to the men below us as they told eachother why they could not match the relative velocities of the two realities, and then we retreated to the back of our little hide to drink a little, try to cool ourselves down and wait for the coast to be clear for our exit.

[Arlen]

Oooooooh. This is getting very exciting indeed!

[Pleater]

Alas the excitement of the next day or so passed me by, for my thoughts were all on the mathematics of the controlling device for the rings.

It is all in Wikipedia, if you need to look it up, but it is really quite simple, if you start with the definition of a circle used by Apollonius of Perga, as a set of points having a constant ratio to two fixed loci.

If you then consider the six possibilities of cross-ratio - the lambda, one over lambda, one over one minus lambda and so on, there are values of lambda which have enhanced symmetry, corresponding to fixed points of the action of S3 on the Riemann Sphere, or equivalently those points with a non trivial stabiliser in this permutation group.

The second set of fixed points, that is minus one, one half and two represent a harmonic/symmetric cross ratio where lambda equals e to the power plus or minus i times one third Pi.

Of course that is visualised as a projection of the Argand plane by stereographic projection, it is the complex number plane wrapped around a sphere, with the addition of a point at infinity - you see - it extends the plane of complex numbers.

Expanding Apollonius' definition of a circle into three dimensions - you only need the two loci A and B, the point P becomes a ring at right angles to the Apollonius circle - or rather it becomes a circle on the surface of a sphere, and if A is the controller and B is the rod at the other end of the hall, then between them you have the point C and altering the angle APB so that the ring at P and C converge or diverge you move the ring perpendicularly over the surface of a Riemann sphere.

However - the rings sould be blue.

It made no sense for the rings to be turquoise. I could see that by moving the values used for lambda just slightly off the harmonic it would put strain on the portal and push the spectrum towards the green - but surely no one would be so silly as to twist the fabric of the multiverse and risk a rupture in reality just to make the ring conform to the colour of a company logo.

Surely not?

[Livingston]

OK, way too early in the day to read that. I'll have to come back this evening after a wee dram or 30 and see if I can make sense out of it then.

[Bugbear]

No... That makes sense. I would have thought the sphere to be mobius in this case, but blue sounds right.

[Pleater]

No... That makes sense. I would have thought the sphear to be mobius in this case, but blue sounds right.

It is mobius, but only when you press the red button.

Glad someone is paying attention.

[Standard]

So is that pie are square, or pie are round to make turquoise sphere?

[Bugbear]

So is that pie are square, or pie are round to make turquoise sphere?

Shhh. We're dealing with elliptic curves and Fermat's Last Fashion Theorem. At least until the red button is pushed...

[Pleater]

Ah now I thought ellipses - definitely ellipses, I thought, but then there is the possibility of getting an astroid or hypocycloid.

That would never do, and it is so easy with the requirement being just

x to the power 2/3 plus y to the power 2/3 equal to 1

- and then you'd have ten singularities, four in the real plane, two at infinity and four complex double points - you could end up anywhere - or perhaps even worse, everywhere.

So you see, it has to be a circle.

[Bugbear]

My apologies. I used to proofread mathematical stone arangements in the last time/world, but it's been so long ago...

I'll be standing over there in the shadows for the rest of this story.