T.H.Levering's result (J.of M.G.S.,1994,vi,344) regarding n-dimensional manifolds in game space has some interesting implications for the modern gameof Mornington Crescent. A team at the University of Gornal, led by Dr K. Latimer, reports on the current state of play.
This model is based on a tentative system put forward by Haringey and others, regarding MC game space as a four-dimensional system with two of the parameters indicating the spatial position of the station in question. However, there are limitations in this approach, not least the way it predicts situations where the only possible move does not correspond to any station. This is contrary to experience, as with the amount of MC being played in the world today one might expect such situations to occur with regularity if they were as common as Haringey envisaged.
In this paper, we hope to demonstrate that a more effective rendering of normal MC game space may be visualised in the form of a twisted 9-manifold (Z-space) existing in a space of 11 dimensions ("-space). Dimensions are quantized and bounded, leading to a finite number of possible moves. At present we have no model for manoeuvres such as the Knoydart Tangle which appears to alter the space by 'kicking' it into a new configuration in which 'abnormal' moves are possible, however preliminary calculations do suggest that any game space produced in this way must pass through MC itself.
We shall call the 9 qualities corresponding to dimensions inside Z-space:
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c = colour,
a = aroma,
p = phase,
g = gradient,
d = delta-iridescence,
p = pi-iridescence,
i = intractability,
l = confoundedness,
Rmx(v).
- Note 1: These are the results of a small competition held between members of the Department here in Gornal, for which the prize was a king size Mars bar. Thanks to Tim, Matt, Sara, Russell, Tim, Tim, Helen, Laura and Tim, and congratulations to me.
- Note 2: "-space will not be considered here. Suffice it to say that it is necessary for Levering's result to hold in this case; we suggest the two extra variables be named x and x to save confusion.
Full details of the proofs of the following statements can be found at our Web site http://www.gornal.ac.uk/maths/~gtmc/zspace.html#proofs, as well as by post or telephone.
Our results are:
1) that in Z-space a move may be considered legal if it satisfies the relation
d/dg ( a2-c2i+pd ) + d2/dg2 (p+d) . X = d/di ( p2l2-2p0.433ei^2 ) . Q
where Q denotes the integral determinant of the matrix
( acgl cgla glac lacg ) ( pipd ipdp pdpi dpip ) ( c.Rmx(v) a.Rmx(v) p.Rmx(v) p.Rmx(v) ) ( g.Rmx(v) i.Rmx(v) l.Rmx(v) d.Rmx(v) )
and X denotes the Q-values of the previous 15 moves, 'blobbed' together in the method devised by Albers,
2) that Knip may readily be spotted, as the X and Q values have a common factor of cpil/Rmx(v).
3) that there exists a similar way for determining the existence of Spoon and Bar which we have not bothered to work out,
4) that all points with a confoundedness value greater than or equal to 4.522253750 (Lhatsang's Constant) must lie on the DLR,
5) that the most confounded station in the game of Mornington Crescent is Mudchute (Poplar being a close second),
6) that, if a move D is possible, there exists a move 1/D which performs the inverse function in the reverse situation provided no Suspect Packages exist within the Circle Line (possible use for this proof in Reverse Crescent?),
7) that if the matrix above is symmetric Mornington Crescent is a legal move,
8) that Lhatsang's Constant also gives the distance of Mornington Crescent from the centre of Z-space in arbitrary units,
9) that the station nearest the centre itself is Dollis Hill. (A possible explanation for its unparalleled gravitic effects?)
We look forward to hearing from any other groups which have derived interesting or useful results from the use of Z-space.
References:
1. Oh my, a proof!: A surprising proof of the continuity of topological tetheredness in (n+2)-dimensional game theory, T.H.Levering et al [J. of M.G.S.,1994,vi,344]
2. 'Mr Greedy', John Hargreaves, Thurman Publishing
3. A new prescriptive coordinate system for Mornington Crescent, D.W.Haringey [J. of M.G.S., 1991, ix,597]
4. Matrix blobbing (Matrischer Verblobbieren, Q.S.Albers [Zeitung der Mornington-Crescenter Spielstheorie, 1987, i, 34]
Dr K. Latimer (Game Theory and Mornington Crescent Group) Rm 102a/b (in the corner behind the pot plant) Dept. of Mathematics University of Gornal Sedgeley West Midlands B74 2GP (0121) 192-6174 (ext. 3.14159)
(reproduced in full on the Delphi-sponsored "Mornington Crescent at Camden Lock" server, and re-reproduced here in the Encyclopaedia. Original poster unknown.)