Mathematics and Number Wisdom

I've at last got round to updating the mathematics section of my website. The "Rational Mathematics", "Alternative Mathematics" and "Geometry" sections are now all under one index page. I've also transferred the pages on "Altairian Arithmetic", "Numerology" and "Numeromancy" from Esoterica to the Alternative Mathematics section, together with a new page on "Arithmosophy".

I'm not sure that Numeromancy and Arithmosophy can really be called Mathematics but they use numbers and require a knowledge of some arithmetic. I base my interpretation of "Wisdom of Numbers" on multiplicative relationships rather than additive properties. This makes it a more disciplined realm.

I have masses of notes on Geometry, but lack of satisfactory software for producing diagrams has held it back. There is nothing particularly new, just improved arrangement and presentation, making use of CSS.

Mathematical Art: A Chessboard Mosaic

Over the bank holiday weekend I spent some time drawing and colouring the pattern shown here. It is a "Chessboard Mosaic" of the type I described in The Games and Puzzles Journal (Vol.1 No.4, March-April 1988, p.64) but on a larger scale. Varied patterns of this type can be formed by first numbering the cells of a chessboard 1 to 64 in some fashion. This example is derived from the numbering of the first Magic Knight's Tour, discovered by William Beverley in 1848. It is a 64 by 64 matrix; when numbered 1 to 64 along the top and left edges, a mark in the square where the r row meets the s column indicates that a piece can move from cell r to cell s. In this example the dark cells indicate rook moves, the yellow cells bishop moves and the red cells knight moves. The pattern is symmetric about the principal diagonal since these moves are all reversible. The rook move pattern is symmetric about the secondary diagonal, but the knight and bishop patterns deviate slightly from this symmetry, due to the nature of the Beverley numbering. The red railway-line pattern down the main diagonal is the result of using a knight's tour numbering, since r is always connected to r+1 by a knight move. The apparent figure "8"s on the main diagonal derive from several zigzag pattern of knight moves in the tour wheren the first and third and second and fourth cells are in the same rank or file and thus connected by a rook move. I'm wondering whether I should exhibit this at the Art Forum; apparently any member can exhibit one item free; but is it really Art?

Edit: I've replaced the original image by an enhanced version, since it came out too gray. The background I used was in fact a sheet of white card.

Euler and Me

As noted before I'm in process of tidying up my notes on knight's tours. One of the sections I've not previously published, as far as I recall, concerns the enumeration of all the smallest knight-tourable boards, at least up to 12 cells. The illustration shows all the centrosymmetric tours on boards of 12 cells.

The first two of these diagrams are not merely centrosymmetric, in the sense of being unchanged by a 180 degree rotation, but are also axially symmetric about the two diagonal axes. The first of these was published by the famous mathematician Leonhard Euler in what waa probably the first scientific paper on knight's tours, presented in 1759. The second is one of my own favourite discoveries, it is the 12-cell tour whose containing rectangle is the largest possible, the 6 by 6 square.