Tiling Stuff of J.L.F. King

All files are compressed and, unless otherwise mentioned, are in postscript. More recent articles are at the end.

In Shape Tiling (6 pages), Kevin Keating and I examined tiling rectangles with other rectangles. It appeared in the Electronic Journal of Combinatorics, in the special 1997 issue in honor of Prof. Herb Wilf, and this version is available in several formats.
3 Nov 1997 Continuing from Shape Tiling, Kevin and I again look at rectangular tilings in Signed Tilings with Squares (8 pages), this time using a tensor product to studying tilings of weighted rectangular regions, W, in the plane. We obtain an if-and-only-if condition for whether W can be Z-tiling by rectangles of specified eccentricities.
It appeared in the Jour. of Combinatorial Theory, Series A 85, (1999) 83-91.
8 April 1996 How does one pack rectangles into rectangles? -more generally, D-dimensional bricks into other bricks? It turns out that there is a mysterious connection with the Dedekind sequence, which is proved -but not understood- in Brick Tiling and Monotone Boolean Functions. Pages 1-14 are in Part ONE, while pages 15-26 have been split off for Part TWO.
Feb 1996 (Updated 23Jun1998) The brick tilings led to a curious connection between tiling and the Frobenius number of a collection of integers (4 pages).
19 Mar 1997 The material on brick packings I have split off (from the …Boolean Functions paper) into Brick Packings and Splittablity (6 pages).
07 May 1998 In late 1996, Hugh Redelmeier wrote a computer program which computed more of the maxrank numbers, which are the basis of the Polynomial Conjecture in …Monotone Boolean Functions paper. I found a proof of the conjectures in early 1997. Currently, I am writing up this proof.
08 June 1998: (Updated 23Jun1998)     A change-of-coordinates from Geometry to Algebra, applied to Brick Tilings (18 pages) will appear in the Proceedings from the First International Conference on Semigroups & Algebraic Engineering, held in Aizu-Wakamatsu City, Japan, during March 24-28 of 1997.
    This is a preparatory paper for proving the Polynomial Conjecture of …Monotone Boolean Functions (on Tilings Page), and describes the transition from the Geometry to the Lattice Theory in a series of steps. The technical proofs of these steps will appear elsewhere.
    The XXX Math Archives has a version which is available in PS, PDF, DVI and other formats.
24 June 1998     deBruijn's Harmonic Brick Condition is computable (6 pages), will appear in the Electronic Journal of Combinatorics. The article has two figures. Figure E4 is also available separately as a jpeg file.
    The article shows the following: Given N integer-sided bricks, each of dimension D (the protobricks), when is it the case that each box which is packable by translates of copies of protobricks, in fact can be parallel packed by copies of a single protobrick?
    Protobricks for which the answer is yes I call a deBruijn brick-set. My article gives an algorithm to detect when a brick-set is deBruijn. The algorithm runs in time big O of DN³. The article ends with two open questions.

    The above version has pagewidth=7.67inches. Also available is a narrower version (width 6.665 inches). A "half-step magnified version" (width 7 inches) uses half-step magnified fonts, which is easier on the eyes than the narrow version. However, if your system doesn't have the half-step fonts, then it will have to generate them, which usually takes several minutes.

TilingStuff (J.L.F. King)

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