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TilingStuff
(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 ifandonlyif condition for whether W
can be Ztiling by rectangles of specified eccentricities.
It appeared in the
Jour. of Combinatorial Theory, Series A
85, (1999) 8391.

8 April 1996
How does one pack rectangles into rectangles? more generally, Ddimensional 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 114 are in
Part ONE, while
pages 1526 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 changeofcoordinates 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 AizuWakamatsu City, Japan, during March 2428 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 integersided
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 brickset. My
article gives an algorithm to detect when a brickset is deBruijn. The
algorithm runs in
time big O
of DN^{3}.
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
"halfstep magnified version"
(width 7 inches)
uses halfstep magnified fonts, which is easier on the eyes than the narrow
version. However, if your system doesn't have the halfstep fonts, then it
will have to generate them, which usually takes several minutes.
WebResources for Tiling
Computing information about tilings led to calculating sequences of integers.
Looking up an integer sequence was made easy by
the email request mechanism, provided by Neil Sloane,
which searches his
Handbook of Integer Sequences.
He has also provided a
web site for looking up sequences.
You can automatically search the
XXX Mathematics Archive, maintained
at Los Alamos, for recent papers on
Tilings
and
Tessellations.
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