From: hoey@AIC.NRL.NAVY.MIL
Newsgroups: sci.math
Subject: Re: Tiling problem, polyomino classification
Date: 10 Apr 92 17:06:21 GMT
Organization: Naval Research Laboratory, Washington, DC

dwilson@cvbnet.prime.com (David Wilson) asks:

>    Will the pentomino

>        **
>         **
>         *

>    tile the plane minus a quadrant?


Yes, as follows:
                                                               .
                                                               .
                       .    .    .    .                        .
                       .    .    .    .                       a
                       .    .    .    .                      aaa b
                     11111222221111122222       regions 1,2: acbbb
                     11111222221111122222                    cccbd
                     11111222221111122222                    caddd
                     11111222221111122222                    aaadb
                     11111222221111122222 ...                acbbb
                     11111222221111122222                    cccbd
                     1x1112x2221x1112x222                    c ddd
                     xxx1yxxx2yxxx1yxxx2y  ...                  d
                     zzxyyy5xyyy6xyyy5xyyy6
    3333333333333333zz77y5555y6666y5555y666
    33333333333333333z77755555666665555566.
 ...3333333333333333377777555556666655555  .             i
    333333333333333333777775555566666555    .          jiii
    33333333333333333zz7777755555666665 .              jjjil
    4444444444444444zz8877777555556666   .              jklll
    44444444444444444z888777775555566 .   .              kkkli
 ...44444444444444444888887777755555   .                  kjiii
    4444444444444444448888877777555     .                  jjjil
    44444444444444444zz88888777775 .                        jklll
    3333333333333333zz77888887777   .                        kkkl
    33333333333333333z7778888877.    .       regions 5,6:     k .
 ...333333333333333337777788888  .                               .
    33333333333333333377777888    .                               .
    33333333333333333zz777778 .
    4444444444444444zz887777   .
    44444444444444444z88877.    .                       mm  
 ...4444444444444444488888  .                           nmm
    444444444444444444888    .                         nnmoo
    44444444444444444  8 .                              nnpoo
           .         .    .            regions 7,8:      ppomm
           .         .     .                              ppnmm
           .         .                                     nnmoo
                                                            nnpoo
                                                             ppo
regions 3,4:    eeffeeff                                      pp.
                 eeffeeff                                        .
            ...  egfhegfh                                         .
                  gghhgghh
                 gghhgghh

In a later message,

>     I was wondering about the ability to tile certain figures with
>     polyominoes.  The figures I am interested in, with cartesian
>     descriptions, are:

>     rectangle (r)		[0, a] X [0, b]
>     halfstrip (hs)		R+ X [0, b]
>     strip (s)			R X [0, b]
>     angled strip (as)		([0, a] X R+) U (R+ X [0, b])
>     quadrant (q)		R+ X R+
>     halfplane (hp)		R X R+
>     three-quarter plane (tp)	(tp)(R X R+) U (R+ X R)
>     plane (p)			R X R

All of these except the plane have versions with diagonal (stairstep)
edges instead of orthogonal edges.

For the r-pentomino, we already know how to do the strip. I have found
out how to tile the diagonal quadrant, as well, so the diagonal
halfplane and diagonal three-quarter plane are corollaries.  I don't
know about the diagonal rectangle, diagonal halfstrip, or diagonal
angled strip.

Dan Hoey
Hoey@AIC.NRL.Navy.Mil
Xref: sparky sci.math:5747 sci.math.num-analysis:1406