PERIODIC
TILINGS AND TILINGS
BY REGULAR
POLYGONS
by
DARRAH PERRY CRAVEY
A thesis submitted
in partia1
requirements
fu1fi11ment
for the degree of
Doctor of Philosophy
(Mathematics)
at the
UNIVERSITY
OF WISCONSIN
1984
- MADISON
of the
© Copyright
by Darrah
Perry Chavey 1984
All Rights Reserved
PERIODIC TILINGS AND TILINGS BY REGULAR POLYGONS
Darrah Perry Chavey
Under the supervision of Professor Donald W. Crowe
Abstract:
We assume a tiling has, under its symmetry group, v orbits
of vertices; e orbits of edges; and t orbits of tiles.
Inequalities
are established relating these parameters, both for arbitrary tilings
and for tilings by regular polygons, and we show that some of these
inequalities are sharp.
In the case of tilings by regular pOlygons,
we classify those tilings with v ~ 3, e ~ 3, or t~
2, and show that
the number of tilings with some fixed number of orbits of vertices
[or edges; or tiles] is finite.
The edge figures which can occur in
a tiling by regular polygons are Classified, as are tilings which
contain at most three different types of these edge figures.
Progress is made towards classifying those tilings by regular
polygons which contain at most two different types of vertex figures.
with respect to tilings by regular polygons which contain only two
types of tiles (two congruence classes of polygons),
possible orbits of each polygon is determined.
the number of
Tilings by regular
polygons in which any two congruent tiles are equivalent under the
symmetries of the tiling are classified, as are tilings which satisfy
a similar condition on the edges.
ii
"We're all in it - we're all tiled, here."
Olga •.
The Grand Duke,
by Gilbert and Sullivan.
"He's got 'em on the list - he's got 'em on the list;
And they'll none of them be missed - they'll none of them
be missed."
Chorus of Men,
The Mikado,
by Gilbert
Dedicated
and Sullivan.
to the two women I love
Peggy and Eunice Chavey.
iii
Acknowledgements
Now that it's almost over, it seems amazing
friends and my thesis committee
managed
to me that my
(which are not exclusive)
to put up with me for the last month or so.
the many people
I wish to thank for helping
have
They are among
to make this thesis
possibl.e.
None of this work woul.d have been possible
survey of the subject by GrUnbaum
them for making
their advance
Donald Crowe and Michael
creating
and sustaining
this seminar
and Shephard,
without
and I wish to thank
copy avail.abl.eto us.
Bl.eicher deserve
thanks
a seminar covering
Professors
for their efforts
that most of these results developed.
are difficult
to pin down, to conversations
Mary Leland discovered
proof of theorem 2.3 (as mentioned
extend
parameters
in tilings.
improvement
out by John Rosenberg.
vol.unteered to draw most of the til.ings in figures
Carnegie-Mellon
aspects
laser printer,
to my
there), and this class helps to
1.3 is a drastic
nal, and this proof was pointed
attributed
one class of tilirlgs used in the
the known range of realizable
proof of fact 1 in section
in ways that
with Don Crowe and Mike
but some of the work can be more directly
colleagues.
in
this work, and it was from
Much of the work in this thesis owes a great deal,
Bleicher;
the excellent
and these figures
of the thesis) would have been impossible
The nice
of my origiElsa Gunter
5.2 - 5.5 on a
(one of prettier
without
her help.
188
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