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Chapter 6: Summary and Suggestions for Further Research.

Most theses seem to contain a certain amount of solid research,

and some filler. This thesis is no exception. The additional

material is intended to fill some of the corners and niches that

arise fairly naturally in studying these tilings, but these results

are definately of less overall importance than some of the other

results. My opinion is that there are three main, important results

in this thesis:

1) The bounds on v, e, and t as functions of each other; both for

general periodic tilings and for tilings by regular polygons;

2) The classification results in chapter 5; and

3) Theorem 4.8, partly due to its classification of e-isotoxal

tilings, but especially due to the unexpecJed strength of its

connection between local and global regularity conditions on the

I

edges.

In addition to these specific results, there are 3 techniques which I

have used (for the first time, as far as I knOj) which appear to be

of significant further interest:

I) The systematic use of the ideas of connected representative sets

of elements in establishing other results (e.g. #1 above);

II) The use of edge types, simple edge types, and (especially) non-

monogonal edge types as a principal step towards the various

classification results;

III) The algorithm implied in the finiteness arguments of section

177

3.2, and the idea of a closure of a fundamental region, as a

possible computer tool for further classification problems.

The ideas of fusing and dissecting hexagons and 12-gons as a

teChnique for studying certain kinds of tilings was used by

Sommerville [1905] and is extended here, especially in the proof of

lemma 4.3. This technique is very useful, and it would seem that it

has potential for further research -- especially if generalized

appropriately. To demonstrate its strength, we note that lemma 4.5

(b) is easily established in a paragraph using this approach.

Without this approach, Levy [1891] investigated this problem and

could not discover the solution. A few years later, still unsuccess-

ful at the solution, he submitted this as a research question (Levy

[1894]).

These are the highlights of the thesis. The rest of this

chapter gives a more detailed survey of the thesis results and their

importance, together with a collection of various unanswered

questions which arise from this research and se'em to merit further

study. The survey is arranged in approximately the same order as the

thesis, and the open questions are numbered seJuentiallY.

Chapter 1:

The facts and lemmata contained here are straight-forward. The

counter-examples to the claims of Grunbaum and IShephard [1983] are at

least as interesting as the pDoofs that these facts hold under the

assumption that the tiling has no singUlar points. My primary

178

interest, and that of other researchers in geometry, is in ti1ings

without singu1ar points. Whi1e lemma 1.1 [connectedness' of graphs

and dual graphs in the absence of singular points] is thus of

interest, lemma 1.2 [on connectedness of dual graphs] is included

only because its statement and proof are quite simple. From this

perspective a comparable result for the connectedness of graphs seems

of less interest, since the examples imply that necessary and suf-

ficient conditions cannot be particularly simple.

Lemma 1.1 [connectedness of graphs and dual graphs in the

absence of singular pOints] seems nearly self-evident, and yet is

more powerfu1 than one might suspect. In particular, it gives rise

to lemma 2.1 [existence of connected, representative sets]. While

1emma 2.1 itself may seem straight-forward, Krotenheerdt [1969] takes

two journal pages to prove the equivalent of part 1 for the very

special case of vertex-homogeneous tilings by regular polygons. Our

proof of part 1, using lemma 1.1, requires only 8 lines.

Theorem 1.3 [equiva1ence of periodicity with the finiteness of

v, e, or t] seems to be of independent interest, but I have found it

quite frustrating not to be able to decide if the assumption of no

vertices of infinite valence is necessary. This raises the first

question I would like to see settled:

1) Are there any tilings with singular points which have only a

finite number of vertex orbits?

It seems unlike1y that such ti.1ings exist.

179

Chapter 2:

Lemma 2.1 [existence of connected, representative sets] is quite

important, as mentioned earlier in connection with lemma 1.1. This

idea is alse, fundamental to many of the classification results.

Nevertheless, it is not a surprising result. Lemma 2.2 [the

comparable situation for locally regular tilings] is one of those

corners I felt was worth mentioning -- primarily because of the

curious gap in part 3. Thus, even though it may not be of great

import to further research in the area, I would like to know:

2) If a tiling T is 3-toxal, does it necessarily have sets of 3

edges which induce connected graphs (dual graphs)? What if T is

4-toxal?

I suspect that the answer is yes for 3-toxal tilings. Another

question mentioned in chapter 2 also points out some of the

differences between e-toxality and the other local regularity

conditions, namely:

3) Does there exist a monotoxal non-isotoxal tiling?

Theorem 2.3 [bounds on v, e, and t as functions of each other]

is the first major result in the thesis, and this has already been

published in Chavey [1984]. As remarked in section 1.2, GrUnbaum and

Shephard [1983] ask if there are triples (v, e, t) that are not

realizable; and this theorem answers that question with a vengeance.

Nevertheless, many open questions remain on the road to a fuller

classification of the realizable triples. Specifically:

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4} Is it possible to bound the value of v or e in a periodic tiling

as a function of t?

5} Establish a lower bound for Eu(T}, where Eu(T} = v - e + t. In

particu~ar, is Eu(T) ~ O?

6}Show that wider ranges of parameters are realizabl.e than those

given in theorem 2.3, (4) and (5).

The examples of (k+1, k+m, k+m+1}-tilings with m > 0 (used in

the proof of theorem 2.3, parts 3 and 4) all used tilings in which

the intersection of some pair of til.es was disconnected. If we

restrict our attention to til.ings in which this does not happen

(which Grunbaum and Shephard [1983] cal.l.normal. til.ings), it may be

possible to improve some of these bounds. In particular:

7) Are there bounds on e or t as a function of v for the class of

normal tilings?

Chapter 3:

The classification of edge types (theorem 3.1) seems like the

sort of thing that should be in print somewhere, and this

classification becomes very useful for many of the later results.

Neverthel.ess, the main contribution of section 3.1 is probabl.y the

introduction of the ideas of edge types and simple edge types. These

ideas are extremely useful in much of the later work. As mentioned

before, the classification of the non-monogonal edge types is of

great use in later classifications. The definition of "edge figure"

in chapter 1 appears to be ne~, and this is possibl.y due to an uncer-

181

1

tainty as to what the correct definition should be. There are two

reasonable choices: 1) the edge and all incident edges (our choice)

which is equivalent to the edge and both incident vertex figures; or

2) the edge and both incident tiles. In the case of tilings by regu-

lar pOlygonn, the first definition corresponds to the (full) edge

type while the second definiton corresponds to the simple edge type.

As we have seen. both of these ideas are quite fruitful in investiga-

tions of these tilings. The homogeneity condition for these two

definitions would correspond to what we have called edge-homogeneous

and strongly edge-homogeneous. In the case of tillngs by regular

pOlygons, the first-definition always gives more information than the

second. but this is not true for more general tilings (hence the term

"strongly edge-homogeneous" is probably inappropriate for more

general tilings). Although we will discuss chapter 4 later, it is

worth noting here that, with respect to the alternate definition of

an edge figure as the edge and the two tiles incident to it (and

'hence with an alternate definition of e-toxal tilings), the proof of

theorem 4.8 shows that all monotoxal tilings are isotoxal. This

proof also shows that the only 2-toxal tilings (under this alternate

definition) are the 2-isotoxal tilings and the following two classes:

A) Tilings by triangles and hexagons with no adjacent hexagons; and

B) The 2-gonal (36; 32.4.3.4) tilings (which are not yet classified).

Thus, although the general question of the classification of 2-gonal

tilings is listed later, this special case has independent interest:

182

8) Classify the 2-gonal (36; 32.4.3.4) tilings.

Equiva1ent1y, by fusing the vertices of type 36 into hexagons:

8') C1assify the 2-gona1 (32.4.3.4~ 3.4.6.4) ti1ings.

The finiteness resu1ts of section 3.2 are of reasonab1e

interest, a1though not particu1ar1y surprising. As discussed in

section 1.2, these resu1ts are closely related to those of

Krotenheerdt [1969] (also, see the discussion of 1emma 1.1 above).

These results answer two questions of GrUnbaum and Shephard [1983]

(see section 1.2). One obvious further prob1em might be:

9) Calculate reasonable upper bounds on the number of v-isogonal

[e-isotoxali t-isohedral] tilings.

The basic technique of theorem 3.3 (with certain straight-forward

improvements) yie1ds the ridiCUlous upper bound of at most

(12v)4v+1.14v. 4; (4v)! v-isogonal tilings. This, for example,

shows that there are at most about 7.7 x 106 isogonal tilings (there

We can also show that there are at least l. 2v-1vare actually 11).

v-isogonal tilings. Improvements in these bounds should not be

difficult. Nevertheless. because of the vast disparity between these

bounds, it would seem that improvements in either direction would

have to be fairly extreme to be of significant interest.

Lemma 3.2 [uniqueness of tilings with a given fundamental region

and labeled closure] raises some questions that are probably less

significant than many others listed here, but which I would still be

interested in:

183

10) Can the word "labeled" be dropped from the statement of l"emma

3.2? If not, would it be sufficient just to label the vertices

in R according to their vertex orbits?

TO a J.arge ext.en t, lemma 3.2 (and the answer to # 10) telJ.s us how

much of a tilling must be drawn in order to guarantee that there is at

most one ext.ensf.on to a tiling of the plane. An obvious related

question is:

11) How much of a tiling must be drawn in order to guarantee that

there is at least one extension to a tiling of the plane?

Since there are many non-monogonal edge figures by regular polygons

which cannot be embedded in 2-isogonal tilings (see section 4.1), it

is generally necessary to draw more than just a fundamental region.

The obvious candidate for an answer to question 11 is the closure of

a fundamental. region.

The final. question that arises from lemma 3.2 is the possibility

of using the techniques of the proof to develop an al.gorithm for a

computer search of certain classes of tilings. There seems to be no

theoretical difficulty with the computer successively generating all

possibl.e centers; fundamental. regions; and l.abel.edcl.osures -- and

then drawing the resulting tiling. The researcher would probably

have to eliminate duplicates by hand, and would either need an answer

to problem #11 or else verify by hand that the computer's tilings can

l be extended to tiJ.ings of the plane.I

Although lemma 3.2 generaltes several questions, the major result

l

184

l

in chapter 3 is probably the improvement of the bounds of theorem 2.3

for the case of ti~ings by regu~ar po~ygons. This resu~t, theorem

3.6, also answers two questions of Grunbaum and Shephard [1977a] and,

together with corollary 3.7 and theorem 3.8, has been published in

Chavey [1984]. Such bounds automatica~~y raise the question of

possible improvements, and we phrase this as two prob1ems to

emphasize a distinction:

12) Improve the bounds of theorem 3.6 (2) and (3).

13) Determine whether or not the bounds of theorem 3.6 (1) can be

improved.

While it seems plausib~e that the bounds of (1) might a~ready be

sharp, it seems inconceivable that this is true for (2) or (3). In

the other direction, one could look for tilings which pull the

equalities of theorem 3.8 closer to the bounds of theorem 3.6. Thus:

14) Can the equalities of theorem 3.8 be improved so as to widen the

known range of realizable triples (v, e. t) for tilings by

regular polygons?

In connection with this problem. it might be worth investigating

tilings with 12-gons (none of the classes used in theorem 3.8 contain

12-gons).

Corollary 3.7 [classification of 2-isotoxal tilings] is

superseded by theorem 4.8; but it is still quite pleasant to be able

to solve an open problem of Grunbaum and Shephard [1977a] in two

sentences.

185

Chapter 4:

Certainly the most interesting result in chapter 4 is theorem

4.8, which shows that e-toxal tilings are e-isotoxal if e ~ 3, and

also classifies these tilings. The partial classification of 2-gonal

tilings is very helpful in the later classification theorems, but

will probably not have a great deal of independent interest unless

this classification can be completed. Thus, the next problem is:

15) Complete the classification of the 2-gonal tilings.

Theorem 4.6 [rough classification of 2-hedral tilings] is not of deep

significance, and is included primarily for completeness. It does,

however, lead naturally into theorem 4.7 [how many orbits of

triangles and hexagons can exist in a 2-hedral tiling]. This theorem

is of reasonable interest, especially since it answers another

question of Grunbaum and Shephard [1983] and contradicts a claim of

theirs. Nevertheless, this would be a ,nicer result if it could be

combined with the solution to another problem of Grunbaum and

Shephard [1983] (Exercise #2.3.9):

16) Determine the triples (t, s, h) such that there is a 3-hedral

tiling with t orbits of triangles, s orbits of squares, and h

orbits of hexagons.

So far I have examples to show that": s ,

h) is realizable so long

as: (a) s > 2; (b) t > 2h - 1; and (c) t == 2h+1 mod 3.I

The fact that so little can be said about 2-hedral tilings seems

somewhat incongruous with the state of affairs for 2-gonal and

186

2-toxa1 ti1ings. To a 1arge extent this can be b1amed on the fact

that, whi1e we may fee1 that vertices and tiles should be dual, this

duality is not fully reflected by the ideas of "tile" and "vertex

figure." Fc)rexample, in a tiling by regular polygons, a vertex

l figure implicitly defines the tiles which are incident to that

vertex; but a tile does not (generally) define the types of its

incident vertices. Thus, to attempt to restore the duality, it seems

natural to define a tile figure as the union of a tile and all the

edges which meet that tile. For edge-to-edge tilings by regular

l po1ygons, a ti1e figure would then fix the vertex types of the

incident vertices. Many questions arise fairly naturally from here;

but the most interesting is probably:

l 17) For what values of t is it the case that any edge-to-edge tiling

by regular polygons with only t tile figures is t-isohedral?

It is fairly easy to see that the answer to #17 includes t = 1 and

t = 2, and that these tilings are precisely the isohedral and

2-isohedral tilings.

Chapter 5:

All four of the classification results in this chapter

[3-1sogonal; 2-isohedral; tile-homogeneous; and strongly edge-

homogeneous] are of independent interest: and all but the last one

answer questions raised by Grunbaum and Shephard [1977a]. The most

important resu1 t is probably t.hec1assification of the 3-isogonal

tilings, and this result is u l.edin several places. This theorem has

l187

a long, tedious proofi but it makes it possible to give the more

elegant proofs of the classifications of strongly edge-homogeneous

tilings and tile-homogeneous tilings. These latter two proofs make

effective use of the earlier classifications to eliminate most of the

Of couz'se we can always try to extend these results to higher

values of v, e. and ti but most further results will probably require

the use of a computer search. The three most interesting questions

are:

work.

18) Classify the 3-isohedral tilings.

19) Use a computer to classify the 4-isogonal tilings.

20) Use a computer to classify the edge-homogeneous tilings.

Since theorem 3.6 tells us that v ~ e. a solution to problem 17 will

also give the classification of 4-isotoxal tilings.

The word "homogeneous" has been used in this thesis, and in the

works of both Krotenheerdt and Grunbaum & Shephard, to imply tilings

where any two objects that look alike are, in fact, equivalent under

the symmetries. We have extended the use of this term in the present

work, but further extensions are also possible. For example, if

applied to the idea of "colored tilings". a homogeneous coloring

would be a coloring where any tile of one color could be carried, by

a (color preserving) symmetry, to any other tile of the same color.