TILINGS BY REGULAR POLYGONS III:
DODECAGON-DENSE TILINGS
Darrah Chavey
Published in Symmetry: Culture and Science, 25(2014), No.3
Mathematician, (b. Flint, Mich., U.S.A., 1954).
Address: Department of Mathematics & Computer Science, Beloit College, 700 College Street, Beloit, Wisc. 53511,
U.S.A. E-mail: [email protected] .
Fields of interest: Ethnomathematics, Geometry, Designs & Patterns, Computer Algorithms.
Publications and/or Exhibitions related to Symmetry:
Chavey, D. (1984a), Periodic Tilings and Tilings by Regular Polygons, [Ph.D. Dissertation], Madison: Univ. of
Wisconsin, 191 pp, 1984.
Chavey, D. (1984b), Periodic Tilings and Tilings by Regular Polygons I: Bounds on the Number of Orbits of
Vertices, Edges, and Tiles,” Mitteilungen aus dem Mathem. Seminar Giessen, 164(2), pp. 37-50.
Chavey, D. (1989), Tilings by Regular Polygons II: A Catalog of Tilings,” Computers & Mathematics with
Applications, 17, pp. 147-165.
Chavey, D., (2010a) “Constructing Symmetric Chokwe Sand Drawings,” Symmetry: Culture and Science, 21(3),
pp. 191-206.
Chavey, D. (2010b), “Strip Symmetry Groups of African Sona Designs”, Bridges: Pécs, pp. 111-118.
Chavey, D. (2011), “Symmetry Orbits: When Artists & Mathematicians Disagree”, Bridges: Coimbra, pp. 337-344.
Chavey, D. (2012), “Depression Glass and Nested Symmetry Groups”, Bridges: Towson, pp. 539-542.
Abstract: In Tilings and Patterns, Grünbaum and Shephard claim that there are only four kuniform tilings by regular polygons (for some k) that have a dodecagon incident at every
vertex. In fact, there are many others. We show that the tilings that satisfy this requirement
are either the uniform 4.6.12 tiling, or else fall into one of two infinite classes of such tilings.
One of these infinite classes can be fully characterized, while the other can be shown to be
equivalent to the class of all tilings by squares and equilateral triangles; i.e. a largely
unconstrained infinite class. This characterization is, however, sufficiently powerful to
determine all such k-uniform tilings for k ≤ 14.
Keywords: Tilings, Patterns, Periodic Designs, Regular Polygons, Dodecagons.
1. INTRODUCTION AND BACKGROUND
In Grünbaum and Shephard’s definitive book Tilings and Patterns (1987), section 2.2 is
devoted to “k-Uniform Tilings.” A k-uniform tiling is an edge-to-edge tiling by regular
p. 1 polygons in which the symmetry group of the tiling acts on the vertices of that tiling so that
they fall into k different transitivity classes, or orbits. The first exercise at the end of this
section reads:
Show that if each vertex of a k-uniform tiling is incident with a 12-gon, then either
k = 1, and the tiling is (4.6.12) or (3.122), or k = 2 and the tiling is (3.4.3.12; 3.122),
or k = 3 and the tiling is well determined. Find its type in the latter case.
As it stands, this exercise asks the reader to show something that is not true. This paper
analyzes their question and attempts to find a characterization of the tilings by regular
polygons that meet their criterion. We refer to such tilings as dodecagon-dense tilings. We
show that if a tiling is dodecagon-dense, that either it is one of the two Archimedean (i.e.
1-uniform) tilings of figure 1, or else it falls into one of two infinite classes. (The tiling of
figure 1a is also a member of the second infinite class.) One of these classes is easy to
characterize. But even with what appears to be a very tight constraint on the tilings, it is not
possible to give a fully satisfactory classification of the second of these infinite classes. In
this second case it is, however, possible to reduce the problem to a more easily understood
one—reducing fairly complicated tilings with dodecagons to simpler tilings that use only
squares and equilateral triangles. Specifically, we will show that in this later case a kuniform dodecagon-dense tiling can be reduced to a k'-uniform tiling by triangles and
squares, where k' ≤ k/3 +2. Thus the complexity of the resulting square-triangle tiling is
approximately one-third that of the original tiling. For example, since all 6-uniform tilings
are known (Galebach, 2002), we know the triangle-square tilings with k' ≤ 6, and hence
implicitly know the dodecagon-dense k-uniform tilings for k ≤ 14. Thus to the extent that we
are interested in such k-uniform tilings for modest values of k, this classification is
successful.
Figure 1: The two 1-uniform
dodecagon-dense tilings, i.e tilings
in which every vertex has an
incident dodecagon, and in which
every vertex is symmetrically
equivalent to every other vertex.
The tiling on the left is (3.122), and
the tiling on the right is (4.6.12).
The (4.6.12) tiling is the only
dodecagon-dense tiling with a
vertex of type 4.6.12.
Throughout this paper, we use the standard terminology of Grünbaum & Shephard (1987)
for tilings: vertices, edges, tiles, adjacency and incidence. We review some less frequently
used terminology. A tiling is edge-to-edge if no vertex of one polygon lies along the inside
of an edge of another polygon, and from here on we assume all tilings are edge-to-edge
unless stated otherwise. (This also means that all polygons have the same edge length.) A
vertex of the tiling has vertex type n1.n2.n3. … if it is incident, in cyclic order, to a regular
n1-gon, n2-gon, etc. Thus the tiling of figure 1a has all vertices of type 3.12.12, which we
will also write as 3.122, and the tiling of figure 1b has all vertices of type 4.6.12. An edge of
the tiling has edge type a.b if it is incident to an a-gon and a b-gon. There are only four
distinct ways to arrange regular polygons at a vertex of a tiling so as to include a corner of a
p. 2 dodecagon, i.e. vertex types 32.4.12, 3.4.3.12, 3.122, and 4.6.12 (we do not view a reversal of
the cyclic order as a distinct type). In describing a k-uniform tiling T, the tiling symbol for T
lists the k vertex types in a sequence, e.g. as (3.122; 3.122; 3.122; 3.4.3.12) (shown in figure
10, #3). In a tiling with this tiling symbol, there would be three orbits of vertices of type
3.122. No symmetry of that tiling would take a vertex of one orbit to the other, even though
combinatorially they look identical. On the other hand, if we chose one representative of
each of the three orbits of 3.122 vertices, then any 3.122 vertex in the tiling can be taken by a
symmetry to one of those representatives. Krötenheerdt (1969) defines a homogeneous tiling
to be a tiling by regular polygons in which all vertices of the same vertex type lie in the same
orbit, in which case this tiling symbol would not repeat any vertex type. If Grünbaum and
Shephard’s exercise used “k-uniform and homogeneous” instead of “k-uniform,” the exercise
would be correct (the answer would be the two tilings of figure 1 and the first two tilings of
figure 10). We believe this is what they meant by their question. Indeed, we have seen some
authors assume homogeneity as part of their definition of “uniform,” although Grünbaum
and Shephard do not, hence it is possible that this exercise was adapted from such a source.
2. CLASSIFICATION OF DODECAGON-DENSE TILINGS
An important class of tilings are the periodic tilings, whose symmetry groups contain
translations in two non-parallel directions, and which are thus k-uniform with k finite. While
our results will partially classify the non-periodic dodecagon-dense tilings, our primary goal
is to analyze the periodic tilings, especially those with modest values of k. To help describe
the tilings, we let t, v, and e represent the number of orbits of tiles, vertices, and edges
(respectively). (Commonly, “k” is used instead of “v”, and we use both here, depending on
the context.) While the greatest attention has traditionally been given to the value of v, some
results here use all three of these orbit parameters, and we list the values of all three in the
dodecagon-dense tilings of figure 10.
2.1 Classification of Dodecagon-Dense Tilings with a Vertex of Type 4.6.12
Of the four possible vertex types that can occur in dodecagon-dense tilings, tilings that
contain a vertex of type 4.6.12 are particularly easy to classify:
Theorem 1: If T is a dodecagon-dense tiling and T contains
any vertex of type 4.6.12, then T must be the 1-uniform tiling
of type ( 4.6.12 ).
Proof: If T is not uniform, then there is an edge connecting a
4.6.12 vertex 1 to a vertex 2 of a different type (see figure 2),
which must then be 32.4.12. Then the required dodecagon at
vertex 3 forces vertex 4 to have three triangles, which cannot be
completed to a tiling vertex that also contains a dodecagon. n
Figure 2: Configuration
forced by a 4.6.12 vertex.
p. 3 2.2 Classification of Dodecagon-Dense Tilings with a Vertex of Type 32.4.12
We next consider tilings with vertices of type 32.4.12. One way to construct an infinite
family of dodecagon-dense tilings that includes vertices of this type is to take the uniform
tiling of type 3.122, split it along any set of the horizontal zigzags (the bold edges in figure
3a), separate the tiling along those selected zigzags by a distance of 1 edge length, and fill in
the gaps with triangles and squares, such as in the middle zigzag of figure 3b. Once this has
been done, additional unsplit zigzags can be moved parallel to each so that their dodecagons
match, opening room to replace the individual triangles incident to this zigzag with
configurations of a square and 4 triangles, as shown with the top zigzag in figure 3b. We
refer to this horizontal alignment as matched zigzags. The next theorem shows that this
process is the only way to introduce such vertices.
Figure 3: The 1-uniform ( 3.122 ) tiling, on the left, with its horizontal zigzags in bold. On the right is the
same tiling after “splitting” the middle horizontal zigzag, to add squares and equilateral triangles, and
“matching” dodecagons on the top zigzag to create space for square/triangle configurations.
Theorem 2: If T is a dodecagon-dense tiling, and T contains any vertex of type 32.4.12,
then T can be constructed from the uniform (3.122) tiling via the “split zigzag” and
“matched zigzag” procedures described above. Furthermore, if T is periodic with z orbits of
these various types of zigzags, then T is k-uniform with: 4z – 4 ≤ k ≤ 6z – 1.
Proof: [Classification of Tilings] Consider the edge ε of a square incident to a vertex of type
32.4.12 (e.g. an endpoint of edge ε in figure 4). Theorem 1 says there are no vertices of type
4.6.12, so both endpoints of ε must be of type 32.4.12, giving the bold lines of figure 4. Let τ
be the square incident to ε, and hence incident to these two vertices. Neither of the other two
vertices of τ can have type 32.4.12, else we create a vertex with three triangles (and a
dodecagon), which is impossible. These other two vertices must then have type 3.4.3.12. The
edges incident to ε of type 3.3 must end in vertices of type 32.4.12, which forces the edges ε'
to have the same properties as ε. Repeating this argument with those edges forces the rest of
the split zigzag.
p. 4 Figure 4: The patch of a tiling forced by a 32.4.12 vertex
in a dodecagon-dense tiling.
Figure 5: The reflections (solid lines) and glide
reflections (dashed line) in a tiling by strips of
various types of zig-zags.
On the zigzag above this one, if either of the vertices ν1 in figure 4 had type 32.4.12,
then they both would and, as shown in the gray on the top right of figure 4, the vertex
marked “?” would then be incident to 2 squares, leaving no room for its dodecagon. If either
of the vertices ν1 had type 3.122, then they both would, and this would force an unsplit
zigzag. Thus we may assume these vertices both have type 3.4.3.12, as shown. If vertex ν2
had type 32.4.12, the argument of the previous paragraph forces a split zigzag. Otherwise, ν2
has type 3.122, which forces the matched zigzag described above.
[Numbers of Orbits] Regardless of the types of zigzags in the tiling, every vertical line
that passes through edges of type 12.12 will be a vertical reflection line (see figure 5 for an
example of these vertical reflection lines). If there are no other symmetries of the tiling, then
the number of vertex orbits contributed by each orbit of these zigzags will be 4, 5, or 6
(respectively), depending on whether the zigzag was unsplit, matched, or split, and this will
account for all orbits of vertices in the tiling. In this case, with a orbits of unsplit zigzags, b
orbits of matched zigzags, and c orbits of split zigzags, the number of orbits of vertices
would then be k = 4a + 5b + 6c.
The existence of even one vertex of type 32.4.12 forces a split zigzag, and this zigzag
prevents the possibility of any rotation of order greater than 2. Combining this with the
known vertical reflections limits the possible symmetry groups to pm, cm, pmm, pmg, or
cmm. Some of these groups have vertical glide-reflections as well as the vertical reflections,
which affects the number of orbits of zigzags, but not the number of orbits of vertices within
a zigzag. These groups can, however, have 0, 1, or 2 orbits of horizontal reflections or glide
reflections, which do affect the number of orbits of vertices on the zigzags containing those
symmetries. Horizontal reflections can only occur centered in the strips of dodecagons or
along a matched zigzag; horizontal glide-reflections can only occur centered on split or
unsplit zigzags. The effect of these symmetries, in addition to reducing the number of orbits
of zigzags, is to reduce the number of orbits of vertices on a zigzag as follows:
1. A horizontal reflection through the centers of dodecagons has no effect;
p. 5 2. A horizontal reflection on a matched zigzag reduces the orbits on that zigzag by 2;
3. A glide reflection on an unsplit zigzag reduces the orbits on that zigzag by 2;
4. A glide reflection on a split zigzag reduces the orbits on that zigzag by 3;
Since there are at most two such symmetries, at most two of these reductions can occur,
hence k ≥ 4a + 5b + 6c – 6. Letting z = a + b + c be the total number of orbits of zigzags,
and knowing that c > 0, since we must have at least one split zigzag, we have:
k ≥ 4a + 5b + 6c – 6 = (4a + 4b + 4c) + b + 2c – 6 ≥ 4z – 4
If a = b = 0, we have the tiling of figure 10 (#2), which satisfies the theorem. Otherwise:
n
k ≤ 4a + 5b + 6c = 6(a + b + c) – 2a – b ≤ 6z – 1
The formula k = 4a + 5b + 6c, which applies when there are no horizontal symmetries, plus
the reductions of 1-4 above when there are horizontal symmetries, gives us much more than
just the bounds of theorem 2. They allow us to convert the calculation of k to an analysis of
the 1-dimensional sequence of zigzags which occur in the tiling. For example, letting u, m,
and s refer to an unsplit, matched, and split zigzag (respectively), consider the tiling
constructed from the infinite length 7 repeating sequence:
|
|
|
|
|
…ss mumssss mumssss mumssss mumssss mu…
Here we have used vertical bars to show one of the many ways of selecting a “pattern” of
zig-zags for this tiling. In table 1, we would list this sequence as ( m u m s s s s ). Then the
horizontal reflections and glide-reflections of the tiling will be located where the symmetries
of this character sequence occur, which in this example will be a glide-reflection directly on
the unsplit zigzag “u” and a reflection between the 2nd and 3rd split zigzags “s”. Looking at
the orbits of characters in this sequence, we see that there is one orbit each of u and m, and
two orbits of s, i.e. a = b = 1; c = 2. Consequently, we can calculate k for this tiling as:
k = 4a + 5b + 6c – 2 = 19
where the “–2” is the adjustment for the glide-reflection in “u”, i.e. case 3 in the list above.
This direct calculation of k from its zigzag character pattern allows for an examination of
character sequences of a specified length. The number of orbits of characters in a character
sequence of length l with mirror symmetries is "( l +1) 2# , so letting z be the number of
orbits of zigzags, i.e. z = a + b + c, then 2z – 1 ≥ l. We can thus use the lower bound of
theorem 2 to conclude:
k ≥ 4z – 4 = 2(2z–1) – 2 ≥ 2l – 2, when the sequence has mirror symmetries; and
€
k = 4a + 5b + 6c ≥ 4z + 2c ≥ 4l + 2, when the sequence has no mirror symmetries.
which gives us an explicit upper bound to the length of the character sequences we need to
inspect for tilings with a given k. It is then easy (with computer aid for larger l) to enumerate
all possible sequences up to some value of l, check them for symmetries, and calculate the
values of k, without actually constructing or analyzing the tilings themselves. The following
table lists this sequence form for all tilings of this type with k ≤ 12 and c > 0. These are listed
according to sequence length and the locations of symmetries, with repetitions removed. For
odd length sequences, one reflection must be on a character, and one between two
characters; for even length sequences either both reflections are on characters or both are
between characters. We use “.” to indicate a character position and “|” to indicate a
p. 6 symmetry position (of either type). We list the sequence of the types of zigzags, and the
calculations of the value of k. Here the “adjustment” is the required correction to the value of
4a + 5b + 6c that is required to find k, as described in the proof of theorem 2.
Minimum length
of zigzag pattern
Location of Symmetries
a, b, c
4a+5b
+6c
Adjustment
k
l =1
( | )
0, 0, 1
6
3
3
( S U )
1, 0, 1
10
5
5
( S M )
0, 1, 1
11
5
6
(S S U)
1, 0, 1
10
2
8
(S S M)
0, 1, 1
11
2
9
(U U S)
1, 0, 1
10
3
7
(M M S)
0, 1, 1
11
3
8
(U S S S)
1, 0, 2
16
5
11
(M S S S)
0, 1, 2
17
5
12
(U S M S)
1, 1, 1
15
4
11
(U U S U)
2, 0, 1
14
5
9
(M M S M)
0, 2, 1
16
5
11
(M U S U)
1, 1, 1
15
5
10
(U M S M)
1, 1, 1
15
5
10
(S S U U)
1, 0, 1
10
0
10
(S S M M)
0, 1, 1
11
0
11
(U S S U U)
2, 0, 1
14
2
12
(S U U S U)
2, 0, 1
14
2
12
(U U U U S)
2, 0, 1
14
3
11
(U M M U S)
1, 1, 1
15
3
12
(M U U M S)
1, 1, 1
15
3
12
( S )
l =2
l =3
l = 4, Type 1
l = 4, Type 2
l =5
( | | )
(.|. |)
(| . | .)
(.|. .|.)
(. .|. . |)
Table 1: A listing of all character sequences that generate the dodecagon-dense tilings of theorem 2 with
k ≤ 12. Each group is shown with the symmetries of the sequence corresponding to horizontal
symmetries, and shows the steps needed for the calculation of k described.
p. 7 2.3 Classification of Dodecagon-Dense Tilings with Vertices of Types 3.122 and 3.4.3.12.
As a consequence of theorems 1 and
2, the remaining dodecagon-dense
tilings can contain only vertex types
3.122 and 3.4.3.12, an example of
which is shown as the solid lines of
figure 6. These tilings are the
hardest to classify, and we can do so
only in the sense of converting such
a tiling T into a simpler tiling T '
built from equilateral triangles and
squares. To quantify the claim that
these tilings are “simpler,” we
establish inequalities relating the
values of the orbit parameters t, v,
and e for T with the corresponding
values t', v', and e' for T '. To
sharpen these inequalities we also
define t'4 as the number of orbits of
squares in T '. Using this notation,
we show:
Figure 6: Reducing a dodecagon-dense tiling to a sub-graph of
the dual tiling, thus creating a tiling by squares and triangles.
Theorem 3: There is a one-to-one symmetry preserving correspondence between the class of
tilings T in which every vertex has type 3.122 or 3.4.3.12 and the class of tilings T ' in
which all tiles are squares or equilateral triangles. This correspondence matches dodecagons
of T with vertices of T '. If the corresponding tilings are periodic and (i) t, v, and e represent
the orbit parameters for T, and (ii) t', v', and e' represent the orbit parameters for T ', and t'4
is the number of orbits of squares in T ', then either T is the 1-uniform (3.122) tiling or:
i. t ≥ t' + v' + 2t'4 – 4
ii. v ≥ 3v' – 6
and
hence
t ≥ t' + v' + 1
v' ≤ v/3 + 2
iii. e ≥ e' + 2t'
Proof: [Classification of Tilings] We construct T ' as a sub-graph of the tiling “dual” to T.
We do this by placing vertices at the centers of each dodecagon in T and connecting the
centers of adjacent dodecagons with an edge. (See the dashed lines of figure 6 for an
example of such a construction.) All such edges will have the same length (which is 2 + 3
times the original edge length). A dodecagon in T cannot be incident to triangles along 3
consecutive edges, else it is incident with two consecutive vertices of type 3.4.3.12, which
forces the existence of a vertex incident to two squares, which cannot
€ then have a
dodecagon. Consequently, at least every 3rd edge is incident to a dodecagon, hence angles in
p. 8 T ' cannot be greater than 90º. A dodecagon in T cannot be incident to two consecutive
additional dodecagons, since three dodecagons cannot fit at a vertex, hence angles in T '
cannot be less than 60º. The constraints that all edges have the same length and all angles are
either 60º or 90º forces the resulting “partial dual” tiling to be a tiling by squares and
equilateral triangles.
It is easy to reverse this process and convert any tiling that uses only equilateral
triangles and squares into a dodecagon-dense tiling with vertices of types 3.122 and 3.4.3.12.
In particular, we need only replace squares and triangles with the tiling patches shown for
them in figure 6. Figure 6 includes all four vertex types possible with triangles and squares,
i.e. 36, 33.42, 32.4.3.4, and 44, and this establishes the first two sentences of the theorem.
[Number of Orbits] We now consider the inequalities (i)–(iii). In addition to the
notation t'4 of the theorem statement, we use tn for the number of orbits of n-gons, and t3(a)
and t3(b) for the number of orbits of two types of triangles in T: type a triangles, surrounded
by 3 dodecagons, and type b triangles, which are adjacent to a square. Similarly, we use ex.y
for the number of orbits of edges of edge type x.y, with similar notation for T '. Since the
symmetry groups of T and T ' are identical, the construction above maps specific elements
of T to other elements of T ', and this mapping tells us that:
(1)
t12 = v'
t4 = t'4
(2)
4t'4 ≥ t3(b) ≥ 2t'4 – 4
t3(a) = t'3
e12.12 = e'
For (2), the construction puts four triangles of type (b) inside every square of T '. For those
squares of T ' that are fixed by no symmetry, these 4 triangles will all be in different orbits,
and we would have 4t'4 = t3(b) (for these tiles). If a square is fixed by a symmetry, then it
contains 1, 2, or 3 orbits of such triangles, hence for these tiles, t3(b) ≥ t'4 ≥ 1. However, a
square of T ' contains only one such orbit only when that square is fixed by a C4 or D2
symmetry group. Inspection of the 17 wallpaper groups shows that are at most 4 such orbits
of squares (which occurs with pmm), hence t3(b) ≥ 2t'4 – 4. With these equations and
inequalities we can establish inequality (i) of the theorem:
t = t12 + t4 + t3(a) + t3(b) = v' + t'4 + t'3 + t3(b) = v' + t' + t3(b) ≥ v' + t' + 2t'4 – 4
To establish (ii) we use a discharging algorithm to relate orbits of vertices of T to orbits
of tiles of T, and thus to vertices of T '. (See figure 7 to follow the definition of this
mapping.). We assign a charge of +1 to every vertex of T, then define the “discharging” as:
(a) A vertex ν of type 3.122(a) is incident to a unique triangle τ, and sends its charge to the
dodecagon of T incident to τ but not incident to ν;
(b) A vertex of type 3.122(b) sends half its charge to each of its adjacent dodecagons;
(c) A vertex of type 3.4.3.12 sends its charge to its unique incident dodecagon.
p. 9 We need to show that this mapping from
vertices to tiles induces a mapping from
orbits of vertices to orbits of tiles. To do so,
note that if a vertex ν of type (a) or (c) is
moved by a symmetry to another vertex ν' in
its orbit, then the dodecagon τ receiving its
charge is moved by that same symmetry to a
dodecagon τ' in the same orbit as τ.
Similarly, the pair of dodecagons receiving
charge from a vertex ν of type (b) will be in
the orbit(s) of the pair of dodecagons
receiving charge from any vertex in the orbit
of ν. The discharging is thus well-defined on
the orbits of these tiling elements. To
demonstrate (ii), we need a lower bound for
the charge that will be assigned to the
dodecagons of T.
If there are no symmetries of T fixing a
Figure 7: The mapping from vertices of T to
tiles of T (other than the triangles).
given dodecagon τ, then τ will receive a charge of at least 6 from its adjacent vertices, no
two of which can be in the same orbit. However, if τ is fixed by symmetries, then this would
lead to double counting of the contributions from some orbits of vertices to the charge on τ.
For example, if τ is fixed by a reflection, then the charge from the vertices on one side of the
reflection are the same as those counted from the other side. If that were the only symmetry
fixing τ, then its charge would be at least 3. Similarly, if τ is fixed by the point group C2, it
will have a charge of at least 3; if it’s fixed by D2, C3, or C4, it will get a charge of least 2;
and if it’s fixed by D3, D4, C6, or D6, it gets a charge of at least 1. Letting “#D3” (etc.)
represent the number of dodecagons fixed by a D3 point group (etc.), then the total charge on
all dodecagons will be at least:
3t12 – [ 1·(#D3 + #C3 + #C4) + 2·(#D3 + #D4 + #C6 + #D6) ]
Examination of the values of this second term for the 17 wallpaper symmetry groups shows
that the largest value of this is 6 (for p3m1). Using the fact that this charge counts the
number of orbits v in T, and t12 counts the number of orbits v' in T ', we get:
v = the total charge ≥ 3t12 – 6 = 3v' – 6
This establishes inequality (ii).
Finally, we establish inequalities (iii). Let e3.12(a) count the orbits of edges of type 3.12
which are incident to a triangle of type a, and similarly for e3.12(b). Every orbit of triangles
in T ' contains at least one orbit of edges of type 3.12(a) in T, and 2 orbits if the triangle is
not fixed by a 3-fold rotation. Each square in T ' contains at least one orbit of edges of type
3.4 and one orbit of edges of type 3.12(b). Thus if there are no three-fold rotations, then the
edges of T contained completely inside the squares and triangles of T ' can be counted as:
p. 10 e3.12(a) + (e3.12(b) + e3.4) ≥ 2t'3 + (t'4 + t'4) = 2(t'3 + t'4) = 2t'
Wallpaper symmetry groups that have 3-fold rotation centers have at most three such
centers, and they have neither 4-fold rotation centers nor D2 rotation centers. In the absence
of these later types of rotations, squares in T ' must contain at least two orbit of edges in T of
type 3.4, and four orbit of edges of type 3.12(b). Using this with the limit of 3 orbits of 3fold rotation centers, we can count these same edges as:
[ e3.12(a) ] + [e3.12(b) + e3.4] ≥ [2(t'3 – 3) + 3] + [4t'4 + 2t'4] = 2t' + 4t'4 – 3 ≥ 2t' + 1
Thus we may assume the former inequality holds. From (1), e' = e12.12, hence:
e = e12.12 + (e3.12(a) + e3.12(b) + e3.4) ≥ e' + 2t' .
n
To fully understand the tilings of theorem 3, we would need to understand tilings of triangles
and squares, a class of tilings which are largely unconstrained. Of course this says that these
dodecagon-dense tilings are also largely unconstrained. The reduction in complexity, though,
says that the existing computer classification for k ≤ 6 will give, via this process, all
dodecagon-dense tilings for k ≤ 14 (since if v' ≥ 7, then v ≥ 3v' – 6 = 15).
3. SOME REMARKS ON TILINGS BY SQUARES AND TRIANGLES
Theorem 3 shows that many dodecagon-dense tilings reduce to simpler tilings by triangles
and squares. This reduction can be quite helpful, but does not give a complete classification
of this family of tilings. Such a classification would require knowing all triangle/square
tilings. To emphasize the difficulty of this problem, consider the special case where every
vertex is incident with at least one triangle and one square. This version of the problem is
equivalent to classifying the tilings that contain only the vertices 33.42 and 32.4.3.4, i.e.
where each vertex contains exactly 3 triangles and 2
squares. This problem was first investigated by
Sommerville (1905), who could not answer it. This
problem has withstood the efforts of several
researchers for a century. One limited result for
triangle/square tilings is a classification of those
tilings that do not contain a vertex of type 32.4.3.4.
We say that a strip of squares or triangles is a set of
those polygons that lie between two parallel lines and
fill the space between them. Figure 8 shows an
example of a tiling by strips of polygons. Assuming
that the strips all use tiles of the same length and that
the resultant tiling is edge-to-edge, tilings by strips
may contain vertices of types 36, 33.42, or 44, i.e. all
combinations of squares and triangles that do not Figure 8: A strip pattern containing 3 of
the 4 vertex types of squares and triangles.
include 32.4.3.4. These tilings have been classified:
p. 11 Theorem (Chavey, 1984a): The only tilings by squares and equilateral triangles that do not
contain the vertex 32.4.3.4 are tilings by strips of squares and strips of triangles.
With these tilings, there are vertical reflections at all vertical edges of squares, and by
inspecting the possible symmetry groups, at most two strips can contain horizontal
reflections or glide reflections. Using this, it is not hard to show that for these tilings,
inequality (ii) of theorem 3 can be strengthened to:
v ≥ 4v' + t'4 – 4
The same inequality appears to be true for all tilings of theorem 3, but the author has not
found a satisfactory proof of that.
4. SYMMETRIC DODECAGON-DENSE TILINGS
The v-uniform dodecagon-dense tilings characterized by theorems 1–2 can be calculated up
to any reasonable value of v. Galebach’s (2002) classification includes all square/triangle
tilings with v ≤ 6, and theorem 3 can then be used to generate all dodecagon-dense tilings of
that type up to v = 14. Combining these results, table 2 lists the number of such tilings for v
≤ 14, figure 1 shows those tilings for v = 1, and figure 10 shows those tilings with 2 ≤ v ≤ 8.
v=
#
1
2
2
1
3
1
4
1
5
2
6
2
7
3
8
2
9
4
10
7
11
8
12
9
13
12
14
13
Table 2: The number of v-uniform dodecagon-dense tilings with v ≤ 14.
5. NON EDGE-TO-EDGE DODECAGON DENSE TILINGS
Traditionally, non edge-to-edge tilings have
attracted less attention than edge-to-edge tilings.
As such, we mention the additional dodecagondense tilings that can be constructed for this
situation, but do not include proofs.
Theorem 4: If T is a dodecagon-dense non edgeto-edge tiling by regular polygons, then it can be
constructed from an edge-to-edge dodecagondense tiling T ' by dissecting some type a
triangles in T ' into 4 equilateral triangles and/or
decomposing other type a triangles into a hexagon
and 3 equilateral triangles.
Figure 9: A 4-isogonal dodecagon-dense tiling
using non edge-to-edge tiles.
p. 12 These two types of dissections of triangles are both shown in the 4-uniform tiling of figure 9.
If we had dissected all of the triangles in this example the same way, we would construct two
different 2-uniform dodecagon-dense tilings.
p. 13 Figure 10: v-uniform dodecagon-dense tilings with 1 < v ≤ 8. Values for t and e (orbits of tiles and edges)
are also listed. Tilings with v = 1 are shown in figure 1.
p. 14 6. DEMI-REGULAR (2-MONOGONAL) TILINGS
Grünbaum and Shephard define a vertex figure to be the set of edges incident at a
vertex. For tilings by regular polygons, this is equivalent to the set of all tiles meeting at that
vertex. They define a tiling to be “monogonal” if all vertex figures are congruent to each
other, and “isogonal” if the symmetry group of the tiling maps any one vertex to any other.
Thus all isogonal tilings are monogonal, but not conversely. They also define “k-isogonal”
tilings as those where the symmetry group divides the vertices into k classes of vertices.
Although they do not use the term “k-monogonal,” it is consistent with their terminology to
think of this as the class of tilings in which the vertex figures fall into k congruency classes.
Using this terminology, the tilings of theorem 3 are examples of “2-monogonal tilings by
regular polygons,” and we use this terminology here. In addition we use a tiling symbol of
the form { 3.122; 3.4.3.12. } to specify the vertices of a 2-monogonal tiling.
A closely related term used by several authors is that of “demiregular tilings” (or
demiregular tessellations). Unfortunately, this term has been poorly defined by most authors
who use it, Aslaksen (2009) discusses several sources using this term and notes that while
most such authors agree that there are 14 tilings that qualify, they disagree as to which are
the 14 tilings! Following Ghyka (1946), most of these tilings include vertex figures from
only two “situations” of vertex types. However, they generally view vertex types such as
32.4.12 and 3.4.3.12, which have the same number of each polygon, as being of the same
“situation.” As such, their tilings are either 2-monogonal or 3-monogonal. And both Ghyka
(1946) and Critchlow (1969) include a single {32.4.3.4; 32.4.12; 36 } tiling with three vertex
types (from three “situations”), with no explanation as to why it’s included, although they
recognize it as different than the others on their list. In addition, such authors seem to assume
some sort of regularity on the appearances of these vertex types. For example, Williams
(1972) says that demiregular tilings are “orderly composits” of these vertex figures.
However, this “orderliness” does not imply that vertices of one type are symmetrically
equivalent, i.e. that they are “homogenous” tilings in the terminology of Krötenheerdt (1969)
and Grünbaum & Shephard (1987). Indeed, in Critchlow’s drawings of the 14 tilings, only 8
are homogenous. So these authors do not appear to have a clear idea of what type of “orderly
structure” on these tilings would lead to such a small set of only 14 “demiregular” tilings.
Of the various authors that use the term “demiregular,” the only one we know of that
seems to have a clear definition is Blustein (2012), who is farily explicit in her definition of
“demiregular” as being a tessellation by regular polygons with two vertex types. For this
paper, we agree with Blustein and say that:
Definition: A demiregular tiling of the plane is a 2-monogonal edge-to-edge tiling
by regular polygons.
The classification of the demiregular tilings, under this definition, has a substantial history.
There are 16 possible types of such tilings (Chavey (1984), Corollary 4.2). Of these, two
were classified by Sommerville (1905), albeit without proofs (proofs can be found in
Chavey, 1984a). Krötenheerdt (1969) classified three other types by showing that any such
tiling must be the 2-uniform tiling of that type. The author’s thesis (1984a) classified five
other types of 2-monogonal tilings. Lenngren (2009) noted that two of the remaining cases
were essentially the same, leaving 5 remaining sub-problems. One of those remaining subproblems was the classification of the { 3.4.3.12; 3.122 } tilings, and that family is classified
by theorem 3. This leaves 4 remaining demiregular tiling classifications unsolved.
p. 15 7. OTHER MEASURES OF COMPLEXITY / SYMMETRY OF A TILING
The most commonly used measure of the complexity of a tiling, or its degree of
symmetry, is the value of v, the number of orbits of vertices in that tiling. On the other hand,
one could also categorize tilings by the values of e or t, the number of orbits of edges or tiles
(respectively) in the tiling. Thus we might wish to find all dodecagon-dense tilings with
moderately small values of these parameters as well as v. Chavey (1984b) shows that for any
tiling by regular polygons, v ≤ e, hence the list of dodecagon-dense tilings with v ≤ 14
includes all tilings with e ≤ 14.
If a tiling meets the conditions of theorem 2, then the dodecagons on each row all lie in
the same orbit, and t12 is determined by the number of such rows. If the repeating sequence
of zig-zags has length l, and there are no horizontal reflections/glide-reflections, then t12 = l.
If there are horizontal symmetries, then l/2 ≤ t12 ≤ l/2 + 1, depending on whether l is odd or
even, and the exact locations of the horizontal symmetries (i.e. on dodecagons or on zigzags). The number of orbits of tiles on the zig-zags depends on the type of zig-zag and
whether or not it is fixed by a horizontal symmetry. The number of such orbits will be:
i)
For an unsplit zig-zag: 1 orbit if there is a glide-reflection, 2 otherwise;
ii) For a matched zig-zag: 3 orbits if there is a reflection, 4 otherwise;
iii) For a split zig-zag: 3 orbits if there is a glide-reflection, 6 otherwise.
Thus t3 + t4 = 2a + 4b + 6c – correction, where the correction factor, as in §2.2, depends on
the exact location of the horizontal symmetries (if any), and is easily calculated from these
sequences as done in §2.2. As such, one can directly calculate the value of t from the
sequence of zig-zags, as done with table 1. For any desired value of t, a computer program
(for example) could generate all possible strings of u, m, and s that might generate
appropriate tilings and calculate directly which tilings have that value of t.
If a tiling meets the conditions of theorem 3, then we cannot always mechanically
calculate the set of tilings with t orbits of tiles. However, inequality (i) of theorem 3 tells us
that t ≥ t' + v' + 1. Chavey (1989) lists all tilings by regular polygons with t' ≤ 3, and
Galebach (2002) lists all such tilings with v' ≤ 6, hence we know all the tilings of theorem 3
unless t ≥ 4 + 7 + 1, i.e. we can identify all dodecagon-dense tilings with t ≤ 11 by
inspection.
8. CONCLUSION
Grünbaum & Shephard (1987) asked for a classification of what we have called
dodecagon-dense tilings by regular polygons. Their answer to that question, as they worded
it, was incorrect. This paper gives a reasonably complete answer to their stated question.
Because the answer involves two infinite families of tilings, the question arises as to how
precise our descriptions of these two families are. Theorem 2, and calculations such as those
of table 1, imply essentially a complete understanding of that infinite family of dodecagondense tilings. The tilings of the form discussed in theorem 3 are limited by our understanding
of tilings by triangles and square, an essentially unconstrained class. We have argued that for
tilings of the form described by theorem 3, we have sufficient detail to identify any
reasonably symmetric tiling, whether the measure we use for that level of symmetry reflects
the number of orbits of vertices, edges, or tiles.
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p. 17