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This paper describes 4-valent tiling-like structures, called pseudotilings, composed of barrel tiles and apeirogonal pseudotiles in Euclidean 3-space. These (frequently face-to-face) pseudotilings naturally rise in columns above 3-valent plane tilings by convex polygons, such that each column is occupied by stacked congruent barrel tiles or congruent apeirogonal pseudotiles. No physical space is occupied by the apeirogonal pseudotiles. Many interesting pseudotilings arise from plane tilings with high symmetry. As combinatorial structures, these are abstract polytopes of rank 4 with both ﬁnite and inﬁnite 2-faces and facets.

An

This article deﬁnes and investigates structures in Euclidean 3-space which are closely related to both normal simple tilings by barrels and normal simple tilings in the plane. The deviation from classical tiling theory occurs when we introduce

Allowing inﬁnite regular polygons as faces, Grünbaum [

Over the last few decades, the development of the theory of abstract polytopes and their realizations provided a common framework for the investigation of regular and other polytopes. This generalization of polyhedra and polytopes to combinatorial objects seemed only natural after a long evolution of the terms

After giving a ﬁrst example of a barrel pseudotiling in

What makes the pseudotilings interesting, among other things, is that they are faithful realizations of certain abstract rank 4 polytopes (see

As a note of caution to the reader, in deviating from the original usage in [

Start with the regular hexagonal tiling (6^{3}) of the plane. Color the tiles in the tiling properly (^{3}) functions merely as a guide in the construction of the columns. The plane in which (6^{3}) is situated serves as a reference plane for applying shifts when positioning adjacent barrel-ﬁlled columns relative to each other in order to match the pentagons.

(

(^{3}); (

The remaining part of 3-space separates into inﬁnite columns associated to tiles of color ∞. Let these columns inherit the boundary structure from the hexagonal barrels in adjacent columns. As a result, the boundary of each column colored ∞ falls apart into three “intertwined” or “stacked” apeirobarrels (see

Moreover, a similar construction is possible for several other plane tilings, including the uniform tilings (4.6.12) and (4.8^{2}), in which the regular convex polygons surrounding each vertex have four, six, and twelve, respectively four, eight, and eight vertices. Naturally, the question comes up which kinds of

Related kinds of tiling-like structures with apeirogons, but in the Euclidean plane instead of 3-space, have been investigated by Grünbaum, Miller, and Shephard in [

In order to keep the exposition as self-contained as possible, we will now review basic notions from several areas of relevance.

A

A

All graphs considered in this paper will be

Let

A

The (open) star of a vertex _{Δ}(_{Δ}(_{Δ}(_{Δ}(_{Δ}(_{Δ}(

For the purpose of this article, we will use the term

The

We will only need some very basic ideas about abstract polytopes, which are combinatorial objects generalizing the previously existing notions of (convex or non-convex) polyhedra and polytopes. For a thorough discussion of abstract polytopes it is recommended to consult the standard reference by McMullen and Schulte [

In order to understand the ﬁrst condition, we need to know that a

The second condition is commonly called the

(1) Each edge is incident to precisely two vertices;

(2) For each 2-face and each of its incident vertices, there are precisely two edges which are incident with both the 2-face and the vertex;

(3) For each 3-face (

(4) There are precisely two facets incident with each 2-face.

A

This section is devoted to the proper, but rather technical, deﬁnition of a

_{0},...,A_{k−1}} of k > 1 mutually non-intersecting apeirogons in ∂Z = ∂D +

_{i}, we have x + ru ∈ A_{l} for some l if and only if r ∈ _{i+j} for each j ∈

Any _{i}_{i}_{i+1}

_{i}

A

Thus, every barrel in a ﬁnite stack has an “impossible staircase” (familiar, e.g., from Dutch artist

M.C. Escher’s print

Observe that translation by

Combinatorially, the faces of the constructed object, together with a unique largest face of rank 4 and a unique smallest (empty) face of rank −1, form a graded poset as required for an abstract 4-polytope. The faces of rank 0 are the vertices, the faces of rank 1 are the edges of the pseudotiling, the faces of rank 2 are the planar polygons and non-planar apeirogons, and the faces of rank 3 are the tiles (barrels) and pseudotiles (apeirobarrels). There are inﬁnitely many faces of each of these kinds. Additionally, we have

This 4-apeirotope is realized faithfully in Euclidean 3-space, such that all ﬁnite faces lie in an afﬁne subspace of the appropriate dimension and coincide with the convex hull of their vertices (straight edges, plane polygons, ﬁnite barrels stemming from right prisms). The size of the ﬁnite barrels is uniformly bounded, so a property very close to normality is retained. All vertices are 4-valent, so, in many ways, barrel pseudotilings are similar to simple normal tilings of 3-space.

We conclude this section with a note on the face-to-face property.

A rough outline of constructing a pseudotiling by polygonal barrels and ∞-barrels is provided by the following ﬁve steps.

First of all, select a plane, simple, normal tiling

The fourth step deserves a more detailed explanation and will be carefully analyzed in the next section: Shift each stack of

In order to assure that each rectangle in a prism’s mantle splits into precisely two 2-faces, it is necessary and sufﬁcient to shift adjacent stacks (_{0},_{1},..., ∗,...). Let us call such a symbol a

The requirement of splitting the rectangles in the mantle of an _{0},_{1},..., ∗,...). Let (∗, (_{0} − _{1} −

It may not be possible to carry out step four consistently, depending on earlier decisions,

Recall that for the remainder of this article, we consider our plane base tiling

Unwrapped: An apeirobarrel winding up upstairs (

To each barrel and apeirobarrel in a barrel pseudotiling

_{1}, _{2}, and _{3} meet there as pictured in _{1},_{2},_{3}) is the listing of tiles in counterclockwise order around _{1}, _{2}, or _{3}. We obtain a repeating pattern since translation by the vertical unit vector

Determining upstairs orientations: (

_{1}_{2}_{3 }_{1}_{2}_{3}_{1}_{1}_{2}_{3}_{1}_{3}_{2}

^{' }of the base tiling

This fact allows us to adopt the convention that, unless otherwise stated, all subsequent barrel pseudotilings are assumed to have a counterclockwise upstairs orientation. We now focus on the properties of barrel pseudotilings with only

Let

Staircase lines on unwrapped mantle of ∞-stack (schematic). Other edges have been omitted for clarity.

It is not known if there is any such correspondence for pseudotilings with adjacent ∞-stacks, which makes it harder to explore them. Therefore, for the remainder of this article, all considered barrel pseudotilings have only isolated apeirostacks.

The following Lemma characterizes when

_{0},_{1},...,_{n−1}) _{i}_{i+1}

_{i} in the neighborhood symbol (by taking the remainder of the height modulo 1, _{i}, w.l.o.g. _{1}, the corresponding contiguous entries in the neighborhood symbol increase as long as the staircase is trapped between _{i} to a strictly smaller _{i+1} in the neighborhood symbol.

Conversely, in constructing an ∞-stack Ω, any apeirogonal faces are constructed as polygonal lines not intersecting the staircases. Any time there is a jump in the entries of the neighborhood symbol from an _{i} to a strictly smaller _{i+1}, the staircase has climbed another unit. Consequently, if there are

Observe that the condition formulated in Remark 6.1 assures that step four of the construction outline can be carried out, whereas the condition in Lemma 6.4 assures that ∞-stacks can be ﬁtted in step ﬁve (as long as

_{1}_{2}_{3}_{1} = _{1})_{2} = _{2})_{3} = _{3}) _{1}_{1} < _{2} < _{3}

_{0},_{1},...,_{n−1}) _{i}_{i+1}

Any normal simple tiling of the plane has a normal simple dual, which is a (topological) triangulation of the plane (

Assume that ^{0 }= (^{−}^{1 }([0, 1)) denote the set of vertices of ^{0 }denote the simplicial subcomplex of ^{0 }(it contains all simplices which have only vertices in ^{0}, and the empty simplex).

^{∞ }:= ^{0 }

This is simply the dual version of the statement that all tiles are either colored ∞, or adjacent to an ∞-colored tile, but not both. As a consequence, ^{0 }contains no complete simplicial neighborhood of a vertex in ^{0 }which lie in its interior (we do not use the term ^{0 }may be a one-dimensional subcomplex of

^{0}| ^{0 }^{0}| ^{∞ }

Furthermore, the function ^{0}, where {^{0}). Furthermore, every arc is ^{0 }has two mathematically positive and one mathematically negative oriented arcs. Note that the resulting directed edge graph of ^{0}, which we denote by

The neighborhood of a regular vertex ^{0}.

A ^{0 }such that the underlying topological space ^{0 }is disconnected. A vertex of ^{0 }which is not a cut-vertex is called a ^{0 }has as underlying topological space a closed disc whose boundary is a simple cycle containing

•

•

•

For cut-vertices

We call two triangles ^{0 }_{0},_{1},...,_{n} = ^{0}, such that for _{i−1} and _{i} are adjacent (^{0 }induced by a set of mutually triangle-connected triangles of ^{0 }a

^{0}

First of all, observe that, by deﬁnition, any (ﬁnite) triangle component

A single triangle, as the base case, has two positively and exactly one negatively oriented arc (wrt an interior point). Suppose now that after _{k}_{k}_{+}_{1} the following cases need to be considered:

• The new triangle is glued on at a single positive boundary arc only. In this case this positive boundary arc (which now is no longer on the boundary) is replaced by two positive boundary arcs in _{k}_{+1}, and _{k}_{+1} retains exactly one negative arc on its boundary;

• The new triangle is glued on at a single negative boundary arc only. In this case this negative boundary arc is replaced by a positive and a negative boundary arc in _{k}_{+}_{1}, and _{k}_{+1} still has exactly one negative arc on its boundary;

• The new triangle is glued to _{k}_{k}_{+}_{1} having only positive arcs in its boundary. This is impossible because it would lead to an oriented cycle in the boundary,

• The new triangle is glued to

• The new triangle is glued on at a single positive (negative) boundary arc, and its third vertex is also glued. However, then the new boundary would (topologically) not be a simple closed curve, as

Thus, in passing from _{k}_{k}_{+}_{1}, the new triangle can only be attached along a single edge on the boundary of _{k}_{k}_{+1}| is a closed topological disc because it is also a bounded region. The component _{k}_{+}_{1} then has one more arc on the boundary, which is positively oriented, and retains one negatively oriented arc. This satisﬁes the induction hypothesis, so we have established that |

Note that if a ﬁnite triangle component

We can now give a dual version of Theorem 6.1, which is not formulated in terms of a precise coloring

^{∞}

^{∞}

^{0 }

For all examples shown in this section, we still implicitly assume that the global upstairs direction is mathematically positive (

A class of base tilings which reproduce the examples mentioned in ^{3}) and (4.8^{2}). Recall that the uniform tilings of the plane are precisely the vertex-transitive tilings by regular convex polygons. Recall further that (_{1},_{2},...,_{k}) is the standard notation for a uniform tiling of the plane where each vertex is cyclically surrounded by an _{1}-gon, _{2}-gon, and so forth, in that order (compare [

Decorated duals of the uniform base tilings (6^{3}) and (4.8^{2}) (

More interesting things happen when the base tiling is not 3-colorable, or if one deliberately uses more than three colors or shift lengths (possibly inﬁnitely many). In this case, the conditions on the neighborhood symbol have signiﬁcant implications, unlike in the previous case. In the decorated dual base tiling we can now see longer paths and possibly nonempty triangle components forming.

Barrel pseudotilings arise from the information encoded in a coloring of (6^{3}) with four (^{2}) with eleven colors (^{2}) (

We conclude the paper with some remarks about barrel pseudotilings and suggestions for further investigation in more general settings. In

We have made many assumptions and chosen the deﬁnitions in this article in a certain way. First, we require that (ﬁnite) barrels stem from a

Second, we selected the deﬁnitions so as to produce structures which are realizations of abstract rank 4 polytopes. Several very similarly arising structures have been discarded on the grounds of not being associated with an abstract rank 4 polytope. Nevertheless, it may be of interest to chemists or crystallographers to identify the corresponding 4-valent atomic networks.

This patch with base (6^{3}) is not part of a barrel pseudotiling, as the face structure of the object does not correspond to an abstract 4-polytope.

In order to further explore the connection with abstract polytopes, one could use a similar construction not only on tilings of the plane, but also on polyhedral maps. This would not ﬁt as nicely into

The author would like to thank Egon Schulte for his careful reading and many helpful suggestions. Also, the author would like to thank the two anonymous referees for their insightful comments which led to the improvement of this paper.