Frieze Patterns: Approaches with Group Theory and Combinatorics

Frieze Patterns: Approaches with Group Theory and Combinatorics Margaret McGuire December 4, 2017

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Introduction In “Classifying Frieze Patterns without Using Groups,� sarah-marie belcastro and

Thomas C. Hull present a proof of the fact that there are exactly seven different types of symmetry combinations that can be found in frieze patterns [6]. Here we prove there are seven different frieze patterns using group theory, and then show how Hull and belcastro prove there are seven types of symmetries in frieze patterns using combinatorics. These two proofs are of interest to compare as they provide slightly different justifications for the same conclusion, the combinatorics proof drawing on images as proof because it does not introduce group theory, and the group theoretic proof drawing on group theoretic axioms to justify claims. While we find the group theoretic approach to be more rigorous as it does not rely as heavily on images as proof, we see the value in having a combinatorial approach in circumstances where group theory should not be introduced, such as in the context of an introductory geometry class.

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Getting Started Before we discuss the proofs, we will begin by defining essential terms and devel-

oping a framework of thinking about symmetries in friezes.

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Friezes

Frieze patterns are traditionally found on entablatures of classical architecture, but also appear in many other forms, such as at the edges of carpets, on bracelets, on the tops of wallpapers, or anywhere else where a repeating band of decoration is found [6] [3]. In the case of mathematical friezes, the pattern continues infinitely.

Figure 1: This is an example of a frieze pattern that we could imagine continuing indefinitely to the left and right. This particular example comes from one of M.C. Escher’s drawings of the Alhambra [7].

In the case of mathematics, there are certain types of friezes which we do not consider, namely those with no symmetry. A non-example of a frieze pattern is included here for clarity.

Figure 2: While this is a frieze, the main subject matter does not contain a pattern, and therefore is not the kind of frieze we discuss in this paper [5].

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2.2

Symmetry Groups

A symmetry group is a group specific to an object, where the elements of the group are functions that render the object invariant (the operation of the group is function composition)[3]. By “render invariant� we mean that applying the function to the object means that the result is identical to the starting object. The functions in the symmetry group are isometries. Isometries are transformations that preserve distance relationships within an object; examples of isometries are translations, reflections, rotations, and glide reflections, whereas non-isometric funtions dilate, shrink, skew or distort an object. Examples of non-isometric functions are summarized in Figure 3.

Figure 3: Transforming a frieze pattern with a non-isometric function leads to a product that is non-superimposable on the original frieze pattern. As such, these transformations are not in the symmetry group of friezes.

2.3

Types of Isometric Functions

We have alluded already to the four types of isometric functions found in the symmetry group of frieze patterns; they are translations, rotations, reflections and

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glide reflections. By definition, all symmetry groups of frieze patterns contain translational symmetry, otherwise they fall into the category of friezes we describe in Figure 2, those that we do not consider in mathematics.

In the case of rotations, the only possible rotation that renders the original frieze invariant is a half turn or a 180 degree rotation [6]. To see why, consider the following illustration:

Figure 4: A non-180 degree turn does not return the frieze to its original position.

Figure 4 demonstrates that any rotation function in the symmetry group must be a half turn, as otherwise the rotation changes the location of the frieze in space.

Now we investigate the possible types of reflections a frieze can have.

Figure 5: There are two kinds of mirror lines about which we can take a reflection of a frieze: a vertical line, or a horizontal line.

Figure 5 shows two ways in which we can reflect a frieze pattern to yield the same frieze pattern again. There are many vertical lines we may reflect the frieze about to yield the same frieze, but there is only one horizontal mirror line that renders the frieze invariant.

We now turn to glide reflections. Glide reflections are defined broadly as translations on a mirror line followed by a reflection about that mirror line [6]. In the case 4

of frieze patterns, the only glide reflection possible is a translation about a horizontal line, followed by a reflection about that horizontal line. Glide reflections about any other mirror line would force us to translate the frieze in some direction other than the horizontal, thus the frieze would not not be in the same position after applying the transformation function.

Figure 6: Glide reflections involve translating a piece of the frieze pattern half a unit to the left or right, and reflecting it about a horizontal line through the frieze.

The green portion indicates what happens after performing the 1/2 unit translation, but before the reflection is performed.

This completes our classification of the types of isometric functions found in the symmetry groups of frieze patterns. We now turn to two proofs about the seven types of frieze patterns.

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A Group Theory Proof of the Seven Frieze Types Hull and belcastro allude to a proof that there are seven distinct types of frieze

patterns in their paper, citing Armstrong, Farmer and Gallian as sources. Armstrong and Gallian discuss frieze groups and show there are at least seven types of friezes, but do not appear to provide complete proofs for why these are the only possible types of frieze groups. Furthermore, while Farmer provides tasks to direct the construction of such a proof, the text does not offer a complete proof for the fact that there are only seven types of friezes [2]. Drawing on the ideas in these sources, we provide what we believe to be a more complete proof via group theory.

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Claim: There are exactly seven frieze patterns, and thus exactly seven possible frieze symmetry groups.

Proof. To begin, we discuss the notation used in this proof. In proving the seven possible frieze symmetry groups, we refer to the symmetry groups according to the possible elements in the symmetry groups, t, r, v, h, and g. That is, if a frieze has translational and rotational symmetry, then that frieze’s symmetry group contains a translation function and a rotation function, and we denote this symmetry group tr. We might also say for example “applying v,” by this we mean applying a vertical reflection to the frieze.

Without knowing the restrictions on the combinations of possible functions in the symmetry group, we know there could be as many as sixteen different symmetry groups, and we list them here: t, tr, tv, th, tg, trv, trh, trg, tvh, tvg, thg, trvh, trvg, tvhg, trhg, and thrvg. We seek to construct rules that will limit the possible number of symmetry groups to seven. We will propose these rules one at a time, proving each as we go, and indicate which possible combinations they rule out.

Rule 1. Given a symmetry group contains t, h and v, it must also have r.

Proof. Page 21 of Farmer’s Groups and Symmetry illustrates a key fact needed to show this rule holds true. The fact states that the composition of two reflections whose mirror lines intersect is equivalent to a rotation where the rotocenter (the pivot point about which rotation is performed) is at the point of intersection of the two mirror lines. To understand this, consider the following diagram:

Figure 7: P is reflected twice, but the end result is identical to a rotation.

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Here the letter P is reflected about a red mirror line, then about a blue mirror line. Then the P is rotated θ degrees about the intersection point of the red and blue lines to yield the same net transformation. Now since this fact holds for all mirror lines that intersect, we can imagine two mirror lines, one a horizontal mirror line, and one a vertical mirror line, on a frieze pattern. These mirror lines must intersect, and they do so in the middle of the frieze pattern. Thus there must be a rotocenter in the middle of the frieze pattern. Furthermore we know the rotation will yield the pattern invariant because if we perform v or h separately the pattern is the same, and if we perform v followed by h (v ◦ h) or vice versa, the pattern is the same. Thus since we know r = h ◦ v = v ◦ h, we know that a rotational symmetry must exist in this circumstance.

This rule eliminates thv and thvg.

Rule 2. Given a symmetry group with t and h, we must have g.

Proof. A glide reflection is defined as a translation along a horizontal mirror line followed by a reflection across that mirror line (though order does not matter, we could perform the reflection first instead). In other words, we can write g as t ◦ h or h ◦ t. Thus whenever we have t and h in our symmetry group, then since we have a group, the group must be closed under function composition. This means h ◦ t and t ◦ h are in the group, thus g must also be in the group.

This rule eliminates th, thr and thvr.

Rule 3. Given t, v, and r, we must have g.

Proof. We know that r must always be a rotocenter within the middle of the frieze pattern and that it must be a half turn, otherwise, the rotation would not yield the pattern invariant. We also know that the rotocenter is a point 7

of intersection of two mirror lines, one of which we already know must be v.

Given that we know the rotation is a half turn and we already know one of the mirror lines, we are forced into requiring the second mirror line to be the horizontal mirror line (if we did not know that the rotation was a half turn, there are infinitely many possible second mirror lines to choose). The argument is illustrated pictorially in Figure 8.

Figure 8: The left diagram shows the location of a rotocenter and the rotation of 180 degrees performed about the rotocenter. On the right is the known information about the existence of a vertical reflection. The ghost R illustrates the final location of the R after the second reflection. The only possible reflection that accomplishes the goal is a horizontal reflection.

This eliminates tvr.

Rule 4. Given t, r, and v, we must have g.

Proof. We will use a constructivist approach for this proof. We pick a shape that has vertical symmetry, such as a U shape. Then we construct a frieze pattern using rotations only. This yields the following pattern:

Figure 9: Vertical mirror lines are illustrated with red lines and rotocenters are marked in green. When we construct a pattern with rotational and vertical reflection symmetry, we discover that a glide reflection also emerges from the pattern.

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We made no assumptions about there being a glide reflection in the pattern, and yet it appears in the final pattern. Repeating this process with other shapes with vertical symmetry yields the same result. Thus we conclude that if our symmetry group contains t, r and v, it must also have g.

This eliminates tvg.

Rule 5. Given t, r, and g, we must have v.

Proof. We will use a similar constructivist approach for this proof. We pick a shape with no symmetry, such as R. Then we rotate the shape by a half turn, then we take the original shape and its rotation together to represent a single unit of the frieze. Then we perform only glide rfelctions on that unit to construct a frieze. This process is illustrated in Figure 10.

Figure 10: When we construct a frieze using only rotational and glide reflection symmetry, vertical symmetry emerges.

We see that when we perform r and g, v always emerges.

This eliminates trg and thrg.

Using the rules we have proved to be true, we can eliminate th, trv, trg, tvg, thv, thg, trvh, tvhg and trhg. This leaves t, tr, tv, tg, thg, trvg and thvrg as the only seven distinct frieze groups.

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A Combinatorics Approach Hull and belcastro present what they refer to as a combinatorial proof of the fact

that there are seven frieze pattern types. They do not employ group theory ideas such as closure or symmetry groups. For the sake of brevity we do not describe the entire proof in detail here, instead we provide an outline, pointing out the structural differences compared to our group theoretical proof.

Hull and belcastro begin by defining each type of symmetry t, r, h, v and g. Then they determine the number of possible combinations of these symmetries. They use a counting argument here, claiming that given translation must be one of the symmetry types, then there are 4 other symmetry types we might choose, and we can choose,

0, 1, 2, 3, or all 4 of these symmetries. Thus there are 40 + 41 + 42 + 43 + 44 = 16 combinations. Alternatively, we can choose to include or exclude each type of symmetry, meaning there are 24 = 16 possible combinations.

After determining the possible combinations, belcastro and Hull proceed by a constructivist approach similar to what we used in proving rules 4 and 5. They consider each type of symmetry, r, h, v, and g individually (of course t always comes along), and construct frieze patterns on the basis of a single symmetry. Then they inspect the resulting frieze to determine if other symmetries emerge. They find that r, v and g produced no new symmetries, but that h also produces g. Then they consider combinations of two symmetry types, again constructing friezes and investigating the resulting patterns for emerging symmetries. Here they find that hv gives r, hr gives g and v, vr gives h and g, vr gives g, vg gives r, and rg gives v. They summarize their results in a table (reproduced below), showing how these constructions and the rules derived from them show that only seven frieze pattern types are possible.

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Figure 11: The final list, showing what combinations have been eliminated, leaving seven remaining. t is omitted for all but the first for simplicity. [6].

Hull and belcastro’s proof relies heavily on the use of images and constructions as evidence that some combinations of symmetries cannot exist, while a group theoretic proof attempts to use group theory axioms to provide justification for rules that eliminate possible combinations. Admittedly, some constructivist techniques entered into our group theoretical proof, but it may be possible to do away with these altogether.

In the future it is of interest to further develop the group theoretical proof in ways that would not rely at all on imagery or constructivist approaches.

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Conclusion In closing, we have provided the technical terminology to develop a framework for

understanding the group theoretic and combinatorial approaches to the proof that there are seven different frieze types, and then presented those proofs. We perceive the group theoretic proof to be more rigorous as it relies less heavily on images to make arguments. Ultimately Hull and belcastro developed their proof with the intention of presenting it to students with no group theory background and as such it seems reasonable and perhaps unavoidable that they would have to employ images as proof instead of appealing to group theory axioms as we have.

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References [1] M. A. Armstrong. Groups and Symmetry, pages 155–165. Springer-Verlag, New York, NY, 1982.

This is chapter 26 from a textbook about groups. At the beginning of the chapter is a description of the internationally recognized crystallography shorthand used to describe wallpaper pattern groups, which we wanted to discuss but did not have space to do so. We might include a description of what the shorthand means and how to use it in the definitions section of our paper. This chapter also appears to provide a formal proof of why there are 17 different wallpaper groups. Unfortunately it is an exhausting, casework-heavy proof that goes on for pages, and any detailed exposition of it would probably not lend itself well to our paper.

[2] D. W. Farmer. Groups and Symmetry: A Guide to Discovering Mathematics, pages 39–42. Springer-Verlag, New York, NY, 1963.

Chapter 4 contains a series of suggestions on how to construct a group theoretical proof for the fact that there are exactly seven distinct types of symmetry combinations in frieze patterns. We will draw upon this text in presenting our version of the group theoretical proof.

[3] Frank A. Farris. Creating Symmetry: The Artful Mathematics of Wallpaper Patterns, pages 50, 59,168. Princeton University Press, Princeton, NJ, 2015.

Farris gives some good definitions and shows that there are 7 distinct frieze types using an approach very similar if not identical to the combinatorial proof given by Belcastro and Hull, though Belcastro and Hull wrote it 13 years earlier. They are not cited in Farris’s book. Farris’s 12

book offers many extensions which we did not have time to discuss, including the naming conventions of frieze groups and color reversing frieze functions.

[4] J. A. Gallian. Contemporary Abstract Algebra, pages 469–475. Houghton Mifflin, Indianapolis, IN, 1998.

Chapter 28 of this text opens with an interesting observation: in most cases it is customary to view two isomorphic groups as the same group since we would not be able to distinguish between them algebraically.

But in the case of frieze groups, some are isomorphic but are geometrically very different. Demonstrating this fact in our paper with an example may be of interest. The source is also referenced as a source for the proof that there are seven distinct frieze types, however the proof is not well described or particularly rigorous. The focus seems to be on demonstrating that seven exist, but a rigorous proof of why other combinations cannot exist is not offered.

[5] The Courtauld Institute of Art. Art and architecture: Friezes (entablature components), relief, arc de triomphe de l’etoile. http://www.artandarchitecture.org. uk/images/full/45bff324862a0eeb15015900a7c1b3c42225cf1e.html.

Ac-

cessed: 2017-11-21.

This is the source for the non-frieze example. This image is used to show what kinds of friezes we do not discuss in the paper.

[6] s. m. belcastro and T. C. Hull. Classifyng Frieze Patterns without Using Groups.

The College Mathematics Journal, ss:93–98., 2002.

This article is the jumping off point for the rest of our paper. The article describes a combinatorial proof that there are exactly seven distinct 13

types of symmetry combinations in frieze patterns. It references several sources we will use in writing up a proof of the seven distinct types of symmetry combinations using Group Theory.

[7] Brian Wichmann. The World of Patterns. World Scientific Publishing, 2001.

This book is only 23 pages and contains instructions on how to use a CD ROM that contains over 4,000 tiling patterns. These tiling patterns are a great resource to use in the discussion of different types of friezes, or other examples, such as showing what functions are not isometries.

The pattern used repeatedly in this paper (Figure 1) is from the World of Patterns.

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