Computer Graphics and Geometric Ornamental Design

Computer Graphics and Geometric Ornamental Design
Craig S. Kaplan
Department of Computer Science and Engineering
University of Washington
[email protected]
December 29, 2001
Horror Vacui: A tendency, sometimes characterized as medieval or primitive, to fill all the
available pictorial space with decorative or other motifs, as if “afraid of a vacuum.” [10]
1 Introduction
This paper is about the expression of the basic impulse of horror vacui in the digital age. The urge to
decorate, to add pattern to the objects around us, lies dormant in our world, baked into the featureless walls
of the modern cityscape. Horror vacui has not been evolved away, it has been placed in deep storage, an
embarrassment to us in a time of high speed, short product cycles, and the need for instantaneous access to
information.
Throughout all of history and around the world humans have used ornament to embellish their buildings,
their tools, their belongings, and themselves [38]. Ornament gives the world perspective and personality at
a human scale. All these opportunities for decoration still exist, and are joined by a new class of artifacts,
those that exist only as information.
The relatively new field of computer graphics has given us the ability to exploit those opportunities with
great speed and precision. Furthermore, in the past hundred years, our understanding of the mathematical structure of ornament has flourished. The tools of modern geometry help us make sense of historical
ornament and create new designs. And yet, except for a small number of deliberate forays into computer
generated ornament, few attempts have been made to examine the fusion of computer graphics, symmetry
and ornamental design.
1
In this paper, I will attempt to rectify the situation by showing the various ways that computer graphics has
already interacted with symmetry and ornamental design, and suggesting directions in which the state of the
art can be extended.
1.1 Overview
I begin with an introduction to the modern theory of symmetry, including an enumeration of the symmetry
classes important for ornamental design (Section 2). I end the section on symmetry with evidence for its
importance in the judgment of the aesthetic quality of an object. Requisite background in the history and
classification of ornament is then presented, along with reasons why symmetry figures so prominently in
ornament (Section 3). Next, I give examples of how computer graphics has been applied to the studies of
symmetry and ornament separately (Section 4). I then focus on the convergence of all three fields (Section 5), presenting specific cases where computer graphics has been applied to the creation of geometric
ornament: Celtic knotwork, Islamic star patterns, tilings, and dynamical systems. Finally, I end with ideas
for extensions to these cases and other ideas for future work (Section 6).
2 Symmetry
A modern approach to the analysis and creation of ornament must begin with a study of symmetry. The
conception of symmetry has evolved in two centuries from an informal sense of balance into a precise
mathematical notion expressing a kind of structured redundancy. The mathematics of symmetry is readily
reduced to algorithmic form, meaning that a computer can be made to carry out the repetition inherent in
this redundancy and allow the human more time to create.
Most ornament possesses some degree of symmetry. By applying the mathematical theory to historical
instances of ornament, we can find ways to recreate the originals, generate new designs, and even invent
entirely new styles of decorative art.
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2.1 The formalization of symmetry
The original conception of symmetry, as reflected by the dictionary definition, is expressed with words such
as beauty, balance, and harmony. The word was and still is used to refer to a balance of components in a
whole.
The contemporary non-scientific usage of the word, as Weyl points out, refers to an object whose left and
right halves correspond through reflection in a mirror [58]. Thus a human figure, or a balance scale measuring equal weights, may be said to possess symmetry.
In light of the formal definition of symmetry to come, we qualify the particular symmetry described above
as “bilateral symmetry”. Bilateral symmetry is certainly a familiar experience in the world around us; it is
found in the shapes of most higher animals. The prevalence of bilateral symmetry can be explained in terms
of the body’s response to its environment. Whereas gravity dictates specialization of an animal from top
to bottom and locomotion engenders differentiation between front and back, the world defines no intrinsic
preference for left or right [58, p.27]. An animal must move just as easily to the left as to the right, resulting
in equal external structure on each side. Indeed, lower life forms whose structure is not as subject to the
exigencies of gravity and linear locomotion tend towards more circular or spherical shapes.
Let us regard the mirror of bilateral symmetry as a reflection through a plane in space. Saying that the
mirror reconstructs half of an object from the other half is equivalent to saying that the reflection maps the
entire object onto itself. We formalize the notion of symmetry by noting two properties of this reflection. It
preserves the structure of space, just as a (flat) mirror preserves the shapes of objects, and it maps the object
onto itself, allowing us to think of its two halves as having “the same shape”. By generalizing from these
two properties, we will arrive at a formal definition of symmetry.
We use the rigid motions as the set of transformations that preserve the structure of space. The rigid motions
are simply those transformations of space that preserve distance. If the plane is represented by a sheet of
paper lying on a table, the rigid motions of the plane can be thought of as all movements of the paper after
which it is still flat on the table. The paper can slide around and turn. It can also be flipped over and laid flat
on the table again. No matter how these motions are carried out, every rigid motion in the plane is one of five
kinds [24, Chap. 9]: the identity, which does nothing, a translation along a vector, a rotation about a point,
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Figure 1 Examples of the seven kinds of frieze patterns.
a reflection across a line, or a glide reflection, a combination of a reflection with a translation parallel to the
line of reflection. These rigid motions are all affine; each is representable as the sum of a linear transform
and a translation.
A symmetry of a set
T
is a rigid motion that maps
T
onto itself. Here, we let what was informally called
an object or figure be represented by an appropriate subset T of some space. A symmetry is a property of a
set, not of the space, and a single set may have many symmetries. Since the identity is a rigid motion, it is a
symmetry of every set; a set which has at least one other symmetry is called symmetric.
2.2 Symmetry groups
For any object T , the set S (T ) of all the symmetries of T has a natural group structure through composition
of rigid motions.
S (T )
is therefore called the symmetry group of T .
Although there are infinitely many symmetry groups, very few fundamentally different groups can arise
from the ornamental patterns we will be considering. The groups can be classified by observing whether
they contain no translations, only parallel translations, or two non-parallel translations.
A figure whose symmetry group has no translations is called a finite figure. Every such group is either
cyclic, the symmetry group of an n-armed swastika, or dihedral, the group of the regular n-gon. When
all the translational symmetries of a figure are parallel, the figure is called a frieze pattern. There are only
seven distinct frieze groups, shown in Figure 1. Finally, when a figure has translational symmetries in two
non-parallel directions, it is called a wallpaper pattern, a all-over pattern, or simply periodic. There are
exactly seventeen wallpaper symmetry groups, a fact proven by the Russian Crystallographer Federov at the
end of the nineteenth century [25, p. 55]. The seventeen wallpaper groups are shown in Figure 2.
A fundamental region (or fundamental domain) of a planar symmetry group is a shape
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F
such that the
Figure 2 Examples of the seventeen kinds of wallpaper patterns [52, p. 157].
images of F under all the symmetries from the group cover space exactly without gaps or overlaps. As was
said, a symmetry group implies a kind of structured redundancy; a fundamental region is a set that contains
exactly one copy of the non-redundant information of a symmetric figure. Finite and frieze groups have
infinite fundamental regions; wallpaper groups have finite regions.
2.3 Extended forms of symmetry
The definition of symmetry can be further generalized in various ways by extending the ideas of preserving
the shape of space and mapping an object onto itself. What follows are some of the generalizations that will
be relevant to ornamental design. Other extensions will be mentioned later, as I discuss ornamental styles
that use them.
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Figure 3 An example of a pattern with simple colour symmetry. In this case, the rigid motion mapping any triangle onto any
oppositely-coloured triangle is an antisymmetry of the pattern.
2.3.1
Figure 4 A pattern with similarity-based
symmetry.
Antisymmetry and colour symmetry
Symmetric patterns are often drawn in multiple colours, which we can express formally by assigning a
colour (a formal symbol) to every point in the pattern. In a strict sense these colours can break symmetries;
motions that would be symmetries in the uncoloured sense can map a point onto a differently coloured point,
negating the sense in which these points correspond.
Colour symmetry permits two motifs with the same shape but different colours to be considered “congruent”
in an extended sense. A colour symmetry is a symmetry that additionally permutes the colours of the pattern.
Thus two differently coloured motifs may be brought into correspondence, provided the motion involved
maps other motifs of the same colours consistently. In the case that there are only two colours, we may
speak of a symmetry that exchanges colours as an antisymmetry. Figure 3 gives an example of a pattern
with antisymmetry.
Colour symmetry has been used extensively as a tool in mathematics, physics, chemistry, and ornamental
design [52, 25, 57].
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Figure 5 A periodic hyperbolic pattern (with symmetry group [7; 3]) projected into the plane via the
Poincaré embedding [31]. The projection masks the fact that every truncated heptagon in the picture has
the same shape.
2.3.2
Similarity
In the plane, a uniform dilation can be interpreted as preserving shape. By combining the rigid motions
with dilations, we obtain an extended set of transforms called similarities, which can take the place of rigid
motions in the definition of symmetry. Symmetry groups including similarities have been well studied [25,
p. 520], and patterns with similarity symmetry often lead to dramatic designs like the one in Figure 4.
2.3.3
Hyperbolic symmetry
One of the greatest revolutions in the entire history of science was the discovery of non-Euclidean geometry,
overturning millennia of dogmatic belief in the “truth” of the geometry of Euclid [24]. Today, non-Euclidean
geometry is a commonplace mathematical and physical tool, even though less than two centuries ago Kant
declared the very idea “inconceivable” [24, p. 182].
By negating the so-called “parallel postulate” of Euclid, we obtain either hyperbolic (or Lobaschevskian)
geometry, or spherical geometry. I will discuss hyperbolic geometry, which presents more opportunities for
ornamental design.
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The hyperbolic plane is founded on the axiom that two lines intersecting at a point may both be parallel
to some other line, a nonsensical situation in the Euclidean plane. This assumption leads to many unusual
properties. The interior angles of a triangle add up to strictly less than
180Æ .
Rectangles cannot exist. We
can still, however, define a concept of distance between points, and from that rigid motions. From rigid
motions we can develop analogues of the (Euclidean) planar symmetry groups described above. However,
the abundance of space that allows multiple parallels to exist also yields infinitely many periodic symmetry
groups. For any
p
and
q
satisfying
(p
2)(q
2)
>
4,
there is a group
[p; q ],
the group corresponding
to the regular tiling by p-gons, meeting q around a vertex [17]. The other hyperbolic symmetry groups are
subgroups of the [p; q ] groups [14, 17].
The hyperbolic plane is realized as an ornamental design by embedding it in the Euclidean plane. The
embedding will necessarily distort objects, but as long as the distortion is systematic and shapes are still
recognizable the result has aesthetic value.
The embedding most suited to ornamental design is the Poincaré model. In this model, points in the hyperbolic plane are represented by points inside the Euclidean unit circle. Hyperbolic lines are circular arcs that
cut the unit circle at right angles – their straightness is clearly not preserved! Hyperbolic rigid motions can
be passed through this embedding as well; reflections across lines become inversions in circles.
Since the model embeds the infinite hyperbolic plane inside the unit circle, it distorts distances considerably,
with objects getting arbitrarily small near the boundary. But angles (and hence shape, to some degree) are
preserved, meaning that congruence between copies of a motif can still be recognized.
2.4 Why symmetry?
Before simply mandating that ornament should be symmetric, it is important to address the question of why
symmetry matters at all. Symmetry is, after all, just a highly structured form of repetition. Why should
repetition create such a positive aesthetic response in a viewer?
The answer may lie in the perceptual process itself. Psychoaesthetics is an experimental science that attempts to quantify our aesthetic response to visual stimuli. Researchers in psychoaesthetics speculate that
our aesthetic judgment of a visual stimulus derives from the arousal created and sustained by the experience
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of exploring and assimilating the stimulus. Moreover, they claim that the intensity and the duration of the
response is tied to the complexity of the scene being viewed. Since a symmetric design contains less information than an asymmetric one, they would argue that the viewer loses interest sooner, and thus has a less
positive aesthetic response.
Locher and Nodine [45] conducted a series of experiments to test these claims. They found that the presence
of symmetry unquestionably affects our perception of a scene. The eye detects the presence of symmetry at
a glance, even before we become conscious of it. Also, the act of viewing is optimized to take symmetry
into account. By tracking eye fixations during viewing of a scene, they showed that the eye will explore
only non-redundant parts (the fundamental region) of that scene.
However, in another experiment, they show that an increase in symmetry is met with a reduction in arousal,
as predicted by psychoaesthetics. When asked to rate appreciation of works of art, subjects rated asymmetric
scenes most favourably and symmetric scenes decreasingly favourably as symmetry increased. While this
result might appear to bode poorly for the effectiveness of symmetric ornament, mitigating factors should
be considered. For instance, the abstract art they used to test the change in arousal with symmetry might
have suffered as they were altered to be symmetric. The aesthetic value of art comes from structure and
composition, and not just symmetry.
Another issue is the nature of the symmetries tested. Motivated by the importance of bilateral symmetry
in biology, Locher and Nodine examined only figures with simple mirror symmetry and the orthogonal
mirror symmetry of the rectangle. We can conclude nothing about our aptitude for detecting other kinds of
symmetries.
On the other hand, the reduction in arousal with symmetry might be appropriate for the purposes of ornamental design. In many cases, particularly in an architectural setting, the goal of ornament is to please the
eye without unduly distracting it. Locher and Nodine support this claim, mentioning that as complexity of a
scene increases, the rise in arousal “is pleasurable provided the increase is not enough to drive arousal into
an upper range which is aversive and unpleasant.” [45, p. 482]
Other research supports the correlation between symmetry and perceived goodness. In the limited domain
of points in a grid, Howe [34] shows that subjective ratings of goodness correlated precisely with the degree
of symmetry present. In a similar domain, Szilagyi and Baird [55] found that subjects preferred to arrange
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points symmetrically in a grid. In their recent review of the perception of symmetry, Møller and Swaddle
simply say, “humans find symmetrical objects more aesthetically pleasing than asymmetric objects.” [47]
I believe the best reason of all is that a symmetric pattern invites the viewer into a “visual puzzle”. We sense
the structure on an unconscious level, and subconsciously struggle to determine the rules underlying that
structure. As Shubnikov and Koptsik say, “The esthetic effects resulting from the symmetry (or other law
of composition) of an object in our opinion lies in the psychic process associated with the discovery of its
laws.” [52, p. 7]
We should not attempt to use the evidence presented in this section as a complete justification for symmetry
in art and ornament. But these experiments reveal that we do have some hard-wired reaction to symmetry, a
reaction that affects our perception of the world. Rather than relying on purely philosophical arguments for
symmetry in art, we can present a physiological argument, backed by experimental evidence. This evidence
provides us with a partial explanation for the historical importance of symmetry in ornament, and some
confidence in its continued aesthetic value.
3 Ornament
Ornament as a form of human expression is older than recorded history. Decorated human artifacts have been
found dating back to the stone age. The extensive history of ornament and the development of different styles
around the world are well summarized by example in encyclopedic works by Jones [38] and Racinet [50].
Christie [9] goes into more detail about the origins of ornament and the various forms it may take. Most
recently, Wong et al. revisit the nature of ornament in the context of computer-generated floral patterns [59].
Despite the body of literature on the subject, few authors seem willing to provide a well-formed definition of
ornament. In the propositions that open Jones’s The Grammar of Ornament, we find many comments on the
structure and common features of ornament, but no definition. Racinet promises to teach “more by example
than by precept.” [50, p. 13] The definition given by Wong et al., “the aesthetic enrichment of the surfaces
of man-made objects in ways not directly contributing to their functional utility,” [59, p. 425] succeeds in
describing a large part of ornament, though it is inadequate in some ways. I would extend their definition
to allow ornament to exist independently of objects to embellish, so that a design painted on a canvas or
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The applications [region of elaboration] of ornament:
The elements [motifs] of ornamental designs:
A. to bands, which have finite thickness in one
dimension and are infinitely repeating in the
other;
1. geometrical elements, such as lines, polygons,
ovals and the like;
2. natural forms, which can be further classified as
B. to half-open borders, which are tightly constrained along one or more edges, but open in
other directions;
1. plants,
2. animal/human forms
C. to panels, which are arbitrary bounded regions
of the plane; and
3. physiographic features; and
3. artificial objects, such as shields, ribbons, or
torches.
D. to the open plane, in which the ornament typically becomes a repeating pattern.
Figure 6 The taxonomy of ornament as presented by Wong et al. [59].
displayed on a computer screen may still be considered ornament.
However, we need not belabour the point of defining ornament. Like the rest of art, ornament will ever evade
definition through exceptions and innovation. It is more instructive to accept Racinet’s gambit and learn by
example. What matters here is that the artifacts and techniques of the digital age still be as much ornament
as the decorations of the past.
Wong et al. provide two useful tools for the modern study of ornament, coalesced and condensed from
earlier treatises. The first is a taxonomy that divides up the vast domain of ornamental designs according
to the kinds of motifs that make up a design and the kind of region over which the design is elaborated.
The taxonomy is reproduced in Figure 6. Note the correspondence from region types A, C, and D to frieze
patterns, finite patterns, and wallpaper patterns, respectively.
The second tool they provide is a discussion of the basic principles of ornamental design. They single out the
sense of order as the defining characteristic of all ornament, and describe how order follows from the three
principles of repetition, balance and conformation to geometric constraints. Later, in a section dedicated to
principles relevant to floral ornament, they also talk about conventionalization, the development of a stylized,
abstract representation of a real-world form. While conventionalization is important for floral ornament, it
comes into play in the rest of ornament as well. Many motifs that seem totally abstract today in fact evolved
over centuries from the shapes of animals and other natural forms.
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3.1 Geometric ornament
The title of this paper qualifies the ornamental design being studied as “geometric”. There are two distinct
and equally important connotations associated with this word. The first refers to the appearance of the
ornament, the second to the manner of construction.
The first case is given by category 1 in the taxonomy of the previous section. According to this interpretation, geometric ornamental designs are made up from plainly geometric structures such as lines, circles,
and polygons. Designs consisting of geometric motifs are an appropriate starting point for the study of
computer-generated ornament, as computers are endowed with the precision necessary to execute nearperfect geometric motifs.
The second case is more concerned with the arrangement of motifs in a design than with the motifs themselves. A geometric ornamental design is laid out according to some order or scheme derived from geometry.
Jones says (somewhat inscrutably), “All ornament should be based upon a geometrical construction.” [38,
p. 5] Wong et al. give a number of geometric constraints to which ornament typically adheres.
To ground Jones’s proposition in modern geometry, I make the assumption that geometric ornament is symmetric ornament. Naturally, this association cannot be expected to reflect the entire body of ornament, some
of which is certainly asymmetrical. But the importance of symmetry in ornament is borne out by historical
artifacts and scholarship. Older texts lack the language of symmetry theory, and struggle to formalize the
“order” and “pattern” exhibited by ornament.
From the prevalence of symmetry in ornament, it would appear that situations where ornament is called for
are best served with symmetric designs. Why should this be so?
One answer stems from the perceptual studies described in Section 2.4. We are innately attuned to the
presence of symmetry in our environment, and its occurrence in ornament would be one source of a positive
aesthetic response. We demand a certain level of visual harmony from the environment in which we are
immersed daily, harmony that can be achieved in part by symmetrical embellishment of objects in that
environment.
A more obvious reason for symmetric ornament is economy of design and production. Ornament is often
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meant to cover large architectural surfaces such as the faces of buildings. The design and execution of
ornament over such a large area is a massive undertaking. It is simpler to design a single motif and build
the complete pattern out of symmetrically-placed copies of that motif. Also, it is less costly to manufacture
many copies of a single unit that expresses the motif.
Ornament predates (and, some believe, presages) the development of a formal theory of symmetry. Ancient
artisans used at best an intuitive understanding of the laws of symmetry. Nevertheless, they managed to
discover through trial and error all the discrete planar symmetry groups.
This does not imply that our modern mathematical methods are not applicable to the analysis of historic
ornament and the creation of new forms. One fruitful application of symmetry theory to ornament in an
anthropological setting is Washburn and Crowe’s Symmetries of Culture [57]. In it, they show that perception
and usage of symmetry are to some degree culturally-dependent processes. The planar symmetry groups are
used in different proportions by different cultures, and can in fact be used as a predictor of a decorated
artifact’s cultural heritage. Abas and Salman give an effective demonstration of this cultural dependence in
the context of Islamic ornament [2, p. 138].
Applying symmetry to the thoughtful creation of ornament, Shubnikov and Koptsik offer advice on the appropriate choice of an ornamental symmetry group for different environments [52, p. 155]. Each wallpaper
group evokes a sense of movement or stillness in certain directions. For example, they argue that a pattern with horizontal translations and no vertical reflections emphasize horizontal motion and are therefore
appropriate for corridors or subway tunnels.
4 Computer graphics
Before proceeding to examine the intersection of the three disciplines being studied here, I examine how
computer graphics has interacted with each of the other two separately. Techniques and ideas from these
two intersections can then be incorporated into the quest for computer-generated symmetric ornament.
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Figure 7 A frame from Not Knot [29], showing Gunn’s visualization software running on a non-Euclidean manifold.
Figure 8 Computer generated floral ornament from Wong et al. [59]
4.1 Computer graphics and symmetry
When computer graphics and symmetry are combined but ornamental design is excluded, we are left with
research in the field of mathematical visualization, where computer graphics is used as a tool to help explore
the nature of symmetry and the structure of symmetric objects. One recent example of computer graphics in
the service of symmetry is Gunn’s system for visualizing three-dimensional manifolds [28].
Manifold theory explores a generalization of the concept of “surface”. Manifolds arise naturally in the
context of ornamental symmetry, since the quotient of a space by a symmetry group is a manifold. The
quotient can then be examined to derive properties about the symmetry group (and vice versa).
Visualization of manifolds is an important but challenging mental task, particularly in the non-Euclidean
case. Computer graphics can assist the weary cortex by providing a window into the world of these exotic
objects. Gunn constructed an industrial-strength viewer of three-dimensional manifolds arising from symmetry groups, which was used to produce Not Knot [29], a classic of mathematical visualization. A sample
frame from Not Knot appears in Figure 7. His system provides a window on the “insider’s view” of a threedimensional manifold, a view that traces the effect of light traveling to a viewer living in the manifold. A
single object may be visible multiple times, and its size may be drastically distorted. These effects help to
reveal the structure of the manifold.
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Impressively, Gunn’s system runs at interactive rates, due to judicious use of the homogeneous coordinates
and transforms provided by most modern graphics libraries. As a by-product of unifying affine transforms
with perspective projection, graphics libraries can express all transforms in the projective general linear
group. This group includes homogeneous representations of Euclidean, hyperbolic, and spherical isometries,
and all necessary projections onto the viewplane1 .
When high quality output is desired, Gunn also provides a shading and lighting model for the different geometries [27], written in the RenderMan shading language. The shader takes into account the non-standard
notion of distance in the non-Euclidean cases.
4.2 Computer graphics and ornament
Computer graphics has had very little direct interaction with ornamental design. It is true that computer
graphics has been used informally in the creation of ornament, but few have taken the more deliberate
approach of attempting to understand the nature of ornamental design and how computer graphics might be
used to create new designs true to that nature.
As discussed in Section 3, Wong et al. provide a modern approach to the analysis and creation of ornament,
including a taxonomy by which ornament may be classified and a “field guide” for recognizing the common
features of designs. Subsequently, they develop a system capable of elaborating floral designs over finite
planar regions (category 2.1-C in their taxonomy).
Their approach combines conventionalization with a growth model, a synthetic method of distributing design
elements over the region of elaboration with an approximately uniform density. Growth is accomplished by
applying rules to extend the design from existing elements into currently empty parts of the panel. Beginning with a set of “seeds”, the algorithm iteratively applies growth rules until no more growth is possible.
The final design can then be rendered by providing explicit drawing code for the different elements, or by
exporting the elaborated geometry to a tool that can decorate the paths with skeletal strokes.
One of the advantages of their approach is that they eschew more traditional botanical growth models such
1
Note that he is forced to use the Klein model of the hyperbolic plane. The Poincaré model does not admit a projection via a
transform in the projective general linear group
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as L-systems. The most compelling reason they give is that L-systems are a powerful tool for modeling real
plants, which is exactly what floral ornament is not. There is no reason to believe that a simulation of the
biological process of growth should lead to attractive designs. They set a strong example for future research
in computer-generated ornament: conventionalization applies to the design process as well as individual
elements. The process should model the stylized result, not the real-world antecedent.
There are two major shortcomings to their technique. The first is the degree of user intervention and ingenuity required to create a successful design. In essence, the only automated part of their system is the decision
of where growth should be attempted. Designing a set of rules for one style involves careful planning. Furthermore, the user is completely in charge of the conventionalization process. They must choose a set of
elements and create stylized graphics for each (or rely on skeletal strokes). There is so much freedom in this
process that it is easy to wander out of the space of floral ornament.
The other missing piece is symmetry. In the paper, they present repetition as a principle of ornament. They
then refer the planar rigid motions as examples of repetition, suggesting that repetition is often carried out
via symmetry. Yet in their algorithm, growth is constrained only by availability of space and the boundaries
of the region of elaboration. The user may encode rules that explicitly (or accidentally) create symmetric
designs, as some of their examples demonstrate, but symmetry is not enforced.
To be sure, asymmetric ornament is plentiful and their results are new, highly successful instances of it.
Their designs would be quite appropriate as decoration for illuminated text or a small man-made artifact.
But in an architectural setting, where the design is to be applied over a wall or floor, the lack of overall
structure could conceivably have a deleterious effect on the viewer’s experience.
4.3 Putting it all together
In the second annual SIGGRAPH conference in 1975, Alexander presented a program for drawing patterns
on a plotter using the seventeen wallpaper groups [3]. Alexander’s work was quite possibly the very first
combination of computer graphics, symmetry, and ornament.
The gradual advance of technology in the past century has been met by a corresponding decline in the use of
ornament. Yet, based on what I have said so far, computers are ideally suited for ornamental design. They
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have the flexibility to enable a wide variety of styles and motifs, combined with the power to elaborate those
motifs over arbitrarily large regions. They can encode discrete structures like symmetry groups, and execute
those structures flawlessly. They can carry out forms of repetition tedious or difficult to do by hand, such as
the Poincaré projection of a hyperbolic pattern.
A software tool that generates ornament should obey certain constraints. Most importantly, such a tool
should attempt to reinvent the paintbrush and not the artist. Creative software should respect the boundaries
between the tool and the creator. It should allow the user to navigate spaces of possible designs quickly and
easily, and it should elaborate a design in a variety of rendering styles. Conversely, creative control (and the
feeling of control) should remain in the hands of the artist at all times.
It is also crucial to find an appropriate breadth of design space. An overly narrow tool presents the rigidity of
a library of clip art. An overly broad tool devolves into the complete generality of a paper and pencil, where
there are no constraints to keep the user within the bounds of a specific ornamental style. An appropriate
system will permit extended exploration, but enforce strong limits on the territory being explored.
5 Topics in ornament
In this section, I provide some details about a few different ornamental styles. Obviously, it would be
impossible to cover the complete range of ornamental design. Therefore, I choose to investigate ornamental
styles satisfying the following conditions:
(a) the style exhibits some form of two-dimensional symmetry;
(b) software tools have been constructed for exploring the style; and
(c) the style is the subject of continuing research.
Conditions (a) and (b) restrict us to the domain implied by the title of this paper, namely the intersection
of computer graphics, geometry, and ornamental design. Condition (c) helps ensure that the style has been
analyzed using the modern tools of symmetry theory, providing a rigorous framework for existing and future
software implementations.
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As is the case with ornament in general, I will not attempt to give a formal definition of each ornamental
style. At best, we can recognize some features common to many specimens of one style, and develop an
analysis or system to explore designs exhibiting those features. The result will be limited with respect to
the style as a whole. But we should not try to deconstruct a style completely; we cannot outpace the human
capacity for rule-breaking and innovation. Our goal, therefore, is to create some ornament rather than all
ornament.
5.1 Celtic knotwork
The art of the Celts was always non-representational and geometric [37]. With the arrival of Christianity
to their region in the middle of the first millennium C.E. came the development of the distinctive knotwork
patterns most strongly associated with the Celts. The knotwork appears carved into tombstones, etched into
personal items, and most prominently in illuminated manuscripts such as the Lindisfarne Gospels and the
Book of Kells. They are formed by collections of ribbons that weave alternately over and other each other
as they cross. Often, human and animal forms are intertwined with the knotwork, with ribbons becoming
limbs and hair.
Scholars of Celtic art agree that the origin of Celtic knotwork is plaitwork, the weave used in basketry [4].
Plaitwork is by no means Celtic; it appears in numerous ornamental traditions. The Celts learned to enrich
the ornamental possibilities of plaitwork by breaking crossovers and rejoining the bands systematically.
With an instance of Celtic knotwork, it is important to distinguish between the abstract design and the style
used to render that design. The design in this case is just a network of curves. The rendering styles are
numerous; many ideas may be extracted from study of the examples in Bain’s book [4] on the subject (see
Figure 10). Most importantly, the ribbons can be interlaced, cut at crossings to suggest that one ribbon is
passing over or under the other. In almost all finished designs, the choice is made so that every ribbon passes
alternately over and under at successive crossings. For well formed designs (such as the ones discussed here),
this choice can always be made consistently.
Cromwell [13], building on the earlier work of J. Romilly Allen, developed a simple mechanical process by
which a wide variety of knotwork designs may be constructed. Two rectangular lattices are drawn, with the
points of each lying at the centres of the rectangles formed by the other. When straight lines are drawn in
18
Figure 9 The relationship between the abstract
design for a Celtic knot and the final rendering.
Figure 10 Celtic knots can be executed in a number of different styles.
Figure 11 The steps in Cromwell’s process for constructing Celtic knotwork. A grid is specified, and break
markers are chosen. Then the abstract design is drawn conforming to the break markers. The ribbons are
thickened and the background is filled in to conceal construction lines. Finally, a rendering style is chosen
and executed.
the paths between the lattice points, a simple plaitwork results. To alter the basic design, break markers are
inserted. Each break marker connects adjacent points in one of the two lattices, and indicates a line through
which the ribbons may not pass. Instead, the two ribbons that otherwise would have crossed at that junction
turn back onto themselves, as shown.
To create a design, draw the lattices and choose a system of break markers. For a finite design, place a
border of markers around a region of the lattice. Begin to draw a plaitwork, but never pass through a break
marker; instead, connect the current ribbon to the other ribbon terminating on the same side of the marker.
The result is a complete design that can be rendered as described above. Figure 11 illustrates this process.
Focusing on frieze patterns, Cromwell then dissects the symmetry properties of Celtic knotwork. The abstract designs behind the patterns belong to one of the seven frieze groups. Furthermore, the frieze group is
19
completely determined by the pattern of break markers.
Celtic knotwork escapes somewhat from the plane, though. The interlacing of the ribbons endows the
picture with an infinitesimal thickness, allowing ribbons to pass over and under each other. To account for
the interlacing property, Cromwell discusses an extension to frieze symmetry which allows the pattern to
be turned over, reversing the sense of every crossing. In the language of Shubnikov and Koptsik [52, Chap.
5], he considers the symmetries of two-sided bands, where the plane is said to be nonpolar: its front and
back can both contain information. There are 24 purely two-sided band groups. Cromwell argues that by
construction, only ten of these can be the symmetry groups of interlaced Celtic friezes. He points out that
historical examples from the ten groups occur with different frequencies. This should not be surprising given
the thesis of Washburn and Crowe, though there is an initial mathematical argument explaining why some
symmetry groups would be preferred over others.
Cromwell’s work adapts readily to the computer generation of Celtic knotwork. Zongker [60] implemented
an interface that draws the lattices and lets the user place the break markers. His system then elaborates the
knot and renders it in one of several styles. Moreover, Cromwell gives a table linking the size of the lattice
and the frieze group of the break markers to the two-sided band group of the finished design. Using this
table, one can easily draw Celtic friezes with any desired symmetry group.
However, in actuality Celtic knotwork is not always based on a rectilinear framework. Cromwell mentions
that sometimes the ribbons are distorted for aesthetic purposes, an act that obscures the original lattices. Bain
gives numerous constructions of irregular Celtic knots (though unfortunately his construction techniques are
not always transparent).
Mercat [46] developed a more flexible technique that can be seen as an extension of Cromwell’s. The
rectangular lattices given above can be regarded as the vertices of a planar graph and its dual. Mercat’s
method starts with an arbitrary planar graph, and uses the graph and its dual to infer a Celtic knot. Just as
before, break markers can be inserted. Each intersection of an edge in the graph and an edge in the dual
graph is the location of a crossing. By breaking the passage of ribbons at a crossing, the knot is restructured
When the graph in Mercat’s construction is a grid of squares, Cromwell’s technique emerges as a special
case. The initial and final steps of the process are illustrated in Figure 12.
Another way to think about the drawing of Celtic knotwork is via a tile-based approach. The rectangles of
20
Figure 12 The initial and final steps in Mercat’s construction of Celtic knotwork [46].
Figure 13 An example of a Celtic knot fitted to an arbitrary curved outline [6].
one lattice in Cromwell’s technique can be taken as a grid. When the knot is subdivided by this grid, the
rectangles contain only a small number of distinct motifs. The motifs can be placed on tiles and used to
“stamp out” the final design, a process much simpler than tracing the paths of ribbons.
Many Celtic knot rendering programs, Zongker’s included, use this tile-based technique. Recently, Browne [6]
used an extended tile-based algorithm to fit Celtic knots to arbitrary outlines (letterforms in his case). The
technique works by filling the interior of a region with a tiling wherein tiles are as close as possible to squares
and equilateral triangles. Using a predefined set of tiles decorated with fragments of Celtic knotwork, he
assigns motifs to tiles in such a way that the fragments link up to form a continuous Celtic knotwork design. In some cases, the result bears a strong resemblance to the illuminated letters of the ancient Celtic
manuscripts (see Figure 13). Interestingly, Browne’s approach is certainly not the one used by the origi-
21
Figure 14 An example of square Kufic calligraphy [20, p. 32].
Figure 15 An example of an arabesque [20, p. 48].
Figure 16 An example of an Islamic star pattern
executed in the Zellij style [20, p. 126].
nal artisans. It is therefore particularly pleasing to see advanced techniques from computer graphics help
revitalize a decorative style that is all but forgotten.
5.2 Islamic star patterns
The artisans of the Islamic cultures of Asia, Africa, Europe and the Middle East were undoubtedly history’s
greatest creators of ornament. Most scholars state that their incredible innovation in non-representational
decoration was borne of a religious taboo against depiction of real-world forms. The truth may be more
complicated: Abas and Salman [2] believe that the geometric purity of Islamic ornamentation was at least
in part motivated by an attempt to find an abstract representation for God’s perfection. In either case, we
know that their efforts were highly successful. Historical and contemporary examples of their incredible
22
achievements litter the landscape from Spain to the Samarkand. Typical examples of Islamic ornament can
be found in Castera’s extensive survey of decorative art in Morocco [20].
Islamic ornament is composed of several different systems of decoration. Note that a given artifact can
be based on one system, the juxtaposition of several systems in close proximity, or even the complete
commingling of systems. Throughout, we encounter a masterful understanding of order and symmetry.
Calligraphy was raised to a high art by Islamic artisans. Inscriptions from the Qur’an cover the walls of
buildings. They invented several writing styles, of which the most remarkable was the square Kufic, wherein
writing was restricted to the “pixels” of a coarse square grid as in Figure 14. Despite these restrictions, they
still created symmetric arrangements of words. Amazingly, in some designs both figure and ground carry
copies of the words [9, p. 75].
Arabesques are stylized plant forms derived from even older depictions of plants such as the acanthus.
In Islamic culture, arabesques found a high point in elegance and grace. A sample arabesque appears in
Figure 15
It is the Islamic star pattern that is both the most remarkable and the most distinctively tied to their culture.
Star patterns like the one in Figure 16 adorn buildings from the ninth century C.E. to the present day. The
high point of the style (and perhaps of all ornament) is the fourteenth century Alhambra palace in Granada,
Spain [53].
A star pattern is typically composed of an arrangement of radially symmetric motifs such as star polygons [39, 25], with additional, usually bilaterally symmetric polygons filling the interstitial (interstellar?)
spaces. A common method of executing these patterns is as a mosaic of enameled terracotta tiles, a technique
known as “Zellij”. Often, the edges separating polygons are thickened and interlaced.
Unfortunately, the original techniques used to design star patterns are lost in history. The techniques were
the secret property of the craftsmen who used them, to be handed down from master to apprentice or else
forgotten. Over the centuries, many mathematicians and hobbyists have attempted to reconstruct the process
used by the original designers. As a result, we have a number of competing construction tools, each capable
of expressing some space of star patterns.
Grünbaum and Shephard apply symmetry theory to the study of Islamic star patterns [26]. They do not
23
Figure 17 The symmetry-based analysis of the design of an Islamic interlacing [26]. The design is shown
on the left, and the contents of one fundamental region on the right.
consider the layered symmetries of the interlaced patterns, as Cromwell does with Celtic knots. Instead,
they immediately disregard all details of the pattern’s execution and focus on the underlying graph structure,
which they call the pattern’s design.
To analyze a design, they determine its symmetry group and extract a fundamental region. The portion of
the design appearing in the fundamental region can then be regarded as a collection of short open or closed
paths. These paths can be used to establish a number of facts about the design as a whole.
In the examples they give, the designs are from the wallpaper groups p4m and p6m, where a fundamental
region is transformed into its neighbours via reflection. To ensure that degree-four vertices in the composed
design can be interpreted as two crossing strands, paths must “reflect” off of the boundaries of the fundamental region. They compare a path in the fundamental region with the trajectory of a billiard ball bouncing
off cushions formed by the edges of the region. Note that the simplicity of this billiard ball approach immediately suggests a software implementation that finds billiard ball paths in a fundamental region and draws
the design from the computed paths.
The work of Grünbaum and Shephard is limited in that the analysis and results are given only for two
of the wallpaper groups. Abas and Salman provide a more exhaustive set of results [2], applying the above
technique to a large collection of Islamic designs. More information is also furnished by Ostromoukhov [49],
24
who derives the mathematical machinery needed to carry out Grn̈baum and Shephard’s analysis on designs
belonging to any wallpaper group.
Although the symmetry-based approach is useful for analysing and reconstructing Islamic star patterns, it
does not account for the complete structure of a design. Usually, the designs have significantly more order
and repetition than is expressible using symmetry. The design can often be decomposed into units which
individually have dihedral symmetry of a high degree, though when assembled most of the symmetries are
lost. It seems likely that when designing star patterns, Islamic artisans visualized the process at a different
level than that of the fundamental region. Abas and Salman [1], criticizing the implausibly elaborate construction lines in early references such as Bourgoin’s Arabic Geometrical Pattern and Design [5], give a
number of simple techniques by which some designs may have actually been constructed. Castera [20, p.
92] presents a method based on forming skeletons of eight-pointed stars and hexagons.
The technique that I feel is most appropriate to designers of the time and most adaptable to a wide range of
patterns is that first proposed by Hankin [30]. His technique is based upon a tiling of the plane by regular
polygons, each polygon filled with an Islamic motif such as a star or a rosette.
In a recent paper [39], I implemented Hankin’s technique using the geometry of stars and rosettes as given
by Lee [43]. The user chooses from a library of tilings and decides what kind of Islamic motif to place
in each tile shape. By linking up the motifs and extending lines into unfilled regions in an obvious way,
typically Islamic designs can be constructed.
5.3 Tilings
Tilings are not the ornamental style of any one culture like the two previous examples. But the set of analyses
and construction techniques associated with tilings is particular enough that they deserve to be discussed as
a unit.
For our purposes, a tiling of the plane is a collection of topological disks that together cover the plane without
gaps or overlaps. Many of the tilings we see every day on floors or walls are made up of a small number of
shapes; in general, a tiling is monohedral if all tiles have the same shape, and multihedral otherwise.
Many ornamental styles may be interpreted as tilings under this definition. The design of an Islamic star
25
pattern is a polygonal tiling, and indeed is often executed with small terracotta tiles. The broken ribbons of
a Celtic knot and the spaces they delineate can be taken together as a tiling. However, in both of these cases
the interpretation adds little to our understanding of the style. In an ornamental tiling, the “visual puzzle”
is embedded in the perfect interlocking of a small number of tile shapes, and does not follow from some
higher-level act of interpretation.
As with symmetry, tilings have been the subject of informal exploration since antiquity, and examples of
great variety decorate ornament from around the world. Yet only recently has a systematic mathematical
theory emerged, built upon a rigorous infrastructure of topology, group theory, analysis, and combinatorics.
The definitive treatise on tiling theory is Grünbaum and Shephard’s Tilings and Patterns [25]. Since its
publication, progress has been made on several fronts, including the connection between tilings and manifold
theory [48] and the extended combinatoric encoding of all periodic tilings [35].
In parallel with the development of theories of symmetry and tilings came the rise of history’s greatest
creator of ornamental tilings, M. C. Escher. Escher was born in Holland a few years after Federov worked
out the three-dimensional crystallographic groups. He was captivated by the regular division of the plane,
especially as a means of expressing the concept of infinity through art [56]. Over his lifetime, Escher
developed a “layman’s theory” of tilings that he used to create over a hundred interlocking designs of human
and animal forms [51]. His work was featured in both artistic and ornamental settings, and even used as a
visual aid for mathematicians, physicists and chemists.
Escher’s work adapts readily to the precision of computer drawing. Indeed, his legacy has been successfully
carried into the computer age by hobbyists and researchers. Many enthusiasts simply take advantage of
the computer’s capacity for geometric precision and repetition, using commercial drawing packages to draw
tilings [12]. Chow used a family of periodic tilings called the Heesch tilings to create ornamental Escher-like
tilings interactively [7, 8]. TesselMania! [11] is a commercial software product based on Heesch Tilings that
offers a richer set of decorative features. I have developed a software library for encoding, manipulating,
and rendering isohedral tilings, a more sophisticated family of periodic tilings that subsumes the Heesch
tilings. The library was used as the basis for an Escherization [40] system, a tool that discovers novel
isohedral tilings whose tiles are as close as possible in shape to a given input shape. Note that while Escher
and his contemporaries have drawn many multihedral tilings by hand, most tiling software is restricted to
some family of monohedral tilings. The mathematical visualization software created by Huson [36] is an
26
Figure 18 An example of a Penrose tiling.
exception, able in theory to handle all periodic tilings of the plane, sphere and hyperbolic plane.
5.3.1
Aperiodic tilings
An aperiodic tiling is one that does not contain any translational symmetries. Many simple shapes such as
squares can be assembled into both aperiodic and periodic tilings. A more interesting problem is to find a
set of tiles that can only be assembled into aperiodic tilings.
Such sets exist [25, Chap. 10], and afford exciting new possibilities for ornamental design. Glassner provides a good introduction to pattern design using aperiodic tilings [22], showing how motifs drawn on
aperiodic tile sets can be assembled into attractive designs. In another article [23], he demonstrates the same
process with the well-known Penrose tiles.
In general, though the tilings are aperiodic, they are not entirely devoid of symmetry. Many contain rotations
and reflections. More importantly, they often have a “local isomorphism” property: the tiling as a whole
cannot be mapped onto itself via translation, but patches of connected tiles appear infinitely often in the
tiling. This not-quite-symmetric sense of order (sometimes referred to as quasiperiodicity) strikes an effective balance between the regularity of pure symmetry and the meandering randomness of floral ornament.
As Glassner says, “Balancing between repetition and randomness can lead to patterns that draw us in and
keep our interest.” [22]
27
Figure 19 Escher’s Circle Limit 4, based on the
Poincaré model of the hyperbolic plane.
5.3.2
Figure 20 One of Dunham’s interpretations of
Escher’s Circle Limit 1, recoloured and based on
a different symmetry group.
Hyperbolic tilings
As discussed in Section 2.3.3, another way to escape the isometric regularity of the wallpaper groups is
to project symmetric designs in the hyperbolic plane into the unit disk via the Poincaré model. The result
displays a strange and absorbing sense of order. Although individual tiles are necessarily distorted in shape
and size, congruence can still be perceived.
Escher struggled for a long time, trying to find a true expression of infinity within the bounds of the printed
page. The breakthrough came when Coxeter showed him projections of hyperbolic tilings [56, p. 42].
Escher was able to adapt the diagrams into his four Circle Limit woodcuts, one of which appears in figure
Figure 19.
Dunham has for many years explored the computer generation of Circle Limit inspired tilings [16, 14, 17,
15]. He has used his software to recreate Escher’s woodcuts, create hyperbolic interpretations of many of
Escher’s plane tilings, and generate some original designs.
In his system, the user specifies a motif to apply within a fundamental region of some hyperbolic symmetry
group. The program applies the symmetries of the group to copy the motif over a substantial fraction of
the unit disk. When the motif and symmetry group are chosen appropriately, the result is a projection of a
hyperbolic tiling. There is also a mechanism for colour symmetry compatible with the replication process.
28
Unfortunately, Dunham’s system does not construct tilings, it merely replicates a motif according to some
symmetry group. The user must have a sufficient understanding of the structure of the symmetry groups that
the motifs they choose lead to tilings. Adding the ability to edit tile shapes directly would greatly enhance
the functionality of Dunham’s tools.
Another area for improvement is interactivity. Dunham’s papers suggest that replication is only carried
out once the complete motif is entered. Continuous visualization of the hyperbolic pattern as the motif
is edited provides the user with helpful feedback. It would be beneficial to apply Dunham’s research on
Escher patterns to a highly interactive hyperbolic exploration tool such as the one developed by Hausmann
et al. [32].
5.3.3
Parquet deformations
A parquet deformation is, roughly speaking, a “spatial animation” of a tiling: a tiling where the shapes of
individual tiles evolve along a strip. They were invented by William Huff as a design project for his students,
and later popularized by Hofstadter [33, Chap. 10]. With parquet deformations, we might say that the visual
puzzle of the ornament’s structure is replaced by a visual story, a sense of order that pulls the viewer along
the direction of change.
Despite the distinctive style and aesthetic appeal of parquet deformations, they have not been well studied. I
have created a prototype implementation that generates parquet deformations automatically by interpolating
between isohedral tilings.
5.4 Fractals and dynamical systems
The computer has not only been used as a tool for recreating preexisting ornamental styles. Computers
have also made possible styles that could not have been conceived of or executed without their capacity for
precise computation and brute-force repetition.
Fractals are probably the ornamental form most closely associated with computers. They have a high degree
of order, but little Euclidean symmetry. The Mandelbrot set, pictured in Figure 22, has but a single horizontal
line of mirror reflection, but such a stunning degree of self-similarity that order is visible at every point and
29
Figure 21 An example of a parquet deformation.
Figure 23 Runga-Cutta, An example of Field
and Golubitsky’s ornamental symmetric chaos.
Figure 22 The Mandelbrot set, the mascot of
fractal art.
at every scale. The correspondence between parts of the Mandelbrot set is always approximate, creating an
engaging visual experience.
Chaos is closely related to fractal geometry. Field and Golubitsky [21] have created numerous ornamental
designs by plotting the attractors of dynamical systems. In particular, they have developed dynamical systems whose attractors have finite or wallpaper symmetry, generating symmetric ornament like the design
in Figure 23 when executed. In this work, we find a true rebirth of ornamental design in the digital age.
It might also be interesting to examine how their symmetric chaos could be applied in a more traditional
ornamental setting, such as textile design or architectural embellishment.
30
6 Future work
The base of computer generated ornament can be extended in various directions. Here, we explore some of
those possibilities.
6.1 Dirty symmetry
Glassner argues that too much order can be just as unappealing as not enough [22]. A floor covered by a
grid of square tiles is so featureless that it ought not be considered ornamental at all.
We have already seen ways to tamper with the perfection of symmetry. Rigid motions are lost in translation
from the hyperbolic plane to the Poincaré embedding, but the projected design still has order. Redefining the notion of symmetry to allow quasiperiodic tilings or fractals questions but ultimately reaffirms our
perception of order.
The perfection of symmetry can be tampered with even more easily. For example, the entire plane can be
passed through a displacement field based on procedural noise [18]. Motifs are distorted slightly, obliterating every symmetry in the design. However, when the magnitude of the displacement is not too great, we
have no problem filtering out the imperfections and seeing the design as symmetric. Moreover, the design
appears looser, less rigid, and more appealing than its perfectly symmetric progenitor. Møller and Swaddle cite psychoaesthetic and sociological evidence that this sort of imperfect symmetry is preferred over
perfection [47].
It would interesting to explore how this “dirty symmetry” affects aesthetic judgment of a pattern, in terms
of the variety and magnitude of the distortion. Can geometric noise be added to any design to increase its
appeal?
6.2 Conventionalization
Research in conventionalization would seek to turn real-world forms into stylized representations automatically, for use in ornament. This capacity for abstraction might be too much to ask from a computer program,
at least in the short term. It seems as if the program would have to “understand” the nature of the forms
31
Figure 24 An inversion by Scott Kim of the word “mathematics”.
it was abstracting in order to be effective. A more feasible goal would be to investigate what sorts of high
level tools might be uniquely qualified to aid the user in the process of conventionalization.
6.3 Inversions
A popular and peculiar form of ornament worthy of mention is the inversion, sometimes known as the
ambigram [42]. First developed by Scott Kim [41], an inversion is a bit of text that can be read in several
different ways. The usual form is a word or two that reads the same when turned upside down. Figure 24 is
an inversion of the word “mathematics”.
With inversions, symmetry must interact with the delicate process of perceiving letterforms and at the same
time produce an attractive ornament. There is a strong historical connection to square Kufic calligraphy,
where Arabic text is subject to extraordinary geometric constraints and emerges instantly legible. An successful inversion realizes a careful balance between constraints of geometry and legibility. As was the case
with Escherization [40], the search for ornamental forms that satisfy conflicting constraints can be a fruitful
area of research.
6.4 Symmetric floral ornament
The usefulness of the Wong et al. floral ornament work (Section 4.2) could be increased if it were easy to
create frieze and wallpaper patterns with floral motifs. A simple approach would be to elaborate a floral
design within a single fundamental region and replicate the region using the appropriate symmetry group.
Unfortunately, this approach would leave unattractive empty stripes along the edges bordering fundamental
regions, because the elaboration process cannot cross those edges.
A more complete solution would elaborate the design over the symmetry group’s quotient manifold, adapting
32
Gunn’s manifold visualization tools (Section 4.1) to two dimensions. This approach might be overkill,
however. The same effect might be achievable by elaborating over the fundamental region, copying elements
to adjacent regions as they are inserted. Copied motifs that extend back into the primary region would
preserve continuity and factor into availability of space for new elements.
6.5 A Pattern Language
Symmetry is a kind of redundancy, but not all redundancy is symmetry. Event a very simple figure consisting
of two flags has obvious repetition but no symmetries. More complicated designs such as Islamic star
patterns have a great deal more repetition than is expressed using symmetry alone. In general, any sort
of local correspondence is ignored by the far-reaching effects of symmetries. I would like to explore the
question of what sorts of structures can encode this asymmetric repetition.
The automatic theory of groups has illuminated an elegant connection between group theory and formal
languages [19]. In many cases, a simple symmetry group can be associated with a finite automaton in such
a way that the automaton can be used to systematically generate unique group elements. For example,
automatic groups have proven effective in replicating a motif through a hyperbolic symmetry group with
great efficiency [44].
If symmetry groups can be turned into formal languages, can formal languages be used to express more
generalized sorts of patterns? Specifically, we let the alphabet of a language represent a finite set of rigid
motions. Words become motions through composition. The language represents a set of motions to apply to
a fundamental region. Under this formulation, the regular languages are capable of expressing a broad class
of patterns, subsuming the planar symmetry groups. It is then natural to ask what sorts of patterns result
from context-free languages. A direction of more theoretical interest is to investigate the connection between
a pattern’s complexity and the complexity of the formal machines that generate it, a kind of Kolmogorov
complexity [54] of patterns.
33
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