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| MAT_007_I_News A newsletter edited by undergraduates of the Department of Mathematics at the University of Toronto. Includes some online copies. |
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| A__Mathematician\'s_Aesthetics In his classic A Mathematician's Apology, G. H. Hardy likened mathematics to poetry and painting. This site elaborates on Hardy's remark with quotations from Stevens, Klee, Fry, and Focillon. Links to |
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A Treatise on the Aesthetics
of Symmetry
Preface
Home
Review
Boring
Grading
Motif
Ratings
Examples
Angles
Creative
Links
I was walking along Commonwealth Drive one day
in Boston admiring a wrought iron fence when I became introspective, curious
as to why I appraised the fence as attractive. It had something to do
with the inherent symmetrical design. With this in mind, I scrutinized
other items that I traveled past exhibiting symmetry. Some were quite
provocative, others were nominally patterned, and still others were rather
tedious and boring. The conclusions I depict in this essay are anything
but scientific.
Even if you concur one hundred percent with everything
I describe, be careful when applying this to your own designs. At times
the theories that we use to constrain ourselves during design come back
to bite us in our rear ends. These ideas have the potential to capture
your visual design processes and hold it hostage, incarcerating all of
your designs to a central structural consistency. Bad plan. Please apply
what you learn from this treatise where its application fits, but don't
force all of your work toward satisfying these observations.
Finally, aesthetics also transpire below, on top
of, and beyond symmetry. Some of the most exquisite art and craft that
I've seen had nothing to with symmetry at all.
Geometric Symmetry:
Quick Review
Symmetry has several variants, but the basic symmetrical
operations are Translation, Rotation, and Reflection. Translation is simply
the movement in a linear direction of a design; here is an example of
a translation:
The design above has been translated four times.
A rotation is the movement in a circular direction of a design; here is
an example of a rotation:
The design above has been rotated six times. Finally
a reflection is the inversion of a design across a line; here is an example
of a reflection:
The design above has been reflected across both
horizontal and vertical lines. Symmetrical operations can be combined;
here is an example of a translation combined with a reflection (this is
called a Glide Reflection):
Here is an example of a translation combined with
a rotation (a Glide Rotation):
What is Boring
Symmetrical operations certainly add some elegance
to an otherwise drab design. You can quite easily, however, have too much
of a good thing. The following design, for example, is something that
I would consider boring:
Too many repeats of the primary simple design
bores us. What constitutes "too many"
or "too few" repeats is a matter of taste and personal opinion.
It also has to do with the complexity of the simple design, or motif.
Furthermore, it might be affected by the symmetry operation:
The above example is similar to the boring example
with too many repeats, however since the symmetrical operation is a Glide
Rotation instead of a Translation, it is considerably more interesting.
Grading Operations
Can we find some arithmetic rules to show whether
a symmetrical design will be visually appealing? Let's put together an
equation where a higher rating indicates complete randomness (or a symmetry
that is too challenging to discern), and a lower rating shows boredom.
We'll call this rating the Interest level. Clearly, the more complicated
the motif, the higher the rating should be. So we will want higher motif
complexity to increase the number. More repeats makes for a less interesting
pattern, so we will want higher repeats to decrease the number.
I therefore propose an equation for determining
the Interest level of a particular symmetrical design: I = M/sn, where
"I" is the Interest level, "s" is a number for the
symmetry operation, "M" is a number for the complexity of the
underlying motif, and "n" is the number of times that the motif
is repeated. After considerable trial and error I have arrived at some
appropriate values for s, the factor for the type of symmetry operation:
Translation . . . . . 5
Reflection . . . . . 2.5
Rotation . . . . . . . 1
Glide Reflection . . 1
Glide Rotation . . . 1
Repeated Motif
Before we ponder some examples and contemplate
the value of their Interest level, allow me to digress to discuss the
M value for the complexity of the motif. I propose a simple measure of
complexity for the motif: count the number of distinct elements. Here
are some sample motifs, with the corresponding count of their "M":
Note that the concept of a "distinct element"
is a bit subjective: it depends on what you perceive as an element. In
some designs this is not readily apparent... does an arc count as it's
own element, or is it part of some larger visual portion? Here you will
need to use your visual design skills as a license to interpretation.
A motif can have its own symmetry beyond that
of the overall design. In the example below, although the overall pattern
exhibits Rotation symmetry, the motif itself has a design with Reflection
symmetry:
In the case such as above where the Motif show
signs of symmetry, the Interest level equation should properly be I =
M/ysn, where "y", the additional divisor, represents the factor
for the type of symmetry operation within the motif itself.
Ratings Guide
Having explained everything that goes
into the calculation, here is how I interpret the computed Interest level,
I:
0-1.5 . . . . Boring
1.5-2.5 . . . Patterned
2.5-4 . . . . Interestingly Patterned
4-6 . . . . . Complexly Patterned
6-9 . . . . . Disturbingly Patterned
>9 . . . . . . Random
These are not hard and fast rankings
and are intended only as a general guideline. When you review the examples
below, see if you agree in your artist's heart and your subjective imagination
with the calculated ratings and interpretations.
Examples
The page that is linked below has examples of various
symmetrical designs, along with the computed Interest level and the factors
that contributed to that rating:
Symmetry
Interest Level Examples
Visual Angle
Constraints
Although the above examples are appropriate for
computer-based displays, design in real-life usually extends well beyond
600 by 800 pixel resolution. Because the human eye has both limited angles
of visual perception and visual acuity, you need to consider the observant
range when designing for larger areas. For example, a long-running fence
may have several hundred repeats of a motif, even though the eyes of any
given observer may only be seeing twenty of the repeats in a single glance.
A more subtle relevance is whether a design element
should be considered as a motif within a larger symmetrical design, or
as an individual item within a motif. This confusion can arise because
the central eight degrees of the visual field are scrutinized at a higher
level of detail than the surrounding background. When standing next to
a large wall painting of the following design, each element is likely
to be viewed as a motif in the overall design:
In other words, on a computer monitor this has
an Interest level of 8/(2.5)(1)(1) = 3.2, but the Interest level standing
next to a large wall painting of the same design would be 14/(2.5)(8)(1)
= 0.7. Hence, be aware of how your designs are viewed in the context of
the visual angles distinguished by the observer.
Creative Ideas
This treatise has discussed the evaluation of an existing
design; if you are creative then you can use your imagination and talents
to build interesting motifs from scratch. The page below flips the Interest
Level formula around and demonstrates how to develop effective motifs
with interesting symmetry in mind as the target.
Developing Effective Motifs
Offsite References
You might enjoy these other sites dealing with
the use of symmetry in design.
The Arts in Victorian Britain
A scholarly review from the National University
of Singapore.
Tiling Plane & Fancy
An explanation of the 17 types of tiling symmetry,
with historical examples.
Symmetry and Ornament
An extensive scholarly book about symmetry and
ornamentation, heavily mathematical, published online by Slavik V. Jablan.
xPlane Art Blog
An annotated list of Internet links related to
art.
