Download Escher`s Tessellations: The Symmetry of Wallpaper Patterns III

Escher’s Tessellations:
The Symmetry of Wallpaper Patterns III
31 January 2014
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In the past two classes we saw many examples of Escher’s tessellations, and
different combinations of symmetry. We will talk briefly about the broad
mathematical structures used to classify wallpaper patterns, and we’ll see
pictorial examples of the different symmetry types which are possible.
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Different Combinations of Symmetry
One can have rotational symmetry (180◦ ) along with glide reflectional
symmetry.
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One can also have rotational symmetry (120◦ , one third turn) but no
reflectional or glide reflectional symmetry.
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Clicker Question
What kind of symmetry does this picture have, besides translational?
A
B
C
D
Rotational only
Reflectional only
Rotational and reflectional
None
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It is also possible to have rotational symmetry and reflectional (rather
than glide reflectional) symmetry. Escher drew this picture with
reflectional symmetry in two perpendicular directions. Doing so forces
the picture to have 180 degree rotational symmetry.
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The following two picture indicates that performing a vertical reflection
followed by a horizontal reflection results in a 180 degree rotation.
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To each wallpaper pattern one can consider the collection of
isometries which, when applied to the picture, superimposes it exactly
onto itself.
To understand a situation, mathematicians often look for some sort of
“structure” on collections of objects rather than working just with the
individual object.
For example, the collection of numbers has the operation of addition,
which takes two numbers and adds them, producing a third number.
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The collection of isometries has the property that two isometries can
be combined, or “composed”, by performing one, then the other,
producing a third isometry.
A glide reflection is an example of two isometries being composed.
We get it by performing a reflection followed by a translation.
As we just saw, if we compose a horizontal and a vertical reflection,
we get a 180 degree rotation.
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In arithmetic, addition satisfies:
The associative property - e.g., 3 + (2 + 5) = (3 + 2) + 5.
An identity 0 - e.g., 3 + 0 = 3.
Additive inverses - e.g., 8 + (−8) = 0.
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Composing isometries also satisfy the same three properties.
The analogue of 0 is the “no motion”, or identity, isometry. Also,
each isometry has an inverse which, when performed after the
original, results in no motion at all.
For example, the inverse of a rotation by 90 degrees counterclockwise
is a rotation by 90 degrees clockwise. A reflection is its own inverse.
That is, performing a reflection twice accomplishes the same thing as
no motion at all.
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Group Theory
Group theory studies collections of objects together with an operation
which satisfies the same three properties mentioned above which
addition satisfies.
Group theory originated in the early 19th century through the work of
Galois, who introduced the concept in order to study solutions of
polynomial equations.
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The collection of isometries associated to a wallpaper pattern is a
group.
There is one important difference between the group of isometries
and the group of numbers with addition. The latter satisfies the
commutative property (e.g., 2 + 3 = 3 + 2), while the former does
not.
To illustrate this, we consider a vertical reflection and a quarter turn
(counterclockwise) rotation.
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The series in blue results from doing a vertical reflection followed by a
90 degree rotation. The series in red results from performing the two
isometries in the opposite order. Since the results are different, the
order in which isometries are performed matters.
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Clicker Question
If you perform a 90 degree rotation counterclockwise followed by a
reflection across a horizontal line on the following figure, which figure do
you get?
A
B
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Classification of Wallpaper Patterns
It is through the study of groups of isometries that the classification
of all possible types of symmetry of wallpaper patterns was made.
It was discovered that there are exactly 17 different types of
symmetry in wallpaper patterns, by seeing that there are 17 different
groups of isometries. This was completed by Fedorov, Schoenflies,
and Barlow at the end of the 19th century.
Escher discovered the classification on his own. He drew pictures for
16 of the 17 symmetry types.
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You can find a PDF file showing pictures of all 17 symmetry types,
including Escher drawings for 16 of them, at the course website by clicking
on Handouts, and then clicking on 17 Wallpapers.pdf.
You can also find a webpage, also available from the Handouts link, titled
Wallpaper Patterns.
We will look at the 17 symmetry types now.
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Crystallographers, interested in the chemical properties of crystals,
studied their symmetry, and in the late 19th century classified the
types of symmetry of crystals. They found that there are 230 different
symmetry types. This is the 3-dimensional analogue of the
classification of wallpaper patterns.
Group theory has been used in encryption, coding theory, quantum
mechanics, and crystallography, among other areas.
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Next Week
We will discuss encoding data and encryption of data next week.
Encryption is what allows us to do internet shopping, while encoding data
in clever is necessary to allow CDs and DVDs to play correctly even if
there is some dirt on the disk. This will appear to be a completely
unrelated topic to what we discussed this week, but there are common
mathematical ideas in both subjects.
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Assignment due next Friday
Do one of the following three activities:
Variant 1: Draw a tessellation starting with a square (or rectangle) by
following the instructions in the link Tessellations from Squares. Also,
determine if the resulting tessellation has rotational, reflectional, and/or
glide reflectional symmetry.
Variant 2: Draw a tessellation starting with a triangle by following the
instructions in the link Tessellations from Triangles Also, determine if the
resulting tessellation has rotational, reflectional, and/or glide reflectional
symmetry.
Variant 3: Look over the pictures of the 17 symmetry types in 17
Wallpapers.pdf. Give a plausible reason why Escher did not draw a picture
for the symmetry type on the third page of that handout (which is
numbered page 59), when he drew (often many) pictures for the other 16
symmetry types? Give some rationale for your opinion.
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