The Four Planes of Symmetry
Kallia Katsampoxaki-Hodgetts
University of Crete 2012
Symmetry

A repeating pattern in a plane. A pattern is
symmetric if there is at least one symmetry
that leaves the pattern unchanged.
Symmetries

In mathematics, the idea of symmetry gives
us a precise way to think about this subject.
We will talk about plane symmetries, those
that take place on a flat plane, but the ideas
generalize to spatial symmetries too.
Types of Symmetry
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Translation
Rotation
Glide Reflection
Reflection
Translation
Translation Properties

A translation is defined by its direction and/or
distance. Two objects are symmetrical under
translation if one can be superimposed on
the other such that their edges match up
perfectly.
A translation

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It is a shape that is
simply translated, or
slid, across the paper
and drawn again in
another place.
It shows the geometric
shape in the same
alignment as the
original; it does not turn
or flip.
Rotation
To rotate an object means
to turn it around. Every
rotation has a center and
an angle.
Rotational Symmetry

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Rotation is spinning the pattern around a point,
rotating it. A rotation, or turn, occurs when an object
is moved in a circular fashion around a central point
(i,.e origin) which does not move.
Formally, rotational symmetry is symmetry with
respect to some or all rotations in m-dimensional
Euclidean space. Rotations are direct isometries,
i.e., isometries preserving orientation.
Rotation
Symmetrical object under Rotation

It appears the same when rotated about its
center some fraction of 360 degrees. The
most common types of rotational symmetries
are 90 and 180 degree rotations, but others
such as 45 and 60 also occur. For example,
the number "6" is symmetrical to the number
"9" under 180-degree rotation.
Reflectional Symmetry

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Reflection symmetry, line symmetry, mirror
symmetry, mirror-image symmetry, or bilateral
symmetry is symmetry with respect to reflection.
A type of symmetry in which one half of the object is
the mirror image of the other.
In 2D there is a line of symmetry, in 3D a plane of
symmetry. An object or figure which is
indistinguishable from its transformed image is called
mirror symmetric.
Reflection
Symmetry about a
vertical line of reflection.
The hexagon has
reflectional symmetry
about both horizontal
and vertical lines of
reflection.
Related Terms for Reflectional Symmetry
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Flip
Horizontal Line
Vertical Line
Mirror Symmetry
Reflection symmetry, line symmetry, mirror
symmetry, mirror-image symmetry, or bilateral
symmetry is symmetry with respect to reflection.
Properties of Reflections
Reflections exhibit
1.
Collinearity
2.
Betweenness
3.
Distance --length or linear measure
4.
Angle measure
Reflections do not preserve orientation.
Reflections reverse orientation.
Solve a Problem
The letter B has reflectional symmetry. Identify
the list of other alphabets with similar reflection.
Choices:
1. A, H, I, M, O, T, U, W, X, Y
2. C, D, E, H, I, O, X
3. F, G, J, L, Q, R, S, Z
4. A, B, C, D, E, H, I, K, M, O, T, U, X, Y
Steps before the Solution

Step 1: A reflection flips the figure across a
line. The new figure is a mirror image of the
original figure.
Step 2: The alphabet 'B' has horizontal
reflection because half a figure is a mirror
image of the other half.
Solution

The other alphabets with similar reflection
are C, D, E, H, I, O, X.
Glide Reflection
Glide Reflection

Glide reflection is the only type of symmetry
that is a combination of two symmetries. It
combines a reflection with a translation along
the direction of the mirror line. It is also the
only type of symmetry that involves more
than one step.
Glide Reflections
In glide reflection,

Reflection and
translation are used
concurrently much like
the following piece by
Escher, Horseman.
There is no reflectional
symmetry, nor is there
rotational symmetry.
Tesselations

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If you look at a completed
tessellation, you will see the
original motif repeats in a
pattern. One mathematical
idea that can be
emphasized through
tessellations is symmetry.
There are 17 possible ways
that a pattern can be used to
tile a flat surface or
'wallpaper'.
Find the symmetry
References
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http://math.hws.edu/eck/math110_s08/symm
etries/index.html
http://mathforum.org/sum95/suzanne/symsus
an.html
http://www.csun.edu/~lmp99402/Math_Art/Te
sselations/tesselations.html
http://www.ehow.com/info_8465413_different
-kinds-symmetries-math.html
Task 1: True or False?
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Plane symmetry means a symmetry of a pattern in
the Euclidean plane; that is, a transformation of the
plane that carries any directioned lines to lines and
preserves many different distances.
A pattern is symmetric if there is at least one
symmetry that leaves the pattern changed.
In 2D and 3D there is a plane of symmetry.
An object or figure which is indistinguishable from its
transformed image is called mirror.
Task 2:Complete the gaps
1. An object that is symmetrical under reflection is the ………..of the other object. Objects
with reflectional symmetry are categorized by the ……….of the axis of symmetry: vertical,
horizontal or slant. For example, the letter "W" has reflectional symmetry with the letter "M"
along a ………axis.
2. Rotation is …….the pattern around a point, rotating it. A rotation, or turn, occurs
when an object is moved in a …………..around a central point which does not move.
3. Symmetry under translation refers to simply moving the object horizontally,
vertically or both ………rotating or reflecting it. A translation is defined by its direction
and/or….. Two objects are symmetrical under translation if one can be ……….on the
other such that their edges match up perfectly. It shows the geometric shape in the same
a…….. as the original; it does not flip.
4. ……reflection is the only type of symmetry that is a …...............of two symmetries.
Two objects are symmetrical under glide reflection if one is a reflection of the other and
is translated …………….the axis of symmetry.
Task 3: Dictogloss
Step 1:
Listen to the instructor reading a short text on
the four planes of symmetry and take notes.
Step 2:
Reconstruct the text after comparing notes with
a partner.