Link
between Math and Art
In
my literature review I mentioned about symmetry and tessellation and
how this can integrate maths and art. In this essay, I'll be talking
about tessellation and the symmetry used behind it. I will also look
at the kaleidoscope how it relates to tessellation and also talk
about the symmetry within it. Then I will create my own tessellated
kaleidoscope pattern to produce an abstract artwork. This will show
that although maths and art have differences, there is still a
connection between two disciplines and both can still be associated
with one another. I chose to do tessellation because I was inspired
by the work of M.C Eshcer with his beautiful abstract work using
tessellation. The art in tessellation started in Escher with his
hyperbolic tessellation of 'Circle Limit III'.
A
tessellation is created when a shape is repeated over and over again
covering a plane without any gaps or overlaps. Tessellate means
forming or arranging small squares in a mosaic pattern.
Tessellation is floors and maybe kitchen. Thus, in creating a
tessellated pattern symmetry is used as it is a geometric
transformation of tiles in making a pattern. There are four kinds
of symmetry used in tessellation; Reflectional symmetry, rotational
symmetry, translational symmetry and
glide-reflectional symmetry.
(Circle Limit III)
Reflectional
symmetry is when one half of the image reflects or a whole image is
mirrored.
(fig 1)
(fig 2)
Rotational
symmetry is when an image is rotated and it looked exactly the same
as it did in its original position after rotating.
(fig 3)
(fig
4)
Glide-reflectional
symmetry is when the symmetry that an image has is translated a given
distance at a given direction and then reflected over a line.
(fig 5)
Using
these types of symmetry a tessellated pattern is formed. Symmetry is
important especially in creating an artwork as symmetry creates
balance. Everything around us have symmetry and the lack of it can
destroy the sense of balance. Symmetry does not have to be exactly
the same in both sides of the plane or each side of the image but to
have an equal amount on both sides of an image. Symmetry is also
important in tessellation because this create interesting patterns by
only repeating a shape, object or image in one big plane.
Most
of the work of Escher in his tessellations are animals being drawn
inside the shape and outside the shape by making images inside and
outside the shapes. Most of them are contrasting animals like the
'Birds and fish' (Image below). It is really interesting how these
images are made by just using tessellation.
This
photo on the left side is one of Escher's work,
the
box on the image is where the pattern been made
from.
The pattern was formed from a rhombus shape
and
he used translational symmetry to make the image.
In
tessellation, not all the shapes can be tessellated. Polygon shapes
are the most common shapes used in tessellation. However, not all
polygon shapes could be (Birds and fish)
tessellated which means these shapes will leave gaps in between
and will not fill the whole plane. Examples of polygon shapes that
can tile the entire plane without leaving a gap are rectangular,
triangular and hexagonal shapes. The shapes that leaves gaps on the
plane are pentagon, heptagon and octagon. However, this is not the
case, these shapes can still be tessellated by adding shapes in
between the gaps. This kind of tessellation is called non-regular
polygon tessellation. There are other kinds of tessellations used to
create tessellated pattern. The image from above which is the 'Circle
Limit III' created by M.C Escher is an example of hyperbolic and
euclidean plane where the tessellation is done in a circle.
For
my tessellation, I was inspired to create a kaleidoscope view images
by using reflectional symmetry then tiling the image produced using
photoshop. With this artwork, instead of having the whole image
abstract I wanted to keep the main image whole and surround it with
the reflection from it. I will also attempt to make a painting of
tessellation for this project by tessellating origami and putting it
on a canvas laying it out to make an interesting pattern.
Kaleidoscope
is a device that produce an overall pattern from the original design
and the reflected image of it in the mirrors. Kaleidoscope produces
different types of pattern depending on how many mirrors are being
used. But because the kaleidoscope pattern that I will be doing does
not have mirrors in it. I have to know where the mirrored image will
produce without looking into a mirror.
To
make a kaleidoscope view images, I have to understand the symmetry
and reflections depending on the shape that I will use for my image.
Kaleidoscope is also incorporated with tessellation as it has
repetitive pattern in it. Behind those beautiful images that a
kaleidoscope produce also incorporates math which comes down to math
and art being associated with each other. Other than symmetry, angle
is also important.
There
are two basic system of mirrors used in kaleidoscope, a two mirror
kaleidoscope and three mirror. A two mirror kaleidoscope is when the
mirrors are in a “V” position and one is blackened. To show what
happen in a two mirror kaleidoscope.
The
image on the left side shows an example of a 2 mirror view
kaleidoscope. If we call the centre of the circle X, the triangle AXB
is where the original image is. The image is then reflected around
the mirror where AXB is reflected to HXA and also to BXC and so on.
For symmetry to be produced on in the kaleidoscope, the angle of the
actual image must evenly divides the 360° of the circle. If the
angle is not evenly divided to 360° the closure of the symmetry
pattern will not work, instead it will overlap and destroy it. Having
the angle of the image evenly divided from 360° is the basic rule on
the two mirror kaleidoscope. An example below shows angles that
could work to produce a symmetry pattern. From the image (Fig 2) the
angle starts with 90° where it has 4 folds symmetry and 4 point
star. The point star are the point edges from the “V” angle of
the mirror produced.
45o
-
8
fold symmetry -
4
point star 36o
-
10
fold symmetry -
5
point star
30o
-
12
fold symmetry -
6
point star
22.
5o
-
16
fold symmetry -
8
point star
15o
-
24
fold symmetry -
12
point star
10o
-
36
fold symmetry -
18
point star
1o
-
360
fold symmetry -
180
point star
However,
the three mirror kaleidoscope is more complicated as one more mirror
involves. The angle of the mirrors must now add to 180°
as this is the sum of the total number of degree in a triangle. The
image of a three mirror kaleidoscope produce does not form a circular
pattern like the two mirror kaleidoscope does. The image reflects
infinitely and in different directions. Also, the three mirror image
produce much more interesting patterns depending on the angles of the
mirror.
The
image in fig 5 shows an interesting pattern and formed a shape of
hexagon. If you look closely, pattern shows tessellation where the
shape formed is used repetitively without leaving a gap in between.
A
kaleidoscope could add more mirrors to create beautiful patterns but
as more mirrors are being added the symmetry becomes more and more
complicated.
In
conclusion, tessellation and kaleidoscope cannot be formed without
symmetry. Understanding the symmetry behind the reflected image
produced by kaleidoscope and the patterns produced in tessellation
can expand the knowledge of more possible pattern produced. Symmetry
has always been incorporated both in math and art. Tessellation and
kaleidoscope are examples of math and art being integrated and
symmetry as being the bridge in unifying both disciplines.
Bibliography:
Barth,
A. (2007, April 27). Tessellation:
Link Between Math and Art. Retrieved
from
ramanujan.math.trinity.edu/math/students/.../amanda.barth.nm.pdf
Karadimos,
C. (n.d.). Kaleidoscope
Mirror System.
Retrieved Oct 16, 2010, from
http://www.brewstersociety.com/mirrors.html
jpg
image of 'Circle Limit III'. Retrieved from
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image of (fig 1). Retrieved from
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image of (fig 2). Retrieved from
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image of (fig 3). Retrieved from
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image of (fig 4). Retrieved from
jpg
image of (fig 5). Retrieved from
