MTE 3103 GEOMETRY Plane Tesselation BRYAN MIDIR IPG KAMPUS KENINGAU 2010 A tessellation or tiling of the plane is a collection of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of parts of the plane or of other surfaces. Generalizations to higher dimensions are also possible. Tessellations are seen throughout art history, from ancient architecture to modern art . MTE 3103: GEOMETRY TOPIC 1: PLANE TESSELATION In Latin, tessella is a small cubical piece of clay , stone or glass used to make mosaics . The word "tessella" means "small square". It corresponds with the everyday term tiling which refers to applications of tessellations, often made of glazed clay. A tessellation is created when a shape is repeated over and over again covering a plane without any gaps or overlaps. 1.1 WHAT IS TESSELLATION A tessellation or tiling of the plane is a collection of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of parts of the plane or of other surfaces. Generalizations to higher dimensions are also possible. Tessellations are seen throughout art history, from ancient architecture to modern art . In Latin, tessella is a small cubical piece of clay , stone or glass used to make mosaics . The word "tessella" means "Small Square". It corresponds with the everyday term tiling which refers to applications of tessellations, often made of glazed clay. A tessellation is created when a shape is repeated over and over again covering a plane without any gaps or overlaps. 1.2 TYPES OF TESSELLATION 1.2.1 Regular Tessellations Regular tessellations are made up entirely of congruent regular polygons all meeting vertex to vertex. There are only three regular tessellations which use a network of equilateral triangles, squares and hexagons.
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MTE 3103 GEOMETRY Plane Tesselation
BRYAN MIDIR IPG KAMPUS KENINGAU 2010
A tessellation or tiling of the plane is a collection of plane figures that fills the
plane with no overlaps and no gaps. One may also speak of tessellations of parts
of the plane or of other surfaces. Generalizations to higher dimensions are also
possible. Tessellations are seen throughout art history, from ancient architecture
to modern art.
MTE 3103: GEOMETRY
TOPIC 1: PLANE TESSELATION
In Latin, tessella is a small cubical piece of clay, stone or glass used to make mosaics. The word
"tessella" means "small square". It corresponds with the everyday term tiling which refers to
applications of tessellations, often made of glazed clay.
A tessellation is created when a shape is repeated over and over again covering a plane without
any gaps or overlaps.
1.1 WHAT IS TESSELLATION
A tessellation or tiling of the plane is a collection of plane figures that fills the plane with no
overlaps and no gaps. One may also speak of tessellations of parts of the plane or of other
surfaces. Generalizations to higher dimensions are also possible. Tessellations are seen
throughout art history, from ancient architecture to modern art.
In Latin, tessella is a small cubical piece of clay, stone or glass used to make mosaics. The
word "tessella" means "Small Square". It corresponds with the everyday term tiling which refers
to applications of tessellations, often made of glazed clay. A tessellation is created when a shape
is repeated over and over again covering a plane without any gaps or overlaps.
1.2 TYPES OF TESSELLATION
1.2.1 Regular Tessellations
Regular tessellations are made up entirely of congruent regular polygons all meeting vertex to
vertex. There are only three regular tessellations which use a network of equilateral triangles,
Escher type drawings are constructed by altering polygons that tessellate. Altering the sides of
various polygons will produce translation tessellations, rotation tessellations, and glide-
reflection tessellations. These tessellations become "Escher-type" when artistic details and color
are added to the basic design.
EXAMPLE
Tessellations
The following steps illustrate a method of altering the sides of an equilateral triangle to obtain a non polygonal figure that will tessellate. These steps can be carried out with pencil and paper or by computer software programs.
Step 1 Draw a curve from A to B.
<a
Step 2 Rotate the curve about point B so that A maps to C.
<a
Step 3 Label the midpoint of as D, and draw a curve from D to C. Rotate this curve about D so that C maps to A.
<a
MTE 3103 GEOMETRY Plane Tesselation
BRYAN MIDIR IPG KAMPUS KENINGAU 2010
Once the lines of the original figure have been erased, the figure that remains will tessellate.
Starting Points for Investigations
1. If the preceding figure is used to form a tessellation, which of the transformations (rotation, translation, reflection) will map the preceding tessellation onto itself?
2. Step 3 produces a curve on one side of the triangle that is said to have point symmetry because it can be rotated onto itself by a 180˚ rotation. Suppose Step 3 is used to produce a curve with point symmetry on all three sides of a triangle. Will the resulting figure tessellate?
3. Suppose Step 3 is used to create a curve with point symmetry on each of the six sides of a regular hexagon. Will the resulting figure tessellate?
Translation Technique very fundamental technique must be discussed, the translation technique. This technique involves redrawing a side of a shape and then translating a copy of the new side to every instance of the original side type. For example, in the following example, the side AB is redrawn as a curvy line segment and then copied to the side DC (an instance of the original side type). When the new side is copied to all instances, a new tessellation results.
First, side AB is redrawn. Then, a copy (shown in red) of the new side is translated to side DC. Repeating this change for every side equivalent to side AB results in the tessellation shown on the right.
Sometimes, the side that is redrawn does not have an instance on the original polygon. For example, in the following example, the side AB is not identical to BC nor AC. Similarly, side BC is not identical to AB nor AC, and side AC is not identical to AB nor BC. Thus, all three sides can be redrawn.
The sides of equilateral triangle ABC can be completely redrawn since sides AB, BC, and AC are all distinct types of sides. Notice that a new type of shape is formed after the sides are translated.
MTE 3103 GEOMETRY Plane Tesselation
BRYAN MIDIR IPG KAMPUS KENINGAU 2010
Another example of a case where the sides to be redrawn are not next to each other:
The redrawn sides, when translated throughout the tessellation, are not adjacent to one another.
1.3 FRACTAL GEOMETRY
A fractal is an object or quantity that displays self-similarity, in a somewhat technical sense, on
all scales. Fractal can also be said as "a rough or fragmented geometric shape that can be split
into parts, each of which is (at least approximately) a reduced-size copy of the whole diagram.
We will also see that fractals are endlessly repeating patterns that vary according to a set
formula, a mixture of art and geometry. Fractals are any pattern that reveals greater complexity
as it is enlarged."
We can identify fractal geometry from its properties as follow:
i. Self Similarities
Even though being magnified countless times, you can still see the same shape or characteristic
of the particular fractal. For example, when looking at a fern leaf, notice that every little leaf –
part of the bigger one – has the same shape as the whole fern leaf.
ii. Non-integer
Fern, an example of fractal in nature.
MTE 3103 GEOMETRY Plane Tesselation
BRYAN MIDIR IPG KAMPUS KENINGAU 2010
Classical geometry deals with objects of integer dimensions: zero dimensional points, one
dimensional lines and curves, two dimensional plane figures such as squares and circles, and
three dimensional solids such as cubes and spheres. While a straight line has a dimension of one,
fractal curve will have dimension between one and two depending on its space taken as it twist
and curve.
The more flat fractal fills a plane, the more closer it approaches to two dimensions.
Example, a "hilly fractal scene" will reach a dimension somewhere between two and three;
fractal landscape made up of a large hill covered with tiny mounds would be close to the second
dimension, while a rough surface composed of many medium-sized hills would be close to the
third dimension
"A fractal is "a rough or fragmented geometric shape that can be split into parts, each of which is
(at least approximately) a reduced-size copy of the whole ..."
"A fractal is an object or quantity that displays self-similarity, in a somewhat technical sense, on
all scales."
"Fractals are endlessly repeating patterns that vary according to a set formula, a mixture of art
and geometry. Fractals are any pattern that reveals greater complexity as it is enlarged."
1.3.2 Comparison between tessellation and fractal geometry.
The Same:
This fractal was created by Melissa D.
Binde. Her website is no longer online.
As you can see, there is an increasing
level of complexity. The black space on
the right become fractals themselves.
Landscape, show the properties of fractal.
MTE 3103 GEOMETRY Plane Tesselation
BRYAN MIDIR IPG KAMPUS KENINGAU 2010
Both tessellations and fractals involve the combination of mathematics and art. Both involve
shapes on a plane. Sometimes fractals have the same shapes no matter how enlarged they
become. We call this self-similarity. Tessellations and fractals that are self-similar have repeating
geometric shapes.
How they are different:
Tessellations repeat geometric shapes that touch each other on a plane. Many fractals repeat
shapes that have hundreds and thousands of different shapes of complexity. The space around
the shapes sometimes, but not always become shapes in the design. The space around shapes in
tessellations become repeating shapes themselves and play a major part in the design.
1.3.3 BINARY FRACTAL TREE
Fractal trees an plants are among the easiest of fractal objects to understand. They are based on
the idea of self-similarity. As can be seen from the example of a fractal tree below
This tree clearly shows the idea of self-similarity. Each of the branches is a smaller version of the
main trunk of the tree. The main idea in creating fractal trees or plants is to have a base object
and to then create smaller, similar objects protruding from that initial object. The angle, length
and other features of these "children" can be randomized for a more realistic look. This method
is a recursive method, meaning that it continues for each child down to a finite number of steps.
At the last iteration of the tree or plant you can draw a leaf of some type depending on the
nature of the plant or tree that you are trying to simulate. This idea can also be applied to the 3rd
dimension by allowing children to be angled in the z-plane as well as in the xy-plane.
A binary fractal tree is defined recursively by symmetric binary branching. The trunk of length 1
splits into two branches of length r, each making an angle q with the direction of the trunk. Both
of these branches divides into two branches of length r2, each making an angle q with the
direction of its parent branch. Continuing in this way for infinitely many branchings, the tree is
the set of branches, together with their limit points, called branch tips.
MTE 3103 GEOMETRY Plane Tesselation
BRYAN MIDIR IPG KAMPUS KENINGAU 2010
In the obvious way, each branch is determined by a string of symbols L and R specifying the
choice of direction taken along the tree to reach the branch. A branch determined by a string of n
symbols has length rn; a branch tip is determined by an infinite string of symbols. Most of the
analysis in [FT] results from converting eventually periodic symbol strings of branch tips into
geometric series for the x and y coordinates of the branch tips, and making appropriate
interpretations.
For example, in the tree on this page the branch tip marked * can be reached in two ways
LRRRRLRLRLRLRLR... and RLLLLRLRLRLRLRL...
Consequently, for both sequences, the corresponding branch tips have x-coordinate 0. With
simple trigonometry these sequences are converted into geometric series, giving r as a function
of q.
1.3.4 KOCH SNOWFLAKE
The first four iterations of the Koch snowflake
The Koch snowflake a mathematical curve and one of the earliest fractal curves to have been
described. The Koch curve is a special case of the Césaro curve where 𝑎 =1
2+
𝑖
12 , which is in
turn a special case of the de Rham curve.
Construction
The Koch curve can be constructed by starting with an equilateral triangle, then recursively