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.

MTE 3103 GEOMETRY Plane Tesselation

BRYAN MIDIR IPG KAMPUS KENINGAU 2010

1.2.2 Semi-regular Tessellations

Semi-regular tessellations are made up with two or more types of regular polygon which are

fitted together in such a way that the same polygons in the same cyclic order surround every

vertex. There are eight semi-regular tessellations which comprise different combinations of

equilateral triangles, squares, hexagons, octagons and dodecagons.

3.3.3.3.6 3.3.3.4.4 3.3.4.3.4 4.6.12

3.4.6.4 3.6.3.6 3.12.12 4.8.8

1.2.3 Non-Regular Tessellation

Non-regular tessellations are those in which there is no restriction on the order of the polygons

around vertices. There is an infinite number of such tessellations.

The three types of regular tessellation

MTE 3103 GEOMETRY Plane Tesselation

BRYAN MIDIR IPG KAMPUS KENINGAU 2010

1.2.4 Demi-regular Tessellation

Demiregular tessellation, also called a polymorph tessellation, is a type of tessellation whose

definition is somewhat problematical. Some authors define them as orderly compositions of the

three regular, eight semiregular tessellations (which is not precise enough to draw any

conclusions from), or tessellation having more than one transitivity class of vertices (which

leads to an infinite number of possible tilings).

Three example of Demi-Regular tessellation.

1.2 TESSELATION AND ART

Tessellation can be used in art to perform decorative design. Applying tessellation can make our

work faster, exact and easy rather than doing art manually. The following is two types of art

using tessellation.

Two example of non-regular tessellation

MTE 3103 GEOMETRY Plane Tesselation

BRYAN MIDIR IPG KAMPUS KENINGAU 2010

Two types of art using tessellation.

1.2.1 ESCHER-TYPE TESSELLATIONS

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

altering each line segment as follows:

MTE 3103 GEOMETRY Plane Tesselation

BRYAN MIDIR IPG KAMPUS KENINGAU 2010

1. Divide the line segment into three segments of equal length.

2. Draw an equilateral triangle that has the middle segment from step 1 as its base and

points outward.

3. Remove the line segment that is the base of the triangle from step 2.

After one iteration of this process, the result is a shape similar to the Star of David. The Koch

curve is the limit approached as the above steps are followed over and over again. Koch

Snowflake is a base-motif fractal with the following base and motif:

During the first iteration, we substitute every side of the triangle with the motif:

During the second iteration, we substitute every one of the 12 line segments with the same

picture:

We continue doing the same infinitely to get the Koch Snowflake:

MTE 3103 GEOMETRY Plane Tesselation

BRYAN MIDIR IPG KAMPUS KENINGAU 2010

Properties

The Koch curve has an infinite length because each time the steps above are performed on each

line segment of the figure there are four times as many line segments, the length of each being

one-third the length of the segments in the previous stage. Hence the total length increases by

one third and thus the length at step n will be (4/3)n of the original triangle perimeter: the

fractal dimension is log 4/log 3 ≈ 1.26, greater than the dimension of a line but less than Peano's

space-filling curve . The Koch curve is continuous everywhere but differentiable nowhere.

Taking s as the side length, the original triangle area is 𝑠2 3

4. The side length of each successive

small triangle is 1/3 of those in the previous iteration; because the area of the added triangles is

proportional to the square of its side length, the area of each triangle added in the nth step is

1/9th of that in the (n-1)th step. In each iteration after the first, 4 times as many triangles are

added as in the previous iteration; because the first iteration adds 3 triangles, the nth iteration

will add triangles

Curve Construction

The curve begins as a line segment and is divided into three equal parts. A equilateral triangle is

than created, using the middle section of the line as its base, and the middle section is removed.

The Koch Snowflake is an iterated process. It is created by repeating the process of the Koch

Curve on the three sides of an equilateral triangle an infinite amount of times in a process

referred to as iteration (however, as seen with the animation, a complex snowflake can be

created with only seven iterations - this is due to the butterfly effect of iterative processes).

Thus, each iteration produces additional sides that in turn produce additional sides in

subsequent iterations.

An interesting observation to note about this fractal is that although the snowflake has an ever-

increasing number of sides, its perimeter lengthens infinitely while its area is finite. The Koch

Snowflake has perimeter that increases by 4/3 of the previous perimeter for each iteration and

an area that is 8/5 of the original triangle.

MTE 3103 GEOMETRY Plane Tesselation

BRYAN MIDIR IPG KAMPUS KENINGAU 2010

1.3.5 FRACTAL DIMENSION

We can find the fractal dimension of this fractal using the similarity method. Since in the

motif, there are 4 identical line segments that are each 1/3 long, the dimension is log 4 / log 3,

which is approximately 1.26.

LENGTH

At every iteration, this fractal becomes 4/3 times longer. After an infinite number of iterations

this fractal, just like all fractal curves, will become infinitely long.

AREA

Suppose the area of the original triangle is 1. Then, after the first iteration the total area of the

three triangles added is 1/3 and the area becomes 4/3:

After the second iteration, the total area of the 12 triangles added is 4/27, and after the third

iteration, the total area of the 36 triangles added is 4/81. After that the areas added become 3

times smaller with every iteration. To calculate the sum of all areas added, we use the formula

for the sum of an infinite geometric sequence, S = a / (1–r), where a is the first number of the

sequence and r is the common ratio. In our case, the sum would be (4/27) / (1–1/3) = 2/9. If you

add this to the area of 4/3 that we had after the first iteration, the area will become 14/9. That is

the area of the entire snowflake.