MTE3103 Geometry

Topic 1Plane Tessellations

1.1 Synopsis The use of mathematics in art and design is very wide. This topic explores the use of mathematics in this area focusing on the creation and analysis of designs in two-dimensional plane. The type of tessellations discussed included regular, semi-regular and irregular tessellations. Escher-type tessellations also will be discussed. A new branch of mathematics, i.e. Fractal Geometry also will be introduce here.

1.2 Learning Outcomes1.Explain the types of tessellations.2. Design simple Escher-type tessellations.3.Calculate the similarity dimension for Fractal Geometry.1.3Conceptual Framework 1.4TessellationA tessellation is a pattern which completely covers a surface or plane without any overlapping of the shapes used.

Simplest example for tessellations include the types of tillings on most bathroom floors. Some exciting examples include Islamic Tiles star pattern and Escher-type tessellation. Escher-type tessellation will be discuss later in this modul.

1.4.1Types of TessellationThere are several way of classifying tessellations. This includes clasification by the number of shapes used in the tessellations or categorized by regular polygons, semi-regular polygons or irregular polygons. This modul will categorize tessellation according to the number of different shapes used.

1.4.1.1 Tessellations Using One Shape We will begin by looking at the simplest type of tessellations, i.e. formed by using only one shape. It can be categorized according to using regular polygons and irregular polygons.

Tessellations using regular polygonsThere are only three regular polygons which alone completely cover the plane, i.e. the equilateral triangle, the square and the regular hexagon.

If we observe the tessellations carefully, we will find that every vertex in the basic shape used meets the vertices of the neighbouring shape. What is vertex? The vertex is the corner or sharp point of the shape. Figure 1.4 (6) show a vertex in square-based tessellation:

VertexIn any tessellation there are at least three polygons whose vertices meet at the same point. Do you know why?A polygon is a plane figure with three or more straight line segments as its sides. Since it is 2-dimension figure, the total angle in each vertex must be 360. For example, one interior angle for a equilateral triangle is 60. There are six equilateral triangles which meet in each vertex. Thus , and equilateral triangles alone can form tessellations. For the square, one interior angles is 90 and four vertices would meet. . Therefore, squares itself can form tessellations.

Polygons with five sided are called pentagons. For a regular pentagon, one interior angle is 108. If we have 5 pentagons place next to each other, , which is less than 360 . In this case, regular pentagons alone cannot produce a tessellation.

Polygon with six sides is a hexagon, and the interior angle for regular hexagon is 120. When three regular hexagons meet in a vertex, . Thus regular hexagons alone can form tessellations. We extend the same idea for regular heptagon, i.e. a polygon with seven sides. Each interior angle is 128.57, when three regular heptagons meet in a vertex, Interception will happen; as shown in the figure below: For polygons with bigger number of sides, having three regular polygons such that the vertices meet without overlapping is impossible. As we have seen ealier, only three regular polygons can on their own form tessellations. Tessellations using irregular polygonsTessellations also can be formed from irregular polygons. Here are some examples:

1.4.1.2 Tessellation using two or more shapesHomogeneous tessellations are tessellations which use two or more regular polygons to tessellate such that the pattern formed at each vertex are the same. Homogeneous tessellations also known as semi-regular tessellations. Several examples of semi-regular tessellations are given below:

Semi-regular tessellations are named according to the number of regular polygons which meet at each vertex. For the first example above, there are two hexagons and two equilateral triangle which meet in each vertex. Hexagon has six sides and triangle has three sides. Therefore, the tessellation is indicated as 3.6.3.6., i.e. a triangle followed by a hexagon, another triangle, another hexagon, in clockwise order. Why it is not read as 6.3.6.3? In some books this is written as 6.3.6.3., which is not wrong, but we try to keep the smaller number first. If we had 3.3.6.6. instead, this would mean something different: two triangles followed by two hexagons. The symbol for semi-regular tessellation is important, as it can show if the tessellation is homogenuous. There are semi-regular tessellations which is not homogenous, as show in the example below below. Can you distinguish the difference between homogenuos tessellations and non-homogenous tessellatons?

1.5Tessellation and ArtIf we relate tessellations with art, we have to talk about the art works that has been developed by a Dutch artist, called, M.C Escher (1898-1972). There are numerous examples of Eschers work, which is highly mathematical, included in books, on T-shirts, jigsaw puzzles and coffee mugs.

Escher derived much of his inspiration during his first visit to Alhambra in Spain in 1922, where he studied Moorish masaics. However, unlike the Moors, who were forbidden to use graven images and only made tiling patterns with geometric shapes, Escher attempted to create tessellations with shapes which represented objects, animals and birds. Several art works by M.C. Escher: 1.5.1 Creating your own Escher-type TessellationsIn this section we will show you two of the simplest ways of creating Escher-type tessellations. From your previous reading, you should be aware that in producing tessellations with animate object is not easy and that Escher had spent incredible amount of time in researching, practising and implementing his designs.

1.5.1.1 Tessellations based on altering opposite parallel sides This method involves altering one side and then altering the opposite parallel side in a similar waysee the following figure:

1.5.1.2 Tessellations based on Rotation We have seen earlier that there are only three regular plane tessellations those using equilateral triangle, square and regular hexagon. While the square and the regular hexagon have opposite parallel sides, the equilateral triangle does not, thus this technique cannot apply for triangle. However, we can alter a side, and alter one of the adjacent sides in the same way through a rotation. This method of altering adjacent sides can be used to produce tessellations based on the regular hexagon as well. The figure above show the design with equilateral triangle based, where we altered a side, then alter one of the adjacent side by rotation. Observe the figure above carefully, you will find out the third side altered half of the side only and then rotating that alteration around the mid-point of the side. Can you see how the tessellation form? Discuss in your group!

1.6Fractal Geometry Have you ever used a computer program to enlarge a portion of a photograph? If the photograph is enlarged too much, the image may become blurred. A mathematician, Benoit Mandelbrot (1924- ) discovered some remarkable methods that enable us to create geometric figures with a special property: if any portion of the figure is enlarged repeatedly, then additional details of the figure are displayed. Mandelbrot called these endlessly repeated geometric figures fractals.

Figure 1.6 (1): Set MandelbrotBenoit Mandelbrot is known as the father of Fractal Geometry. He was not the first person to create a fractal, but he was the first person to discover how some of the ideas of earlier mathematicians such as George Cantor, Giuseppe Peano, Helge Von Koch, Waclaw Sierpinski dan Gaston Julia could be united to form a new type of geometry. Mandelbrot also recognized that many fractals share characteristics with shapes and curves found in nature. For instance, the leaves of a fern, when we compared with the whole fern, are almost identical in shape, only smaller in the size.

At the present time, there is no universal agreement on the precise defination of a fractal, but we can define fractal as follows. A fractal is a geometric figure in which a self-similar motif repeats itself on an ever-diminishing scale. Figure 1.6 (3): Self-similarity is shown in Sierpinski GasketFractal generally constructed by using iterative processes in which the fractal is more closely approximated as repeated cycle of procedures is performed. For example, a fractal known as Koch Curve is contructed as follows: Stage 0: Start with a line segment. Stage 0 in a fractal is called the initiator of the fractal.Stage 1: On the middle third of the line segment, draw an equlateral triangle and remove its base. Stage 1 in the fractal is called the generator of the fractal.Stage 2: Replace each initiator shape with a scaled version of the generator to produce the next stage of the Koch curve. The width of the scaled version of the generator is the same as the width of the line segment it replaces. Continue to repeat this step for the additional stages of the Koch curve.Example 1.6(1): Koch Curve Stage 0:

Stage 1:

Stage 2: Example 1.6(2): Sierpinski gasket

Stage 0:

Stage 1:

Stage 2:

Example 1.6(3): Box Fractal

Stage 0:

Stage 1:

Stage 2:

Example 1.6(4): Peano curve Stage 0:

Stage 1:

Stage 2:

1.6.1Strictly self-similar fractals All fractals show a self-similar motif on an ever-diminishing scale; however, some fractals are strictly self-similar fractals, according to the following defination.

Example 1.6(5):

Determine whether the following fractals are strictly self-similar.

a. Koch snowflake.

b. Koch curve

Solution:

a.Koch snowflake is a close figure. Any portion of the Koch snowflake (as shown in the circled in figure 1.6 (4)) is not a closed figure. Thus the Koch snowflake is not a strictly self-similar fractal.

b.It is quite obvious that any portion of Koch curve replicates the entire fractal, the Koch curve is a strictly self-similar fractal. The figure is shown in the following figure.

1.6.2Replacement Ratio and Scaling RatioThere are two numbers that are closely related to many fractals are the Replacement Ratio and Scaling Ratio.

Example 1.6(6) :

(i) Koch CurveStage 0: Stage 1: (ii) Sierpinski GasketStage 0: Stage 1: Find the replacement ratio and scaling ratio for (i) Koch curve (ii) Sierpinski Gasket

Solution:(i)The generator of Koch curve consists of four line segments and the initiator consists of only one line segment. Thus the replacement ratio for Koch curve is 4 : 1, or 4.

The line segment in initiator is three times longer than the replica line segments in the generator. Thus the scaling ratio of the koch curve is 3 : 1, or 3.

(ii)The generator of the Sierpinski Gasket consists of three triangles and the initiator consists of only one triangle. Thus the replacement ratio of the Sierpinski Gasket is 3 : 1, or 3.

The triangle of the Sierpinski Gasket in the initiator has a width that is 2 times the width of the replica triangles in the generator. Thus the scaling ratio of the Sierpinski Gasket is 2 : 1, or 2.

1.6.3Similarity DimensionA number called the similarity dimension, D, is used to quantify how densely a strictly self-similar fractal fills a region.

Example 1.6(7):Find the similarity dimension of the,

(i) Koch curve (ii) Sierpinski gasketSolution:

(i)Koch curve is a strictly self-similarity fractal, thus we can find its similarity dimension. From calculation on example 1.6(6), replacement ratio for Koch Curve is 4 and scaling ratio is 3. Thus the Koch curve has a similarity dimension of

D =

(ii)Also from example 1.6(6), Replacement Ratio for Sierpinski gasket is 3 and scaling ratio of 2. Therefore, the Sierpinski gasket has a similarity dimension of

D =

1.6.3 Fractal Geometry in Daily Life

1.6.3.1 Fractal in Natural Life

Rivers

Cloud

Lightning

Leaf veins

1.6.3.2 Fractal in Buildings

Ba-Ili in Afrika

Menara Eiffel in ParisFor further information about this chapter, you are encouraged to do your extra reading and surfing in the internet. You will feel excited how mathematics can be so beautiful! Happy reading!Reminder: Please make sure all the printed materials, included your notes and your solutions is kept properly in your portfolio. Figure 1.4 (1): Islamic Tiles star pattern Tessellation

Figure 1.4 (2): Escher-Type Tessellation

Figure 1.4 (3): Equilateral triangle-based tessellation

Figure 1.4 (4): Square-based tessellation

Figure 1.4 (5): Regular hexagon-based tessellation

Apart from these three regular polygons which alone can form tessellations, is thare any other regular polygon which can tesselate? Why?

Figure 1.4 (6)

Figure 1.4 (7): There is a gap with three regular pentagons.

Figure 1.4 (8)

Figure 1.4 (9): Rectangular based tessellation

Figure 1.4 (10): Trapezium based tessellation

3.6.3.6

4.8.8

3.3.3.4.4

3.3.3.3.6

Figure 1.4 (11)

Prepare a pair of scissors, glue and colour papers, spend thirty minutes to create homogenuos tessellations by combining several shapes of equilateral triangle, square, regular pentagon, regular hexagon, regular heptagon or regular octagon.

Enjoy!

Reading material : EscherGeometry meets art by B. Ansell, The Magic Mirror of M. C. Escher by B. Ernst dan M. C Escher at work by G.A. Escher.

Figure 1.5 (1): A simple tessellation based on a square.

Figure 1.5 (2)

Create an Escher-type tessellation by using one of the methods discussed above.

Figure 1.6 (2): Fern leaves

Draw Stage 3 and Stage 4 for all the example 1.6(1-4) above.

Definition:

A fractal is said to be strictly self-similar if any arbitrary portion of the fractal contains a replica of the entire fractal.

Figure 1.6 (4): The portion of the Koch snowflake shown in the circle is not a replica of the entire snowflake.

Figure 1.6 (5): Any portion of the Koch curve is a replica of the entire Koch curve.

Determine whether Sierpinski Gasket and Peano curve are strictly self-similar fractal.

Replacement Ratio and Scaling Ratio:

If the generator of the fractal consists of N replicas of the initiator, then the Replacement Ratio of the fractal is N.

If the initiator of a fractal has linear dimensions that are r times the corresponding linear dimensions of its replicas in the generator, then the Scaling Ratio of the fractal is r.

Find the replacement ratio and scaling ratio of the

a. Peano curveb. Box Fractal

The similarity dimension (D) of a strictly self-similarity fractal is given by

D = QUOTE

where N is the replacement ratio of the fractal and r is the scaling ratio.

Compute the Similarity Dimension of the

a. Peano curveb. Box Fractal

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