NUMERICAL LITERACY

1.0 ABSTRACT

Tessellation and polyhedral are two types of art. Tessellations have been

around for centuries and are still quite prevalent today.  However the study of

tessellations in mathematics has a relatively short history. 

In 1619, Johannes Kepler did one of the first documented studies of

tessellations when he wrote about the regular and semi regular tessellation, which

are coverings of a plane with regular polygons.

Through history, polyhedral have been closely associated with the world of

art. The peak of this relationship was certainly in the Renaissance. For some

Renaissance artists, polyhedral simply provided challenging models to demonstrate

their mastery of perspective.  For others, polyhedral also were symbolic of deep

religious or philosophical truths. For other artists, polyhedral simply provide

inspiration and a storehouse of forms with various symmetries from which to draw

on.  This is especially so in twentieth century sculpture, free of the material and

representational constraints of earlier conceptions of sculpture.

In this assignment, we are going to provide two models of polyhedral and two

designs of tessellation for each member. Besides that, we are ordered to provide the

notes of both. In this report, we also will show the procedures or steps to create the

models. The explanations comprised here are so helpful. Hopefully the whole

contents of this particular assignment could help those deeply want to know about

polyhedral and tessellation.

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2.0 POLYHEDRAL

2.1 NOTES OF POLYHEDRAL

A polyhedron (plural polyhedra or polyhedrons) is often defined as a

geometric solid with flat faces and straight edges.

The word polyhedron has slightly different meanings in geometry and

algebraic geometry. In geometry, a polyhedron is simply a three-dimensional solid

which consists of a collection of polygons, usually joined at their edges. The word

derives from the Greek poly (many) plus the Indo-European hedron (seat). A

polyhedron is the three-dimensional version of the more general polytope (in the

geometric sense), which can be defined in arbitrary dimension. The plural of

polyhedron is "polyhedra" (or sometimes "polyhedrons").

A convex polyhedron can be formally defined at the

set of solutions to a system of linear inequalities.

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2.1.1 DEFINITION

Any polyhedron can be built up from different kinds of element or entity, each

associated with a different number of dimensions:

3 dimensions: The body is bounded by the faces, and is usually the volume

enclosed by them.

2 dimensions: A face is a polygon bounded by a circuit of edges, and usually

including the flat (plane) region inside the boundary. These polygonal faces

together make up the polyhedral surface.

1 dimension: An edge joins one vertex to another and one face to another,

and is usually a line segment. The edges together make up the polyhedral

skeleton.

0 dimensions: A vertex (plural vertices) is a corner point.

-1 dimension: The nullity is a kind of non-entity required by abstract theories.

A defining characteristic of almost all kinds of polyhedra is that just two faces join

along any common edge. This ensures that the polyhedral surface is continuously

connected and does not end abruptly or split off in different directions.

A polyhedron is a 3-dimensional example of the more general polytope in any

number of dimensions.

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2.1.2 TYPES OF POLYHEDRAL

There are three types of common polyhedral which are platonic solid or

known as regular polyhedral, prisms and pyramids.

2.1.1.1 Platonic Solid or Regular Polyhedral

There are only five regular polyhedral known to mathematics as the Platonic

Solids which are; tetrahedron (four equilateral triangles); hexahedron (aka cube, six

squares); octahedron (eight equilateral triangles); dodecahedron (12 pentagons);

and icosahedrons (20 equilateral triangles).

Types of Platonic Solid Characteristics

Tetrahedron

Four faces

Four vertices

Six edges

Cube

Six faces

Eight vertices

Twelve edges

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Octahedron

Eight faces

Six vertices

Twelve edges

Dodecahedron

Twelve faces

Twenty vertices

Thirty edges

Icosahedrons

Twenty faces

Twelve vertices

Thirty edges

As a conclusion, it will call Platonic Solids or Regular Polyhedral if it is vertex-

transitive, edge-transitive and face-transitive (this implies that every face is the same

regular polygon; it also implies that every vertex is regular).

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2.1.1.2 Prism

A prism is also a polyhedron, which means the cross section will be polygon

( a straight-edged figure). A prism has the same cross section all along its length.

That is mean, all sides will be flat. It has no curve sides. For example, a cylinder is

not a prism.

There are two kinds of prism which are regular prism and irregular prism. A

regular prism is when the cross section is regular or in the other words, it is a shape

with equal edge lengths. And the irregular prism is when the edge lengths of a shape

not equal. A prism is named according to the shape of its base. Below are the

examples of regular prism:

1) Regular Prism

Types of Regular Prism Cross-section

Triangular Prism

Square Prism

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Cube Prism

Pentagonal Prism

2) Irregular Prism

These models are called irregular prism because the pentagon is not regular

in shape. But, the models are still known as prism and polyhedral.

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2.1.1.3 Pyramid

A particularly popular polyhedron is the pyramid. If we restrict ourselves to

regular polygons for faces, there are three possible pyramids which are triangular

pyramid, square pyramid and pentagonal pyramid. All the pyramids are named after

the shape of their base.

Types of Pyramids Base

Triangular Pyramid

Square Pyramid

Pentagonal Pyramid

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2.2 DESIGNS OF POLYHEDRAL

In this assignment, we had created two types of polyhedral. For this part, we

will show the procedures or steps to make the models and also a brief explanation

about the both models. Besides, the procedure also have diagram and picture in

order to simplify others to create these polyhedral too.

2.2.1 SPINNER (OCTAHEDRON)

One of many wonderful designs by the late Lewis Simon, this model combines

two standards bases, the Waterbomb base and the Preliminary base, to form a rigid

modular construction. Use 12 sheets of fairly sturdy paper. The outer colour of the

entire model will be the same as the outer colour of the Preliminary bases.

2.2.1.1 PROCEDURE

Step 1:

Fold six Waterbomb and six Preliminary bases. Open

out one of each base slightly, and allow the

Preliminary base to wrap around the outside of the

Waterbomb base, lining up the creases in the two

bases.

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Mountains fold each of the four corners of the Preliminary base inward, over the

outer raw edges of the Waterbomb base, locking the sheets of paper together.

Step 3:

Allow the Waterbomb base to reform, the

two sheets folded as one. Repeat for the

remaining bases.

Step 4:

Join any two units together by slipping the raw

(Waterbomb) point of the first unit over the raw

point of the other, but underneath the raw edge

created by the Preliminary base. Push in all the

way, until the two edges of what were the

Preliminary bases meet.

Step 5:

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In the same way, join adjacent points of the first two units into similar points of a third

unit, forming a triangular section in the centre, as shown.

Step 6:

Add the final units in the same way the last unit is

the most difficult to achieve. To activate the model

place the points of the completed spinner into the

centre of your palms, holding firmly. Blow hard

against the top point of the model and the spinner

will really spin.

2.2.1.2 DESCRIPTION ABOUT THE MODEL

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For the first model, we had constructed the octahedron. To construct this

model, we used 21cm x 21cm colour papers in order to construct the model.

Besides, to construct this octahedron we need to cut 12 pieces of square paper and

in order to construct this model, we made 6 units of module and join it together.

Actually, the model named as Spinner. It is because the model is easy to be

spin. It also was functioning as a toy for kids. They love to spin and play with

because it is easy to hold and not too hard. So, teachers can do origami for this

model to use in the class.

Besides, this model also suitable to be a decoration element. It is suit to be

placed anywhere except in the toilet. The most appropriate place to be shown is on

the table especially on the study table. It is because the pattern and shape are

suitable to be laid on. And of course the colour itself can make the watcher feels

happy due to the colouring style. So, hopefully this model is helpful to use in any

condition as an art material.

Octahedron

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See the octahedron rotating around the x, y, z axis.

Properties of the Octahedron

Faces: 8 triangles

Vertices: 6, each with 4 edges meeting

Edges: 12

Dihedral angle: 109°28'

The Symmetry

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Surface Area

Let r = the distance from centres to one vertex.

The length (a) of edge, by the Pythagoras Theorem, = r√2.

Then the area of one triangle is (a × h) / 2, where h = √[a² - (a/2)²].

And the area of the octahedron is 8 × the area of one triangle.

Volume

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The octahedron can be divided into two pyramids.

The volume of one pyramid = (base area × height) /3. In the case of the regular

octahedron, the base area = a².

And so, the volume of the octahedron = 2 × the volume of pyramid.

V = (√2 / 3)a³

Introduction to regular octahedron:

In geometry, an octahedron is a polyhedron with eight faces. A regular

octahedron is a Platonic solid composed of eight equilateral triangles, four of which

meet at each vertex. An octahedron is the three-dimensional case of the more

general concept of a cross polytope. A regular octahedron is shown in the below

figure.

                                              

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The properties and Cartesian coordinates of a regular octahedron are given below in

detail.

                                                                                    (Source: Wikipedia)

Properties and Cartesian of Regular Octahedron:

Number of faces in an octahedron = 8. i.e., it has 8 equilateral triangular

faces.

Number of vertices in an octahedron = 6

Number of edges in an octahedron = 12

An octahedron is a regular convex deltahedron.

An octahedron has a Schlafi symbol of {3, 4}

In octahedron, the vertices has the cartesian coordinates of                                    

                              (±1, 0, 0);

                             (0, ±1, 0);

                              (0, 0, ±).

The area and volume of regular octahedron is explained below with examples

which help you to learn more about regular octahedron.

2.2.2 JAPANESE BROCADE (HEXAHEDRON)

This design, by Minako Ishibasi, makes attractive earrings, when folded from

extremely small squares, say 4 x 4cm/ 11/2 x 11/2in. for a practice version, begin

with six squares of fairly sturdy paper. This modular design is very original, featuring

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curved surfaces, which seem to create the illusion of rings encircling the central cube

shape. Only one shows on the finished unit, so begin with this colour face down. Two

squares each of three different colours can also be used, as here.

2.2.2.1 PROCEDURE

Step 1:

Begin by folding the square in half in one direction, to

establish the centre line. Fold upper and lower edges

into meet this crease.

Step 2:

Fold the right vertical edges upward on a diagonal

crease to lie along the upper edge. Fold the left

vertical edge downward to lie on the lower edge.

Step 3:

Unfold the paper completely.

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Step 4:

Fold all four corners inward to lie on the horizontal

quarter creases. Two of these creases will already

have been made.

Step 5:

Hold the upper and lower edges into lie along the

horizontal quarter creases.

Step 6:

Fold the lower right flap inward on existing crease,

made in step 2.

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Step 7:

Peel back the inner vertical edge of the flap folded in step 6, on an existing parallel

crease, while at the same time folding up the lower edge, also on the existing

crease. This perform on both a swivel and a squash fold.

Step 8:

Step 7 completed.

Step 9:

Repeats step 6-7 for the top left flap.

Step 10:

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Pull out the lower border, and allow the flap folded in the step 9 to tuck in behind it.

Flatten the model once more.

Step 11:

Step 10 completed.

Step 12:

Turn the model over, and fold each of the

sharp points to obtuse angles of the

parallelogram, as shown.

Step 13:

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Allow the flaps folded in step 12 to be unfolded slightly, to rest at right angles to the

central squares shape. Make five more identical units.

Step 14:

To assemble, slide the point of any one unit under the

central section of another unit as shown.

Step 15:

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Continue by adding a third unit,

assembling the central cube piece by piece.

If holding with two units of three

different colours, you should add unit of the

same colours opposite each other. All the

units are joined in the same way, all the way

around the model.

Step 16:

Under construction.

Step 17:

Assembly finished.

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Step 18:

Finally, squeeze together the four flaps

that appear on each of the six faces

of the central cube, allowing them to

project upward slightly and form the

circular “bands” around the model.

2.2.2.2 DESCRIPTION ABOUT THE MODEL

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The second model is the cube or hexahedron. To make this cube, almost any

type of paper can be used. As long as the paper is square, it can be folded. But, in

our cube design, we used colour paper.

Actually, the name was given for the model is Japanese Brocade. We wonder

how they got the name, but the most important thing is the shape of the models.

Whether it is interesting to watch or vice versa. It shapes like cube and clearly can

be seen as cube after done. And our cube also can transform to another shape. It

looks like a flower when it is opened. That’s meant, the model have an ecstatic value

on it. That is why we decided to choose it and felt happy after finished it.

The model is appropriate to be placed anywhere. It is because it is look more

natural and peace. Moreover, if we are great in colouring, it will become more

priceless. All people love flower. Even animals also make the flower as a part of their

life. So, we thought that we have to expose a natural phenomenon for this

assignment. Finally, we felt closer to nature. Perhaps, we can teach students the

hexahedron origami so that they know how to make friend with nature in order to

save our world from being destroyed.

Hexahedron

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Cube

A cube is a region of space formed by six identical square faces joined along

their edges. Three edges join at each corner to form a vertex. The cube can also be

called a regular hexahedron. It is one of the five regular polyhedrons, which are also

sometimes referred to as the Platonic solids.

Parts of a cube

Face Also called facets or sides. A cube has six faces which are all squares,

so each face has four equal sides and all four interior angles are right

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angles. In the figure above, drag the 'explode' slider to see the faces

separated for clarity.

Edge A line segment formed where two edges meet. A cube has 12 edges.

Because all faces are squares and congruent to each other, all 12 edges

are the same length.

Vertex A point formed where three edges meet. A cube has 8 vertices.

Face

Diagonals

Face diagonals are line segments linking the opposite corners of a face.

Each face has two, for a total of 12 in the cube.

Space

Diagonals

Space diagonals are line segments linking the opposite corners of a

cube, cutting through its interior. A cube has 4 space diagonals.

Properties of a cube

Volume The volume is s3 where s is the length of one edge.

Surface Area The surface area of a cube is 6s2, where s is the length of one edge.

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The Cube

The cube might also be called a regular hexahedron. It has 6 square faces, 8

vertices, and 12 edges. Three faces meet at each vertex.

Of the Platonic solids, the cube does not have the least number of faces or

vertices, but it is surely the simplest by any other measure. For that reason, this

page will not waste too much effort on tedious derivations of obvious measures. In

the following, we consider a cube of edge length s.

  area of each face      =    s2 Each face is a square of side

length s.

  A    =    6s2 because there are six faces

 

Each edge is perpendicular

to the adjacent edges, so this

follows from the definition of

the diheral angle.

 

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  The distance between two opposite faces can be

measured along an edge, so s is the diameter of

the inscribed sphere. The inradius must be half

of that.

 

 V    =    s3

Other Properties

The cube has 48 symmetries.

The cube is the dual of the octahedron. Connect the centres of the adjacent faces of a

cube results in an octahedron, and vice verse.

One other Platonic solid can be found inside the cube. Pick four of the vertices and

fit a tetrahedron.

A cross-section of a cube can be an equilateral triangle, a square, or a regular hexagon.

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A planar projection of a cube can be a square or a regular hexagon.

3.0 TESSELLATION

3.1 NOTES OF TESSELLATION

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In Latin, tessella is a small cubical piece of clay, stone or glass used to make

mosaics. The word "tessella" means "small square" (from "tessera", square, which in

its turn is from the Greek word for "four"). It corresponds with the everyday term tiling

which refers to applications of tessellations, often made of glazed clay.

Tessellations are tiling of the same shape. The tiles join together without gaps

or overlaps and with a pattern that allows the tessellation to continue in both

directions infinitely. If two shapes are used, it is called a semi-tessellation.

Tessellations work because angles at a point add up to 360°. Certain regular

polygons will tessellate and other special polygons will also tessellate. Triangles and

quadrilaterals can tessellate as well as other shapes.

Often the special shapes that tessellate are actually based on transferring a

section of a tessellating shape from one side to another so that the area is

conserved. These are often called nibble tessellations and they were used as

starting points in Escher’s art. When two regular shapes are used together to cover

space, they are called semi-regular tessellations. Tessellations are important as they

form the basis of area measurements, they are needed to make walls and floors of

buildings, they are found in nature, and they can be used effectively in artistic

creations. Many cultures have developed interesting tessellating and other patterns.

3.1.1 DEFINITION

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A tessellation is called when a shape created is repeated over and over again

covering a plane without any gaps or overlaps. In geometrical terminology a

tessellation is the pattern resulting from the arrangement of regular polygons to

cover a plane without any interstices (gaps) or overlapping. The patterns are usually

repeating.

Tiles with certain designs can be tessellated to produce a myriad of interesting

patterns and continuous curve designs. For example, see the diagram below.

Tessellations of 2D shapes are a way of considering area. Tessellations of 3D

shapes are a way of considering volume.

Example: Curve design.

3.1.2 TYPES OF TESSELLATION

There are three types of tessellation which is regular tessellations, semi-

regular tessellations and non-regular tessellations.

3.1.2.1 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. Since the regular polygons

in a tessellation must fill the plane at each vertex, the interior angle must be an exact

divisor of 360 degrees. This works for the triangle, square, and hexagon and this is

the reason only they can tessellate the Euclidean plane.

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These three tessellations have interesting features. For the triangular

tessellations, it can be folded up into three of the five platonic solids which are

tetrahedron, octahedron and icosahedrons. The square tessellations can be folded

up into one of the five platonic solids which is cube. For hexagons, it cannot fold up

on itself into a 3D object without the introduction of key shapes such as the triangle

or the pentagon.

By the same token, if one were try to create a regular tessellation on a plane

with other shapes such as octagons (eight sided figures) , dodecagons (twelve sided

figures), decagons (ten sided figures) or dodecagons (nine sided figure) or

pentagons (five sided figures), there will always be gaps no matter how there are

arranged.

Example: Regular tessellations.

Rules of regular tessellations :

o The tessellation must tile a floor (that goes on forever)

o The tiles must be regular polygons and all the same

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o For vertex, we need to use regular polygons that add up to 360.

What is a vertex?

Where all the

“corners” meet

(90 + 90 + 90 = 360)

Example 1: Hexagon Example 2: Triangle

3.1.2.2 Semi-regular tessellations

A semi-regular tessellations is made of two or more regular polygons. The

pattern at each vertex must be the same. There are only eight semi-regular

tessellations which comprise different combinations of equilateral triangles, squares,

hexagons, octagons and dodecagons.

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Example 3: Triangles and hexagons. Example 4: Octagons and triangles.

3.1.2.3 Non-regular tessellations or demiregular tessellation

There are also “demiregular tessellations, but mathematics disagree on what

they actually are. These types of tessellations are those in which there are no

restrictions on the order of the polygons around vertices. Some people allow shapes

(not just polygons) so you can have tessellations like these:

Example 5: Curvy shapes Example 6: Eagles

3.1.3 BASIC OPERATIONS IN CREATING TESSELLATIONS

3.1.3.1 Rotation

Rotate the polyiamond in the plane. The rotation operation can be applied to

all polyiamonds which do not possess circular symmetry, for example the hexagonal

hexiamond, which remains unchanged following rotation through 60 or multiples

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From the picture, we see that tessellation are same shape but have been

turned or rotation. One side of a square can be changed and this change can be

rotated 90° to an adjacent side. This same change can be rotated 180° and 270° so

that it appears on all sides. Start with a square and change one of the sides, trace

the change and rotate the change to the remaining sides as shown below. Each

rotation requires that the center of rotation be the vertex between the source side

and the destination side.

3.1.3.2 Reflection

Reflect the polyiamond in the plane, as if being viewed in a mirror. The

reflection operation is limited to polyiamond which are enantiomorphic. An

enantiomorphic polyiamond is one which cannot be superimposed on its reflection,

its mirror image. Sometimes, a shape does not change when it is reflected. This

property is called line symmetry. The example is:

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3.1.3.3 Translation

The transformation that changes one of these shapes into any other one in

the set is called a translation. This is sliding without turning, from one position to

another. To describe such as translation, it need concepts of direction, such as up,

down, left, right, forwards and backwards.

Slide the polyiamond along the plane. The translation operation can be

applied to all polyiamond. Example of translation is:

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3.2 DESIGNS OF TESSELLATION

In this section, we are going to show various designs of tessellation. Every

single member will create two types of tessellation which is simple and complex.

The procedure and short explanation also had been note here.

3.2.1 MUHAMMAD AMIR AL-HAFIZ

PROCEDURE OF MY TESSELLATION (A)

For my presentation tessellation, I had done the tessellation by using the

basic shape of square. For my presentation I had done by using a computer. Let

figure out how my tessellation had been finished.

Step 1:

First, I had draw a square shapes like this:

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Step 2:

Then I continue it by cut the shade region.

Then I get the new shape just like above.

STEP 3

For next step, I had made the reflection to the shape.

image

object line of reflection new shape

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STEP 4:

Then, I had made the same reflection to the new shape.

object

line of reflection

new shape

image

STEP 5

Then, I had made the same reflection to the new shape.

object line of reflection image

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STEP 6

Then repeat 5 until get shape like below .

STEP 7

Then reflect the shape in the step 6.

object

line of

reflection

Image

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STEP 8

Repeat step 7 until get the full tessellation.

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STEP 9

Colour the tessellation.

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EXPLAINATION OF MY TESSELLATION (A)

The basic shape of my tessillation is square. From square, I cut the shade

region. Then, I get a new shape, from that the shape was reflect to the side. The

step was repeted for few time. Then I get a new shape. Again, from here the shape

was reflect to the below. Then I get a real shape of my tessillation. To get the full

tessillation, the new shape was reflacted to the below for a several time. Then I

colour it.

My shape of the tessellation just look like a star. It also look moden

desing.The desing also look like a islamic desing. My desing also are used at the

mouque as the decoration of the wall.

This dising, also beautiful and simple. For your information, if you are

interested in a dising of shape you can made my tessillation as a wallpaper to the

laptop or computer. This tessillation also can become more beautiful if you made a

combination of colour. For example combination colour, green, blue, yellow, and

purple.

Last but not list, many people use my tessillation desing as a wraping paper.

This is all about my tessillation.

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PROCEDURE OF MY TESSELLATION (B)

For my presentation tessellation, I had done the tessellation by using the basic

shape of square. For my presentation I had done by using a computer. Let figure out

how my tessellation had been finished.

Step 1:

First, I had draw a triangle shapes like this:

Step 2:

Then I continue it by made a 90º unticlock wise rotation to the basic shape.

Object

Point of rotation

Image

Then I get the new shape just like above.

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STEP 3

For next step, I had made the same a 90º unticlock wise rotation to the color of

shape .

Image

object

STEP 4:

For next step, I had made the same a 90º unticlock wise rotation to the color shape

image

Object

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STEP 5

Then, I had made a reflection to the new shape.

STEP 6

Then repeat 5 until get shape like below .

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STEP 7

Then reflect the shape in the step 6.

object

line of reflection

image

STEP 8

Repeat step 7 until get the full tessellation

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STEP 9

Colour the tessellation.

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EXPLAINATION OF MY TESSELLATION (B)

My shape of the tessillation just look like a star. It also look moden desing. It

was create by a rotation 90º anticlock wise of the triangle. The step was repeted for

three time. Then I get a new shape. To get the full tessillation, the new shape was

reflacted to the right side, then to the below. Then I colour it.

The desing also look like a islamic desing. My desing also are used at the

mouque as the decoration of the wall.

This dising, also beautiful and simple. For your information, if you are

interested in a dising of shape you can made my tessillation as a wallpaper to the

laptop or computer. This tessillation also can become more beautiful if you made a

combination of colour. For example combination colour, green, blue, yellow, and

purple.

Last but not list, many people use my tessillation desing as a wraping paper. This is

all about my tessillation

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3.2.2 MUHAMMAD RIDHUAN

PROCEDURE OF MY TESSELLATION (A)

This first tessellation usually can be finding in ‘batik’ design or wood craft and

others. There are some steps to form this tessellation look like diamond and stripe.

But, to get this shape, this tessellation must be arranged properly to make sure the

line are correct to another.

Step 1:

This is the basic shape of this tessellation. From this shape, I had made some

reflection to get the diamond and stripe.

Step 2: [ line of reflection ]

From the basic shape, I had made a reflection to get the second shape. This step I

repeat for several times to form such a line.

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Step 3:

[ line of

reflection ]

In this step, I had made reflection from the upper shape to get the bottom shape.

This reflection had form a diamond shape as shown.

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Step 4:

[ diamond

shape ]

[ stripe ]

This is my tessellation after had arrange from the reflection process. There

are diamond and stripes as shown. Some colours are used to make a different

between the shapes to another.

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EXPLAINATION OF MY TESSELLATION (A)

This tessellation is form from the basic shape that shown in procedure. Then

this basic shape is reflecting to its side and its bottom to build some design like

diamond and stripes. The diamond shape clearly can be seeing between the basic

shape and the reflection shape at the bottom reflection.

Usually, we can see this design at batik craft and wood craft. ‘Rumah

Kampung’ usually arranges their wall that had built from wood. From this

arrangement we can see the design likes a line.

To make like stripe, more color need to use to show the different between

another. But this color most suitable if use to design batik craft and not too suitable

for house. It is because the stripe produces really very beautiful if the colors are truly

matching.

Now a day, if we see at the new building, there are some traditional craft on

their built. Some of them are using wood. Sometimes they used this arrangement to

make their building look familiar.

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PROCEDURE OF MY TESSELLATION (B)

This second tessellation most referred to simple tessellation. Comes from the square

as the basic shape, I also use the reflection technique to get another shape. The

step in making this tessellation is shown below.

Step 1:

The basic shape is square. In the square there are six triangles that form one one

square at the center. But I had use circle in both triangles’ angle.

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Step 2: [ line of reflection ]

Second step, I had used the reflection technique to get another shape. I used the

color to show the shapes that are form from this reflection process.

Step 3:

[Line of reflection]

From step 2, I made once the reflection to the bottom.

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Step 4:

So, come from both reflection from step 2 and 3, the arrangement of this tessellation

as shown.

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EXPLAINATION OF MY TESSELLATION (B)

This tessellation showed more similar like chemical bonding. Besides, it looks

simpler than the first tessellation. To form this design, the same techniques are used

that is reflection. There is some basic shape that had been combining together and

then reflected to get another shape.

Although this tessellation looks simpler than the first one, this tessellation

need more process including in reflection process for side reflection and bottom

reflection. That’s why don’t judge the book by its cover.

I don’t think that this shape suitable for batik handicraft like first tessellation.

But, I think, if this tessellation is using for pictures frame in the house, it also quite

beautiful. The arrangement makes this tessellation like wave if using suitable color.

But may be for a person that obsesses with design like that, they used this

tessellation design to paint their house or class and others. It had different thinking

between another.

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3.2.3 MUHAMMAD AMINUL KHALIL

PROCEDURE OF MY TESSELLATION (A)

Tessellation is created when a shape is repeated over and over again

covering a plane without any gaps or overlaps. For this part, I would like to show my

work on tessellation. Actually, it had been decorated by using the basic shape of

triangle and trapezium. Then, it was shaped become other object. So, let’s we study

how the design work and produced.

Step 1:

First, draw a triangle shapes like this:

Step 2:

Add a trapezium shape at the bottom of the triangle.

+ =

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Step 3:

To create something different, double the object oppositely.

+ =

Step 4:

To form a tessellation, do a reflection of the object. So, it becomes more and

multiple.

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Line of reflection

object image

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Step 5:

To construct another side of reflection, the image also can be formed to lower side.

Step 6:

Actually, the image will become wider when it is translated. Let say, it is translated

one step to the right, and one step down (1, 1). The image will be seen like:

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Line of reflection

object

image

1

2

object

second translation / image

first translation

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Step 7:

Besides, to form another rules of tessellation, the object also can be rotated. It will be

described as rotation clockwise or anti clockwise for 90˚ and so on. The most

important thing is the point of the rotation. Let say, we rotate the object likes:

Step 8:

The rotation, reflection and translation as shown in step 5, 6 and 7 could be drawn

anywhere as long as there has a suitable point, line and coordinate placed. Finally,

the image of the repeated process of the tessellation might cover or occupied the

space of the plane as shown like the diagram below:

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point of rotation

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Step 9:

Since a tessellation should be clear and attractive, try to play with colour to give the

impact. Colour the whole design properly.

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EXPLAINATION OF MY TESSELLATION (A)

As shown in the procedure before this, it was clearly seen that the pattern is

simple and typical. It also always been seen at our surroundings like tiles, wallpaper,

mat and so on.

Actually, the design was started from a triangle and a trapezium to form a

lamp or ‘pelita’. Since the Eid Days just around the corner, I felt the lamp is suitable

to be drawn over the plane. But, after alter it to produce something interesting by

doing some rules of tessellation like rotation, reflection and translation; I noticed that

the original shape could become other attractive object. So, the design undergoes

transformation from a typical shape into marvellous shape. Finally, after it joined

together and also coloured well, the art was appeared.

I think this design of tessellation is suit to be placed at wallpaper especially for

the wall of brown or pink. It is because, the drawing could be seen alive when

combines with the colours. The shape will look highlighted with the fantastic

colouring on the wall. But, perhaps it is not suitable to be drawn at the tile especially

in toilet because it would be weird and strange. I hope I can produce more beautiful

design one day in order to improve my skills and virtual spatial tense.

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PROCEDURE OF MY TESSELLATION (B)

For the second design of tessellation, I had created a bit complex to try a new

skills plus to test my ability. It was about a bird flying in the air. But, I emphasized at

the head of bird part in the tessellation design. So, let’s we try the new one.

Step 1:

First, draw a triangle shapes like this:

Step 2:

Add up two more triangles at the bottom and above. Make sure it is joined closely.

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2

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Step 3:

Now, draw two trapezium fit to the below of the triangles formed on step 2 like:

+ =

Step 4:

Then, double the object oppositely or reflect it to form a side of the bird eye.

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Line of reflection

image reflection

object reflection

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Step 5:

Then, double the eye by doing reflection rules also. Make sure the eyes will be

formed.

Step 6:

Next, after get the shape of the eyes, add two diamond shape and place them at the

bottom and above of the eyes. Put them in the middle to show the shape of bird’s

crown and its forehead.

+ =

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Line of reflection

image

object reflection

forehead

bill

1

2

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Step 7:

In the making of tessellation, the head of bird object do a reflection of the object. So,

it becomes more and multiple.

Step 8:

Actually, the image will become wider when it is translated. Let say, it is translated

one step to the right, and one step down (1, 1). The image will be seen like:

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Line of reflection

image

obejct

1

2

first translation

second translation / image

object

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Step 9:

Besides, to form another rules of tessellation, the object also can be rotated. It will be

described as rotation clockwise or anti clockwise for 90˚ and so on. The most

important thing is the point of the rotation. Let say, we rotate the object likes:

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point of rotation

image

object

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Step 10:

The rotation, reflection and translation as shown in step 7, 8 and 9 could be drawn

anywhere as long as there has a suitable point, line and coordinate placed. Finally,

the image of the repeated process of the tessellation might cover or occupied the

space of the plane as shown like the diagram below:

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Step 11:

Since a tessellation should be clear and attractive, try to play with colour to give the

impact. Colour the whole design properly.

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EXPLAINATION OF MY TESSELLATION (B)

As shown in figure of the design, the tessellation had formed bird’s head and

it seems like flying on the air. The blue colour background of the plane showed the

element of cloud. It is clearly to see the arrangement of the birds on the blue air.

Actually, I inspired to design this after seeing a bird flying at the hostel.

Coincidently, I was struggling to get any idea to create the complex tessellation. So,

why not I trying to create a new vision of art like that. Then, I felt so eager to

complete my imagination plus I never did something artistic before this.

This design is able to be placed at high. For example, it can be hang up at the

door to show the movement of the bird perhaps. I think people will attract to watch it

many times to get the real meaning of the design. It was so artistic and natural. I

hope, I can do more interesting design that can give affective impact or emotions to

others so that the value of art would be glowing.

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4.0 REFLECTION

4.1 MUHAMMAD AMIR AL-HAFIZ

On 13 August 2010, I and all my classmates had been given an assignment

from our Numerical Literacy lecturer, Pn. Nazimah Binti Abdullah. This assignment is

including in the Spatial Sense topic and consisted of two different tasks. In the first

task, we have to create a 2-D tessellation design using any suitable tools and give a

description of our design and explain how we created the design. We have to do this

task, individually. In the second task, in groups of 3, we have to construct physical

model of two different types of polyhedral. In order to complete this task, we have to

search the internet for information on various types of polyhedral. We also need to

give a description of your models and explain how we constructed the models.

Basically, when I got this assignment, I was wondering and quiet not

understand about what I suppose to create. It is because the words tessellation and

polyhedral are new for me. Before this, I am never hear the words of tessellation and

polyhedral. I am also did not know the actual feature of tessellation and polyhedral.

In order to overcome this problem, I had taken some actions that can help me

understand more about tessellation and polyhedral.

First, I was listening to the explanation that had been given by Pn. Nazimah

about the definition and the features of tessellation and polyhedral. From this

explanation, I am already know about tessellation and polyhedral. Besides that, Pn.

Nazimah also show us the real model of tessellation and polyhedral and from that I

can see the actual features of tessellation and polyhedral that I suppose to construct.

Besides that, in order to increase my knowledge and information about tessellation

and polyhedral, I had surfed the internet. From the research, I had found many new

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information about tessellation and polyhedral, their features and the examples of

tessellation and polyhedral models from all over the world that were very interesting

for me.

From the information that I got from the lecturer and the internet, I began to

create my own tessellation design and polyhedral model. Both of these tasks teach

me about the actual meaning of patient and diligence. I have to sacrifice a lot of my

time in order to create the tessellation design and construct the polyhedral model.

Sometimes, I feel give up to continue this works because they are so complicated.

However, when I see the tessellation design and the polyhedral model that I created,

I think my efforts had paid off. It is quiet difficult for me to believe that I had created

my own design. At the same time, I feel proud of myself because I had successfully

overcome my giving up feeling. From this I had learned that, if we want to create

something that is good in the future, we must dare to face any obstacles that come.

It is because, behind the bad thing, there must be a good thing.

Besides that, this task also had give me many information about tessellation

and polyhedral. If in the pass I did not know what are tessellation and polyhedral, but

from now on, I had know that both of tessellation and polyhedral are the formations

that are unique in their features. I also already know the definition of both of the

formation and their differences.

Lastly, I hope that all the experiences that I gain from doing this project will be

useful for me when I become the teacher next time. I am also hope to do the

interesting project like this project again and hope that this project will become a

guide for everybody to gain more information and knowledge about tessellation and

polyhedral.

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4.2 MUHAMMAD RIDHUAN

Firstly, a lot of thank to Allah because giving his chance to finished this

assignment in time given successfully. This short term assignment needs to do with

group of three. So, I, Amir and Khalil had work on collaboration during the time given

in way to finish this assignment. A lot of memory had I written during working

towards this assignment. Memory that I hope can be used in school soon.

On 13 August 2010, I and all my classmates had been given an assignment

from our Numerical Literacy lecturer, Pn. Nazimah Binti Abdullah. This assignment is

short term course that including in the Spatial Sense topic and consisted of two

different tasks. In the first task, by individual we need to create a 2-D tessellation

design using any suitable tools and need to give a description and explanation on

how the tessellation are formed. For the second task, by groups of 3, we need to

construct physical model of two different types of polyhedral.

Basically, when I got this assignment, I was quiet not understand about what I

suppose to do. It is because the words tessellation and polyhedral are very new for

me although I had studied about translation in secondary school. So, to make sure

our assignments are correct, we need to collaborative do some research on that.

For both work, to get the description, I had make some research including all

source such as internet, book and sometimes by asking our senior. From this, I had

decided the best tessellation that I can draw for individual task. Although there are

references, to get my best tessellation, I need to draw about ten shapes. These

individual tasks really teach me who the patient teachers in teaching are.

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From the explanation that had been given by Pn. Nazimah also, I had use it to

apply in our assignment such as about the definition and the features of tessellation

and polyhedral. From this explanation, I got some idea about tessellation and

polyhedral. Besides that, Pn. Nazimah also show us the real model of tessellation

and polyhedral that had been done by senior and from that I can imagine the actual

features of tessellation and polyhedral that we suppose to construct.

From zero information about polyhedral, I really very-very happy on our

design. This is because, before this we don’t know anything about this polyhedral,

but with good collaborative, finally in two days, we can finished that. Other team

were jealous with our group because in short time, we can finish the polyhedral

model than their team. This situation, I think, when there are good collaborative, may

be in one day two or three model can be finished. So to get this target, we need to

work more collaborative soon.

Finally, I hope, from effort that I give as individual such as forget sleeping

time, short rest and others can be my best experiences to me. It is because I very-

very confident, the target of this assignment to add more input to student had

reached.

That’s all. Thank You.

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4.3 MUHAMMAD AMINUL KHALIL

The task consists of two parts of questions. The first question required us to

create a 2-D tessellation design using any suitable tools. We have to create them

individually and also give a very brief description for the designs besides explain the

procedures or steps of all the designs. The second questions need us to constructs

models of two different types of polyhedral.

For tessellation as individual task, I noticed that it make me more concern

about art. I know I am very poor on that, so I feel nervous before going start do all

the projects. But, after learned from my group members which are Amir and

Ridhuan, I feel like it was not too difficult and enjoy doing it as long as we always

practice frequently. To complete this task, we need to search the internet to find the

information on the various designs of tessellation.

Actually, there are so many types of tessellation. Either it is easy or vice

versa, it is still have to do properly and patient. I found so many beautiful and

colourful designs on the internet, but I did not feel satisfied if do the same pattern like

them. It was proven when I felt eager to complete the designs from the beginning

until the last dot. So, I decided to create my own designs even though it was look

simple and typical. At least, I can be proud of myself.

The most important thing I had learned from the tessellation task is the usage

of them. Although tessellation is part from mathematical field, it also could be used

for decoration. It is because tessellation is art. That is why we can found them on

clothes, tiles, drawing and so on. Actually, almost all the designs created in this

world comprising tessellation element. Even in the toilet.

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Secondly, the question asked to produce two physical models of polyhedral.

Frankly, I never heard about polyhedral before this. But, after the great explanation

and lesson from our lecturer, slowly it comes to my mind and heart. But it is still

difficult if we did not try to do it and practice. Luckily, I had matched with two persons

that so creative and good in virtual spatial quotient. They always help me and teach

me how to become softer when work on art.

After doing research on the internet, we found two models of polyhedral that

we felt suitable and interesting to be shown. Finally, we decided to choose Spinner

and Japanese Brocade. Both are represented types of polyhedral of octahedron and

hexahedron respectively. Totally, it is difficult to do but by referring to many sources,

we can do it better. Actually, there are so many sources and references in the

internet and also for books. So, we will study the procedure and steps well without

feeling confused.

Polyhedral is defined as a geometric solid with flat faces and straight edges.

Polyhedral can be built up from different kinds of element or entity; each associated

with a different number of dimensions for example 3-D dimension, 2-D dimension

and 1-D dimension. But, it will be seen attractive when shown by 3-D dimension.

That is what we do in the project. Paper folding skills are essential in order to make it

perfect. Moreover, colour also need to emphasize to gleam the models.

Lastly, again I would like to thank to all individual person that involved in this

project. Because this course work is more to communicative and constructive

learning that enable students do it by hands-on. So, for sure all members of both

classes PISPM 2B and PISMP 2C involving together. Hopefully, this assignment

would help others to know about the art in mathematical lesson. Thank you.

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5.0 COLLABORATION

RECORD OF COLABORATION

GROUP MEMBERS:

i) MUHAMMAD AMINUL KHALIL BIN ZAINI

ii) MUHAMMAD RIDHUAN BIN GHANI

iii) MUHAMMAD AMIR AL-HAFIZ BIN ZULKIFLI

DATE BRIEF DESCRIPTION OF DISCUSSION

SIGNATURES OF GROUP MEMBERS

SIGNATURES

OF LECTURER

Briefing from Puan Nazimah

about assignment.

Discussing all of the outlines to

decide the models.

Search information about

polyhedral and tessellation.

Discussing with group members

to divide the tasks.

Show our polyhedral models to

lecturer.

Make the models for polyhedral.

Print out the assignment.

Submit the assignment.

6.0 REFERENCESKERJA KURSUS PENDEK

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6.1 BOOKS

Albert B. Bennett, Jr. and L. Ted Nelson. (2004). Mathematics for Elementary

Teachers, A Conceptual Approach, Sixth Edition. New York: Mc Graw-Hill

Companies, Inc.

Derek Haylock, Anne Cockburn. (2003). Understanding Mathematics in the Lower

Primary Years. Paul Chapman Publishing.

John A. Van De Walle. Elementary and Middle School Mathematics: Teaching

Developmentally, Fifth Edition. Pearson.

6.2 INTERNET

___________. (2010, June 2). Prism (geometry). Retrieved August 18, 2010, from

Wikipedia The Free Encyclopedia:

http://en.wikipedia.org/wiki/Prism_(geometry)

___________. (n.d.). Polyhedron. Retrieved August 12, 2010, from Math Is Fun?:

http://www.mathsisfun.com/geometry/polyhedron.html

___________. (n.d.). Tessellation. Retrieved August 17 , 2010, from Math Is Fun?:

http://www.mathsisfun.com/geometry/tessellation.html

Hart, G. W. (1996). Pyramids, Dipyramids, and Trapezohedra. Retrieved August 15,

2010, from Virtual Polyhedra:

http://www.georgehart.com/virtual-polyhedra/pyramids-info.html

Hatter, K. (n.d.). How To Draw a Tessellation. Retrieved August 16 , 2010, from

eHOW: http://www.ehow.com/how_5158327_draw-tessellation.html

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