tess el lations

WAJ3105 Numerical Literacy Tessellations

Defination of Tessellations

Patterns covering the plane by fitting together replicas of the same basic shape have

been created by Nature and Man either by accident or design. Examples range from

the simple hexagonal pattern of the bees' honeycomb or a tiled floor to the intricate

decorations used by the Moors in thirteenth century Spain or the elaborate

mathematical, but artistic, mosaics created by Maurits Escher this century. These

patterns are called tessellations.

What is a tessellation?

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. There are three types of tessellation.

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.

Those using triangles and hexagons-

Amir & As-Sakari


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.

Amir & As-Sakari


Those using triangles and hexagons-

Non-regular Tessellations

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.

Taking account of the above mathematical definitions it will be readily appreciated that

most patterns made up with one or more polyiamonds are not strictly tessellations

because the component polyiamonds are not regular polygons. The patterns might

more accurately be called mosaics or tiling patterns. Regular tessellations in the

mathematical sense are possible, however, with the moniamond, the triangular

tetriamond and the hexagonal hexiamond. Semi-regular tesselations are possible with

Amir & As-Sakari


combinations of the moniamond and the hexagonal hexiamond. Nevertheless I will

apply the term tessellation (as other authors have) to describe the patterns resulting

from the arrangement of one or more polyiamonds to cover the plane without any

interstices or overlapping.

The following definitions and descriptions refer to tessellations of polyiamonds.

Examples are restricted , with some noteable exceptions, to tessellations of individual

polyiamonds.

Tessellations can be created by performing one or more of three basic operations,

translation, rotation and reflection, on a polyiamond (see Figure).

Translation - sliding the polyiamond along the plane. The translation operation can be

applied to all polyiamonds.

Rotation - rotating 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 60o or

multiples thereof.

Amir & As-Sakari


Reflection - reflecting the polyiamond in the plane, as if being viewed in a mirror. The

reflection operation is limited to polyiamonds which are enantiomorphic. An

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

mirror image.

I propose the following classification of polyiamond tessellations which is based on the

operations performed on the polyiamond being tessellated..

Simple tessellations are those in which only the translation operation is used.

Complex tessellations are those in which one or both of the rotation and reflection

operations is used with the translation operation.

A single or multiple of a polyiamond may be combined to form a figure which is

capable of tessellating the plane using only the translation operation. This figure will

be called the unit cell.

A particular unit cell may be filled by multiples of different polyiamonds. Gardner

described how five pairs of heptiamonds could be used to fill the same unit cell

tessellation pattern.

Amir & As-Sakari


Tessellations may be further classified according to how the unit cells containing one

or more polyiamonds are arranged. If the unit cells are arranged such that a regular

repeating pattern is produced the tessellation is termed periodic. If the arrangement

produces an irregular or random pattern the tessellation is termed aperiodic. Another

arrangement which produces a tessellation with a centre of circular symmetry is

termed radial - such tessellations, with the exception of special cases, are complex

and will comprise two three or six unit cells each containing an infinite number of

poyiamonds.

Amir & As-Sakari


All tesselations which are regular belong to a set of seventeen different symmetry

groups which exhaust all the ways in which patterns can be repeated endlessly in two

dimensions.

The reader should realise that polyiamonds of odd order cannot provide simple

tessellations. Every polyiamond of odd order is by definition unbalanced. The

rotation and reflection operations must be used in order to provide balanced unit cells

for tessellation.

All of the polyiamonds of order eight or less, with the exception of one of the

heptiamonds will tessellate the plane. The exception is the V-shaped heptiamond.

Gardner (6th book p.248) posed the problem of identifying this heptiamond and

reproduced an impossibilty proof of Gregory. However, in combination with other

heptiamonds or other polyiamonds, tesselations using this V-shaped heptiamond can

be achieved.

Kerja Kursus Project.

Amir & As-Sakari


1. Kami memilih untuk menggunakan bentuk segiempat sama seperti Rajah

1.

Rajah 1

2. Setelah memilih bentuk yang diingini kami lakarkan pada segiempat sama

tersebut. (Rajah 2)

Rajah 2

3. Garisan segiempat sama tersebut kemudian dipadamkan supaya bentuk

yang dikehendaki nampak jelas kelihatan. (Rajah 3)

Rajah 3

4. Bentuk yang telah perhalusi ini ditukar kedalam bentuk digital dengan cara

menukar ke dalam bentuk imej dengan mesin pengimbas (scanner). Imej

digital ini kemudiannya ditindan untuk mendapatkan

Amir & As-Sakari


skala yang lebih besar. (Rajah 4)

Rajah 4

5. Imej digital yang telah ditindan sekali lagi dibersihkan garisan-garisan yang

tidak berkenaan untuk menyerlahkan bentuk yang dikehendaki seperti

Rajah 5 dibawah.

Rajah 5

6. Sekali lagi imej ini ditindan untuk menjadikan imej teselasi ini menjadi lebih

besar seperti pada Rajah 6 di bawah.

Rajah 6

Amir & As-Sakari


7. Proses ini diteruskan sehingga bentuk teselasi tersebut memenuhi sehelai

kertas bersaiz A4 seperti Rajah 7 di bawah.

Rajah 7

8. Selepas itu proses mewarna teselasi berkenaah dijalankan dengan berhati-

hati untuk mendapatkan hasil yang menarik dan bersih. (Rajah 8)

Rajah 8

Amir & As-Sakari


9. Selepas proses mewarna selesai maka hasil teselasi bercorak seperti pada

Rajah 9 di bawah.

Rajah 9

10. Corak lain yang boleh dibuat berdasarkan bentuk teselasi yang sama

adalah seperti pada Rajah 10, Rajah 11 dan Rajah 12.

Rajah 10

Amir & As-Sakari


Rajah 11

Rajah 12

Amir & As-Sakari


Rujukan:

Charles Ashbacher, (July 4, 2008) Excellent Introduction To The Principles Of Plane Tessellation And A Good Resource For Activities, Iowa United States,

Dale Seymour, Jill Britton, (January 1990), Introduction to Tessallations, New York, Dale Seymour Publications,

John Willson, (December 1, 1983), Mosaic and Tessellated Patterns: How to Create Them, with 32 Plates to Color (Dover Pictorial Archives), Dover Publications

Pam Stephens, Jim McNeill, (April 1, 2001), Tessellations : The History and Making of Symmetrical Designs, Crystal Productions.

Internet (Laman Sesawang)

http://gwydir.demon.co.uk/jo/tess/index.htm

http://mathworld.wolfram.com/Tessellation.html

http://www.coolmath4kids.com/tesspag1.html

http://www.tessellations.org/

http://www.mathcats.com/explore/tessellationtown.html

Amir & As-Sakari

http://www.mathcats.com/explore/tessellationtown.html

http://www.tessellations.org/

http://www.coolmath4kids.com/tesspag1.html

http://mathworld.wolfram.com/Tessellation.html

http://gwydir.demon.co.uk/jo/tess/index.htm

tess el lations

Documents