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Tesselation
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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-
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WAJ3105 Numerical Literacy Tessellations
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.
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WAJ3105 Numerical Literacy Tessellations
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
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WAJ3105 Numerical Literacy Tessellations
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.
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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.
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WAJ3105 Numerical Literacy Tessellations
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.
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WAJ3105 Numerical Literacy Tessellations
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.
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WAJ3105 Numerical Literacy Tessellations
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
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WAJ3105 Numerical Literacy Tessellations
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
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WAJ3105 Numerical Literacy Tessellations
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
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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
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WAJ3105 Numerical Literacy Tessellations
Rajah 11
Rajah 12
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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
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