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let’s explore
contents what are they? history of the
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tessellations escher’s tessellations
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make tessellations!
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what are they?
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tessellations
what are they?
A tessellation is created when a shape is repeated over and over again covering a plane without any gaps or overlaps. Another word for tessellation is tiling. The word “tessellate” is derived from the Ionic version of the Greek word “tesseres,” which in English means “four”
(the first tilings were made from square tiles) and in Latin, “tessella” which is also from the Greek word “tesseres” is a small cubical piece of clay, stone or glass used to make mosaics. It corresponds with the everyday term tiling which refers to applications of
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tessellations, often made of glazed clay. There are a few rules when it comes to Regular Tessellations: • RULE #1: The tessellation must tile a floor (that goes on forever) with no overlapping or gaps. • RULE #2: The tiles must be regular polygons - and all the same. • RULE #3: Each vertex must look the same.
What’s a regular polygon? A polygon with all sides and all angles equal e.g a square or an equilateral triangle:
What’s a vertex? Where all the “corners” meet:
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There are only 3 types of Regular Tessellations:
Hexagons
Triangles
Again look at each vertex!
Notice what happens at each vertex!
120 + 120 + 120 = 360 degrees
60 + 60 + 60 + 60 + 60 + 60 = 360 degrees
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Squares
What happens at each vertex? 90 + 90 + 90 + 90 = 360 degrees
So, in order for it to be a regular tessellation the regular polygons all need to add up to 360 degrees and for the vertex’s to meet.
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Semi-Regular Tessellations: These tessellations are made by using two or more different regular polygons. The rules are still the same. Every vertex must have the exact same configuration. Here are examples of some:
3, 3, 3, 3, 6
3, 3, 4, 3, 4
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3, 6, 3, 6
3, 3, 3, 4, 4
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These tessellations are made up of hexagons and triangles, but their vertex configuration is different. That’s why we’ve named them!
To name a tessellation, simply work your way around one vertex counting the number of sides of the polygons that form that vertex. The trick is to go around the vertex in order so that the smallest number possible appear first.
That’s why we wouldn’t call our 3, 3, 3, 3, 6 tessellation a 3, 3, 6, 3, 3!
More examples of SemiRegular Tessellations: (try and work out there tessellation name)
tessellations
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history of tessellations
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history of tessellations 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. Some two hundred years later in 1891, the Russian crystallographer E. S. Fedorov proved that every tiling of the plane is constructed in accordance to one of seventeen different groups of isometries. Fedorov’s work marked the unofficial
beginning of the mathematical study of tessellations. However, the most famous contributor was the Dutch artist, M. C. Escher (18981972). M.C. Escher was a man studied and greatly appreciated by respected
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mathematicians, scientists and crystallographers yet he had no formal training in science or mathematics. He was a humble man who considered himself neither an artist nor a mathematician. He is famous for the many tessellation pieces of artwork he created which don’t follow typical regular or semi-regular tessellations conventions.
escher’s tessellations
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escher’s tessellations Escher recognised 4 ways of moving a motif to another position in the pattern. These were four Types of Symmetry in a Plane: •Glide Reflection •Translation •Rotation •Reflection
Translation A translation is a shape that is simply translated, or slid across the paper and drawn again in another place. The translation shows the geometric shape in the same alignment as the original; it does not turn or flip.
Glide Reflection
Rotation
In glide reflection, reflection and translation are used concurrently much like the (see right)piece by Escher, Horseman. There is no reflectional symmetry, nor is there rotational symmetry.
Rotation is spinning the pattern around a point, rotating it. A rotation, or turn, occurs when an object is moved in a circular fashion around a central point which does not move. A good example of a rotation is one “wing” of a pinwheel
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which turns around the centre point. Rotations always have a centre, and an angle of rotation. Tessellation using glide reflection
Reflection A reflection is a shape that has been flipped. Most commonly flipped directly to the left or right (over a “y” axis) or flipped to the top or bottom (over an “x” axis), reflections can also be done at an angle. If a reflection has been done correctly, you can draw an imaginary line right through the middle, and the two parts will be symmetrical “mirror” images.
Tessellation using translation
Tessellation using rotation
Tessellation using reflection
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More examples of Escher’s work
tessellations
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make tessellations!
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make tessellations! YOU WILL NEED •The tessellation grids provided in this pack •Colouring pens or pencils
Using the grids provided colour in your own tessellation patterns. Another alternative is to make your own tessellations from scratch by cutting out regular polygon shapes and sticking them down on a piece of paper. See how creative you can be, can you make great creations like the famous Escher??? Try using other images not necessary shapes just be experimental!
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