Tessellations Year 8

Eisteddfod Competition Tessellations Create a tessellating pattern suitable for tiles in a bathroom or kitchen. Computer generated images are acceptable. A5 size is recommended (that is half A4) Please ensure that your name, form and house name are put on the back of the design. For further information please see your teacher or Mrs Thomas (Maths). 9/19/2018

Eisteddfod 2015 All entries must be submitted to your Maths teacher by Tuesday 2nd March, 2015 Maths Eisteddfod Lunch Time – Room 25 Mrs Thomas 1.30 – 2.00 every day See your maths teacher for a lunch pass

What are Tessellations? The word 'tessera' in Latin means a small stone cube. They were used to make up 'tessellata' - the mosaic pictures forming floors and tilings in Roman buildings Tessellations refers to pictures or tiles, mostly in the form of animals and other life forms, which cover the surface of a plane in a symmetrical way without overlapping or leaving gaps. A Roman floor mosaic

M. C. Escher 1898 - 1972 Escher is known as the father of tessellations. He was born in the Netherlands in 1898. He is known as for being an artist as well as a mathematician. Escher created thousands of tessellating shapes in the form of fish, birds, insects, and other beasts. http://library.thinkquest.org/16661/gallery/thumbnails.html

M. C. Escher Escher – regarded as the 'Father' of modern tessellations Famous for his impossible depictions like Waterfall Master of lino and wood cuts and produced many superbly crafted landscapes as well. 'Waterfall' - 1961

M. C. Escher Escher produced '8 Heads' in 1922 - a hint of things to come. Turn the picture upside-down if all the heads are not apparent. He took a boat trip to Spain and went to the Alhambra. There, he copied many of the tiling patterns. '8 Heads' - 1922

The Alhambra Palace, Granada, Spain, 12th-13th Century

Escher’s Last Tessellation His last tessellation was a solution to a puzzle sent to him by Roger Penrose, the mathematician. Escher solved it and, true to form, changed the angular wood blocks into rounded 'ghosts'. Penrose 'Ghosts' - 1971

Tessellation by M. C. Escher China Boy, 1936 Tessellation by M. C. Escher

Tessellation by M. C. Escher Squirrels, 1936 Tessellation by M. C. Escher

Tessellation by M. C. Escher Fish, 1938 Tessellation by M. C. Escher

Design for Wood Intarsia Panel for Leiden Town Hall, 1940 Tessellation transitions by M. C. Escher

Tessellation by M. C. Escher Horsemen, 1946 Tessellation by M. C. Escher

Tessellation by M. C. Escher 4 Motifs, 1950 Tessellation by M. C. Escher

Tessellation by M. C. Escher Scarabs, 1953 Tessellation by M. C. Escher

Tessellation mural by M. C. Escher Fishes, 1958 Mural Tessellation mural by M. C. Escher

Tessellation by M. C. Escher Pegasus, 1959 Tessellation by M. C. Escher

Tessellation by M. C. Escher Birds, 1967 Tessellation by M. C. Escher

Realism & Tessellations Combined Sometimes, M. C. Escher would combine realism and tessellations. Reptiles is an example of this combination. 'Reptiles' - 1943

by M. C. Escher Realism & Tessellation Combined Metamorphosis I, 1937 by M. C. Escher Realism & Tessellation Combined

Realism & Tessellation Combined Cycle, 1938 by M. C. Escher Realism & Tessellation Combined

by M. C. Escher Realism & Tessellation Combined Day and Night, 1938 by M. C. Escher Realism & Tessellation Combined

A Full Life Escher died on March 27, 1972. He had produced 448 woodcuts, linocuts and lithos and over 2,000 drawings. M. C. Escher

What is a tessellation? Any repeating pattern of shapes that cover a plane without overlap is considered a tessellation. The pattern must be able to go on forever. Link Tessellations and tilings can be found in many cultures, both ancient and modern. Japanese Egyptian Islamic Link

Patchwork quilts (strong tradition in Wales)

Tessellations are all around us. Can you think of think of some examples?

Tessellations cover many of our school subjects. Geography PE Art RE

Tiles

Pavements or roads Roman tessellated pavement found in Mumbles in Swansea.

Fishing nets

A honeycomb Each hexagonal section tessellates with the rest.

Fish scales

Footballs

Soles of trainers etc.

The Giant’s Causeway in Northern Island

Eaglehawk neck tessellated pavement Natural salt pools Connecting the Tasman Peninsula to Tasmania is covered in a pattern of regular rectangular saltwater pools.

The Eden Project Cornwall

Tessellation planning In Tessellation Planning, the pattern and colours are not just for visual effect, they represent spaces and their uses.

Tessellations A tessellation is a repeating pattern that tiles the plane without leaving any gaps. Which of these will tile the plane? A regular tessellation is made from a single regular polygon. Square 4 Regular Pentagon 5 Equilateral Triangle 3 Regular Hexagon 6 Regular Octagon 8

Equilateral Triangles: Do tessellate 

Squares: Do tessellate 

Regular Pentagons: Don’t tessellate They make a nice ring.

Regular Hexagons: Do tessellate 

Regular Octagons: Don’t tessellate: This is called a semi-regular tessellation since more than one regular polygon is used. Regular Octagons: Don’t tessellate: But they make a great kitchen floor when put with squares.

Consider the sum of the interior angles about the indicated point. There are only 3 regular tessellations. Can you see why? 60o 60o 120o 90o 90o Consider the sum of the interior angles about the indicated point. 120o 6 x 60o = 360o 4 x 90o = 360o 3 x 120o = 360o 108o 135o 36o 90o 2 x 135o = 270o 3 x 108o = 324o

Interactive There are 8 distinct semi-regular tessellations. The tiles for 4 of them are shown below. Click button to view Click to exit Interactive

Non-Regular tessellations. Other shapes such as rectangles will tessellate.

In fact any quadrilateral will tessellate.

Drawing Tessellations Show that the kite tessellates. Draw at least 6 more on the grid

Drawing tessellations Show that the hexagon tessellates. Draw at least 6 more on the grid Drawing tessellations

Drawing tessellations Show that the trapezium tessellates. Draw at least 8 more on the grid Drawing tessellations

Investigating Polygon Tiles 6. 6. 6 3. 3. 3. 3. 3. 3 4. 4. 4. 4 6 x 60o = 360o 4 x 90o = 360o 3 x 120o = 360o By considering the interior angles about the indicated node, what are the conditions needed to produce a tile. 135o 135o 90o A tessellation can be described by its Schlfli symbol. 4. 8. 8 2 x 135o + 90o= 360o

By completing the table below for regular polygons and by considering the different combinations of interior angles can you find other tiles. Only 8 of these tiles will form semi-regular tessellations. Can you find these? A semi-regular tessellation is one in which the same polygons in the same order appear at every node. Dodecagon Hendecagon Decagon Nonagon Octagon Heptagon Hexagon Pentagon Square Triangle Interior Exterior Sides Shape

By completing the table below for regular polygons and by considering the different combinations of interior angles can you find other tiles. Only 8 of these tiles will form semi-regular tessellations. Can you find these? A semi-regular tessellation is one in which the same polygons in the same order appear at every node. Shape Sides Exterior Interior Triangle 3 120o 60o Square 4 90o Pentagon 5 72o 108o Hexagon 6 Heptagon 7 51.42…o 128.57…o Octagon 8 45o 135o Nonagon 9 40o 140o Decagon 10 36o 144o Hendecagon 11 32.72…o 147.27…o Dodecagon 12 30o 150o 2 x 135o + 90o

Do it yourself! http://www.tessellations.org/slicemethod-ex2-1.htm

Use different coloured card for each polygon Polygon cut outs Use different coloured card for each polygon