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Tessellations - for teachers


Index --- grids --- squares (examples) --- triangles (examples) --- Escher-style (examples) --- for teachers

Tessellations are regular patterns of shapes fitting snugly together. They are studied at Primary School at about 9-11 years old, but they make attractive patterns, so are suitable for older children as well. Patwork quilters or DIY tile layers may be interested as well.

Probably the best way to start looking at tessellations is to use kits of shapes: triangles, squares, pentagons, hexagons, octagons, etc. These can be made of card or you can buy them made of plastic. However, you need quite a few shapes to experiment with - say a dozen of each shape, and half a dozen different shapes. Then each child or pair of children need their own kit, which means several hundred shapes floating round a classroom! If this fills you with horror, then this website tries to give an alternative approach.

Tessellation grids

The grids webpage shows regular grids of the basic regular shapes, plus a few ideas of how to colour them in. It also has links to big versions of these grids. There are also grids for combinations (such as squares and triangles, or octagons and triangles). Finally there are a few more irregular patterns, just for fun. You can print out the big patterns directly, or save them onto your own computer, where you can fill in the patterns using any paint software. Paint exists on every Windows computer as part of the basic operating system (I give instructions on each large grid page as to how to find and use it) or you can use other similar software. This means if you have enough computers, the children can fill in these grids on the computer very quickly, and able to correct any mistakes. However, if you don't, you can print off each grid and copy it enough times for each child, so they can use felt-tips or crayons. For either method, you should encourage them to make tessellations - that is, they should colour it in to produce regular patterns which repeat. The examples should show what needs to be done. Probably the whole of a big grid would be too boring to colour in completely, but they could experiment with different tessellations on each page. One advantage of this colouring in technique over the plastic or card shapes is that you can colour in the shapes in any colour. This produces completely different effects. The kits tend to make all the squares one colour, and all the triangles another, and so on. Rather dull!

Tessellation designer webpages and examples

However, the grids provide no discipline to stop children colouring them in any old how. (One favourite is to colour them in as a mandela, with a centre and the pattern spreading outwards. Very pretty but not a tessellation, alas!) So there are interactive pages which provide this tessellation discipline. Essentially, you fill in one 'tile' and the webpage repeats this to create the entire pattern. This makes a pattern very quickly, as the computer does most of the work. There are three webpages for this. The squares webpage has a square grid. This is the easiest to work with, as most people understand squares. While playing with the webpage, I found that many of the patterns had strange effects visually, so I made an optical illusions webpage. That gives some examples of the type of tessellations that this webpage can produce, and the illusions are interesting in their own right. The triangles webpage has a triangular grid, which is a little trickier to handle than squares, but can produce some very attractive patterns. You can do hexagons or diamonds with the triangles grid. There is also a webpage of triangle examples. The triangle page is changed from how I used to do it. If you want the old page, click here.

Both the squares and triangles tessellation pages repeat each tile with exactly the same colours, and this is certainly the easiest way to understand tessellations. However, there is a different type of tessellation where alternate tiles are different colours. A chess board is the simplest example. The most famous tessellations were created by M.C. Escher, and he used this technique to create wonderful animals and fishes and monsters, all of which fitted together as tessellations. The Escher-style tessellations webpage makes patterns with these alternate colours, so when you make a sticky-out bit on one edge, a bite mysteriously appears on the next. I must admit that it is extremely difficult to make anything that looks like an animal, but it's fun trying! The Escher-style page is changed from how I used to do it. If you want the old page, click here.

The square tessellation designer has an option of tessellation type. The default value is the simplest - that each tile is repeated the same way round to complete the row, then each row is repeated. However, there are options for other ways to repeat the tiles. Each tile could be a mirror of the previous, for example. I suggest that the children start on the simplest tessellation type, so they can understand what's going on. Then they can 'play' with the other types to see what happens. Some may even work out what the different types mean! But it doesn't matter if they don't - this is all really just play. The Escher-style designer also has options, but this time, the reflect and turn happens within the tile. This is because it is intended for making animals and people, who tend to be symmetrical! The triangle tessellation designer has options which change the shape of the tile rather than producing different symmetries.

Printing a tessellation

This website lets you design a tessellation online. It uses JavaScript to do this. This means that your finished tessellation is not a conventional picture, so you can't save or print it in the normal way. However, there are other ways to do this.

The easiest way to print a tessellation is to print off the whole webpage in the normal way. Cut out the tessellation if you wish.

Copying a tessellation to a word processor

There are two ways to copy the tessellation to a word processor so you can print or save it to print later.

You can high-light the tessellation with click-and-drag. Once you have done this, click on Edit/Copy (top of screen). Leave the webpage and open a word processor file, such as Word. Clink on Edit/Paste in the word processor file and wait a bit, and the tessellation should appear. The format may look a little ragged. Now it is part of the word processor file, you can save it or print it as you wish. I cannot guarantee that this will work with all word processors.

The second way is to screen print. On standard Windows keyboards, there is a key (top right) with Print Scrn or similar on it. When you have the tessellation on screen, click on this. This dumps whatever is on the screen into the 'copy' area (which is called the clipboard). Now go into a word processor file, or any program that handles pictures (such as Paint). Now click on Edit/Paste, and the screen image will appear. This will have more than just the tessellation. If you are in a Paint program, then you should be able to trim it. Whether you do or not, you can now print or save it as you wish.

Saving the tessellation to continue working on it

The methods above preserve the finished tessellation. But you may wish to save a particularly complicated tessellation, and continue working on it in future.

All the design pages have a white box below the tessellation itself (you may need to scroll down). This contains lots of letters. As you design the tessellation, you will see that the letters change. At any time, you can copy-and paste this collection of letters (using click-and-drag in the normal way) into a word processor or Notepad or any other file that accepts writing. There you can save it in the normal way. When you wish to continue working on the maze, you can copy-and-paste it back from the word processor file to the white box in the tessellation design page (make sure you get the right one!), and click on Regenerate maze. Wait a short while, and the maze will re-appear!

You could save all your tessellations this way if you wish, to keep a record of them.

Downloading the webpage to your own computer

If you want, you can download any of these webpages onto your own computer. This will not save any design that you have been working on. But it does mean that you can work on the page even if you're not connected to the internet, and you will avoid internet problems.

To use the webpage:

Remember this will only save one webpage. You cannot move from one webpage to another via the links.


Links to other websites

Other people's websites about tessellations

Tessellations - information and examples
Escher pictures, including tessellations
The official M.C.Escher Website, with gallery
Patterns from the Alhambra
How to make tessellations with pencil and paper
Information about tessellations, and more links
Hop's Escher Tiles
17 Wallpaper Groups
Penrose tiles (these never repeat!)
Hyperbolic Tessellations (another extension of the idea)
General Maths websites

Mathagony Aunt - for those who want to enjoy maths
Keith's Think Zone - Fun and interesting miscellany
Ask Dr. Math
Interactive Mathematics
Mathworld
MegaMathematics
NRICH - enriching mathematical experiences
Plus - living Mathematics