# Square Graph Paper

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There are lots of ways to tessellate the plane. The most regular tessellations are with squares, hexagons or triangles. The tessellations with triangles and hexagons form a pair - if you draw a dot in the center of each hexagon, and join neighboring dots, you get the tessellation with triangles, and vice-versa. The tessellation with squares stands alone - of you draw a dot at the center of each square and join neighboring dots, you get the same square tessellation, shifted up and across by half an edge length!

It's easy to see why these are the only tessellations with regular figures - the angles inside a regular polygon with n sides are 180-360/n degrees each. For the triangle, with n=3, these angles are 180-120=60 degrees, and six of these angles fit around a point. For the square, the angles are 90 degrees, and you get four squares around a point because 4 times 90 degrees is 360 degrees. Likewise, you can fit three regular hexagons around a point. For any other regular polygon, the angles at each corner don't go evenly into 360 degrees, so they can't be arranged to fit around even a single corner, let alone a whole plane.

If you let yourself use more than one type of regular polygon, you can fill the plane in many more ways. You can download and print out some of these at my page of archimedean graph paper. These patterns are very pretty.

There are all sorts of ways you can use square graph paper - obviously, you can use it to draw graphs! When moving house once, I drew up the plans for our new house on graph paper and made cut-out cardboard scale copies of our furniture. Sitting down with a supply of Blu Tack, my wife and I planned where all our furniture would go. It made moving day so much simpler! Many games are played on a square grid. You could use these printable square graph paper downloads to print yourself some game boards.