![]() ![]() First you need to measure the mid-point in between the free ends of your new, translated pair of blue segments. Copy the blue line, shift the copy downwards, and join it up to the bottom end of the glide reflected pair or segments, as at top right in the figure. Draw a wiggly line, like the blue one centre top here, connected to the top end of the glide reflected, red pair of segments. Next we move the flipped copy downwards, and join up the two red segments. We draw a wiggly line, like the red one here, make a copy of it, and flip it over horizontally, as top left in the figure. To explore that a bit, let’s make up a cell using the one recipe out of the 28 that combines all three types of repeat, a rotation, a glide reflection and a translation. – how the whole cell repeats, to produce the tessellation. – the sequence, and sometimes the orientation, in which the pairs of segments connect. ![]() Still others repeat by a kind of reflection, called glide reflection, of which more below. Other segments repeat to form a pair just by shifting along bit. Some repeat as in the example above, just by rotation, rotating either 60, 90, 120 or 180 degrees. ![]() – how each single segment in the boundary repeats to form a pair. – the number of pairs of segments that make up the boundary of the tile There are actually 28 recipes like this for tessellating patterns that don’t include regular reflections. If there are any junctions where just three segments meet, you’ll need three colours if you want to party colour your pattern). That’s possible whenever the segments in tessellation meet at either six way junctions, or four way junctions. (You’ll see that in this pattern there’s a black/white reversal between adjacent cells. Not quite up there with Escher? But it does give us an example of a completely different pattern, but following the same recipe. It’s magical! And once you are practiced, you can end up with cells that represent something, as in the famous tessellations by M.C.Escher.Īctually, the tessellating creatures Escher came up with were really brilliant – they’re not easy to devise. Now it is a fact that any wiggly loop made up of one pair of identical segments set at 60 degrees to one another (always measuring the angle that will end up inside the loop), and one pair of identical segments rotated 180 degrees from one another, will tessellate perfectly. Only this time the angle between them is 180 degrees. Its two segments are also identical, and also arranged like the hands on an old style clock face. Its two segments are identical, and are joined like the hands on an old-fashioned clock-face, so that one is just rotated in relation to the other. When we isolate this fundamental domain, to the right, note next that we can snip its edge into two pairs of segments, one blue, one red. The rich yellow shape is the so-called fundamental domain out of which the entire pattern is made. Notice that the rosette it’s part of, with the paler infill, is made just by repeating the rich yellow shape at different orientations. Start from the shape with a rich yellow infill in the pattern on the left. There are a whole set of recipes, but to get an idea of how they work, take a look at just one. To make cells that tessellate, you have to follow a recipe. If you try to make a pattern like that out of any old shape, you will either end up with gaps or overlaps: In a tessellation, the cells can have wiggly edges, but still fit together like jig-saw pieces. ![]() However tiles are usually geometric shapes – rectangles or squares as a rule, though triangles or hexagons would be possible too. An obvious example would be tiles on a wall. Note added in March 2011! If you’re new to tessellations, before tackling this post, first watch my later post with an animation of how tessellations work.Īny regular pattern consists of identical areas, which repeat without overlaps or gaps. ![]()
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