Nothing transforms an object more than being incorporated into a pattern. Nobody knew that better than the mid twentieth century artist Mauritz Escher, and you can see many examples of his experiments with patterns on the Escher Foundation website http://www.mcescher.com (click on gallery and then on symmetry).
Patterns seem simple - a matter of repetition of shapes. But there turn out to be surprisingly fascinating rules and constraints.
At the heart of every regular pattern there is an element that is not made up of regular repetition within itself. So for example, a whole face would not qualify, because it would include repetition by reflection - only a half face, like the one to the left, will do. There are actually just four ways that such an element can then repeat, combining with itself over and over, to cover the plane. It can just move in regular steps (translation); it can be reflected (like the two halves of a face); it can be reflected but offset (in what is called a glide reflection); or it can be rotated round a point (like the hands on an old analog clock face). Several of these kinds of repetition may combine.
For example, starting to the left with a letter P here, we make a pattern by first reflecting it, then repeating the result three times in a rotation through 360 degrees, and then repeating that result by reflection, into a hexagonal scheme. There are actually just seventeen combinations of translations, reflections and rotations that give a regular pattern across the plane.
To go into all the seventeen possibilities, see Peter S. Stevens, Handbook of Regular Patterns, an Introduction to Symmetry in Two Dimensions (get it from a library unless you are rich, it's excellent but not cheap). If you have access to Adobe Illustrator, Artlandia sell a plug in for making symmetry patterns called Artlandia Symmetry Works http://www.artlandia.com/products/symmetryworks/index.html (I haven't tried it, but it looks good).
Even without a dedicated symmetry design package, computers make the exploration of regular patterns vastly easier - though it can still prove mind-bending stuff. But all you need for some experiments is a graphics package that let's you copy, reflect and rotate your basic pattern element. It helps a lot if your graphics package let's you specify the angle of rotation exactly, so that you can rotate by 120 degrees for a threefold rotation, or 60 degrees for a sixfold one).
However, note that whereas the pattern elements to the left are separated by geometric edges, the edges of the tiles in comic book picture 16 are curved. They only fit snugly together after a change in shape from the edges of the isolated tiles. The rules for getting tile edges to fit together like that are introduced in the next commentary, to comic book picture 18.
Note the striking brilliance of the light from the lighthouse. That's an illusion, see the commentary to comic book picture 5, The Island.
& Comic Book Picture 16, Metamorphosis
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Nothing transforms an object more than being incorporated into a pattern. Nobody knew that better than the mid twentieth century artist Mauritz Escher, and you can see many examples of his experiments with patterns on the Escher Foundation website http://www.mcescher.com (click on gallery and then on symmetry).
Patterns seem simple - a matter of repetition of shapes. But there turn out to be surprisingly fascinating rules and constraints.
At the heart of every regular pattern there is an element that is not made up of regular repetition within itself. So for example, a whole face would not qualify, because it would include repetition by reflection - only a half face, like the one to the left, will do. There are actually just four ways that such an element can then repeat, combining with itself over and over, to cover the plane. It can just move in regular steps (translation); it can be reflected (like the two halves of a face); it can be reflected but offset (in what is called a glide reflection); or it can be rotated round a point (like the hands on an old analog clock face). Several of these kinds of repetition may combine.
For example, starting to the left with a letter P here, we make a pattern by first reflecting it, then repeating the result three times in a rotation through 360 degrees, and then repeating that result by reflection, into a hexagonal scheme. There are actually just seventeen combinations of translations, reflections and rotations that give a regular pattern across the plane.
To go into all the seventeen possibilities, see Peter S. Stevens, Handbook of Regular Patterns, an Introduction to Symmetry in Two Dimensions (get it from a library unless you are rich, it's excellent but not cheap). If you have access to Adobe Illustrator, Artlandia sell a plug in for making symmetry patterns called Artlandia Symmetry Works http://www.artlandia.com/products/symmetryworks/index.html (I haven't tried it, but it looks good).
Even without a dedicated symmetry design package, computers make the exploration of regular patterns vastly easier - though it can still prove mind-bending stuff. But all you need for some experiments is a graphics package that let's you copy, reflect and rotate your basic pattern element. It helps a lot if your graphics package let's you specify the angle of rotation exactly, so that you can rotate by 120 degrees for a threefold rotation, or 60 degrees for a sixfold one).
However, note that whereas the pattern elements to the left are separated by geometric edges, the edges of the tiles in comic book picture 16 are curved. They only fit snugly together after a change in shape from the edges of the isolated tiles. The rules for getting tile edges to fit together like that are introduced in the next commentary, to comic book picture 18.
Note the striking brilliance of the light from the lighthouse. That's an illusion, see the commentary to comic book picture 5, The Island.