By John Casey
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Extra info for A treatise on the analytical geometry of the point, line, circle, and conic sections, containing an account of its most recent extensions, with numerous examples
I, 1 The nonreflecting elements of any of these groups for m its rotation subgroup, at the bottom of t he column. Interlude: About Kaleidoscopes 41 In the same vein, the symmetries of a pattern with signature *2222 are generated by the reflections in the sides of a quadrilateral whose four angles are ~ -that is to say, a rectangle. Here the set of symmetries is no longer unique up to scale; any one version can be continuously reshaped into any other by gradually varying this rectangle. The result is that one set of symmetries can be continuously transformed into the other while consistently maintaining its type.
Mirror Lines The floral pattern to the left has many symmetries. For example, the pattern is left-right symmetric: it has the vertical mirror line shown on the left below. The figure in the middle shows another mirror line, which is a different kind because, unlike the first one, it runs between, rather than along, the petals. Drawing all the mirror lines we can, we get the figure on the right, which is at first sight rather confusing. Fortunately, the small part we've highlighted in the margin contains enough information to reconstruct the whole pattern.
1. Symbol 0 2 Cost ($) 2 Symbol * or x Cost ($) 2 2 4 2 1 1 1 1 3 3 3 3 4 3 4 3 8 4 4 2 5 5 5 5 6 6 5 6 12 N ;;;1 N- 1 N N- 1 00 ? 1. Costs of symbols in signatures. opposite page) The magic theorem no t only classi fies signatures. but helps us calculate the signature of a pattern. The signature 22 x of this pattern. like that of all planar patterns. costs exactly s2. 29 30 3. The Magic Theorem Why is t his? Because, as we shall see in t he next few chapters, t here are Magic Theorems that describe the possible signatures in terms of their costs.
A treatise on the analytical geometry of the point, line, circle, and conic sections, containing an account of its most recent extensions, with numerous examples by John Casey