INTRODUCING PLANE SYMMETRY TYPES (PART 2) ----------------------------------------- Today we met 6 more plane symmetry types in two batches. The first batch, namely 4*2, 3*3, 22* and 2*22 could be understood as maps of partially open pillowcases. Specifically: A pattern with type 4*2 maps a 45-45-90 pillowcase open along one side. A pattern with type 3*3 maps a 30-60-90 pillowcase open along the side opposite the 60 degree angle. A pattern with type 22* maps a rectangular pillowcase open along any one side. A pattern with type 2*22 maps a rectangular pillowcase open along two adjacent sides. Important points: 1) Opening a 45-45-90 pillowcase along its hypoteneuse, rather than one of its sides, leads to a surface which actually has the same geometry (and the same name, 2*22) as that a *square* pillowcase open on two adjacent sides, just folded differently. 2) Opening a 30-60-90 along the side opposite the 30 degree angle (or along the hypoteneuse) leads to a surface, BUT NOT ONE FOR WHICH WE CAN DRAW A FLAT MAP. In either case the edge will contain a point where we find an angle of 120(=2x60) degrees. If you take one 120 degree region and try to make its sides reflection lines, you will need 720 degrees before the thing closes up, not the 360 you find around points in a flat map. A little crude typewriter art illustrates the point. The 7 below reflects across the line pointing to the the lower left to make the >, which reflects across the vertical line to make the <. But reflecting the < across the remaining line, the one pointing to the the lower right doesn't return us to the 7 (it's pointing the wrong way). We'd have to go all around TWICE to come back where we started. | > | < | | /\ / \ / \ / 7 \ The point is that were not interested in the geometry of every sort of surface, just the ones for which flat patterns might be maps. Terminology: A *cone point of order n* is a point on a surface around which the total angle measure equals 360/n. When you map a surface, the image of a cone point becomes a point of gyration of order n. A *corner point of order n* is a point on the edge of a surface where the total angle measure (on one side) equals 180/n. When you map a surface, the image of a corner point of order n becomes a corner point of order n (place where n reflection lines cross). Observe that strictly speaking, we use the term ``corner point of order n'' to mean two separate but related things. Originally we used to to refer to a feature of a symmetric plane pattern, today we used it to refer to a different sort of feature of a surface. Using the same term two ways causes no harm because the two usages occur in different contexts and the corner points in one sense correspond to the corner point in the other sense when we relate those contexts. All these modified pillowcases have a single edge. The naming scheme works as follows: the number before the * give the orders of the cone points; the numbers after the * give the orders of the corner points along the edge. The * itself represents the (single) edge. The second batch (of which we really discussed just two today) consists of *x, xx and 22x. A pattern with type *x maps a surface called mobius strip; you can make one by taking a long thing rectangle and taping together its short sides with a twist. (A mathematician would say ``identifying its short sides.'') A mobius string has just one edge and one side. A pattern with type xx maps a surface called a Klein bottle, made (at least conceptually) by taping together two mobius strings along their respective edges. If you try to do this you will run into physical problems, but mathematically it makes sense all the same. A pattern with type 22x maps a surface called the projective plane (with two cone points). We will discuss all this further on Monday. -------------------------------------------------------------- I gave a simple scheme for remembering the list of names of the 17 types: 1) Just learn 2222, 632, 442, 333, ** and o. 2) If a name consists just of numbers, you may add a * in front; this gives you *2222, *632, *442 and *333. 3) If a name has two equal numbers after a *, you may ``meld'' them and bring them in front of the *. This way you get 2*22, 22*, 3*3, and 4*2. 4) If a name has a * with no numbers after it, you may replace it by an x. >From ** you get *x and then xx; from 22* you get 22x. ---------------------------------------------------------------- In the last minutes of the lecture we saw that a grid of parallelograms with sides of unequal length (they didn't particurally look that way, but we assumed the lengths were not quite equal!) has type 2222. When the parallelgrams are rectangles, such a grid maps a conventional pillowcase. Otherwise the grid maps a pillowcase which pull tight to a tetrahedron, as demonstrated in class. ==================================================================== Finally, where is all this going? Here's the plan: Step one: We meet the 17 types and the surfaces which they map. We learn to draw patterns with a given type. Given a pattern, we learn to construct the surfaces associated to it. Step two: We build a procedure by which we can quickly recognize the type of a given pattern. Step three: We discuss surfaces in general and systematize our naming conventions. Step four: We learn two classical (and highly non-obvious!) facts about surfaces, due to Euler and Gauss, two of histories most famous mathematicians. Step five: We use these facts to see that our list of 17 is complete.