FIRST INTUITIONS ABOUT TOPOLOGY ------------------------------- Today's lecture introduced, in a nontechnical way, certain key ideas from the branch of mathematics called topology. Topologists study objects called _(topological) spaces_. We won't give a formal definition of _topological space_ because that would carry us to a level of generality far beyond what we need. Instead we shall concentrate on a very, very special class of spaces called ``surfaces.'' Eventually we must make precise when to call a mathematical object a ``surface'' and when to count two surfaces as essentially the same. All the same, the place to start is with some interesting examples. Our first example today was the so-called Mobius strip. I drew two pictures of Mobius strips, which looked something like the following: ___ ___ /\ \ /\ \ / \ \ / \ \ / / \ \ / / \ \ /__/___\__\ / /___\__\ \ / \ / / \_______/ \_______/ Mobius strip with Mobius strip with 1/2 twist 3/2 twist Pictures like this are called ASCII art (ASCII was the original standard for typewriter style text in a computer file, before the days of word processors). ASCII art has its limitations, as you can see. Understanding what's the same about these pictures and what's different gives us a good start at understanding what matters in topology and what doesn't. What's different is that these pictures do indeed represent different physical objects. That means that if you take the trouble to cut a long strip of paper and tape its narrow ends together as indicated in the first picture, then no matter how you twist (and even stretch, if paper could be stretched!) you will *never* end up holding something that looks like the second picture. Regardless of whether you find this claim plausible or not, this claim is *not* obvious. To a mathematician, such a claim requires _proof_, and for now we won't go into the complete argument. Nevertheless, let's take the first step. Here's what it looks like if we draw just the edges (or boundaries) of the two strips: /\ /\ /\ /\ / \ \ / \ \ / / \ \ / / \ \ /_________\ /_ /______\ / \ / \ /_______\ /_______\ You can see that the edge on the left can easily be unknotted. The edge on the right forms a trefoil knot. Probably you aren't surprised to learn that the trefoil knot can not be unknoted (without breaking the loop), but this is not so easy to prove and that is the step we omit. If one could twist a 3/2-twist Mobius strip into a 1/2-twist Mobius strip, the very act of doing so would twist the trefoil knot boundary of the 3/2-twist Mobius strip into the unknoted boundary of 1/2-twist Mobius strip. So if you believe the latter is impossible, you must believe the same for the former. So that's a bit about what's different about the two pictures. Here's what's the same. For both surfaces we can form an _atlas_ consisting of 6 rectangular _patches_ (that's actually the technical term) ___ ___ /\ \ /\ \ X \ X X \ X / \/ \/ \ / \/ \/ \ X / \ X X / \ X /_\/_____\/_\ / \/_____\/_\ \ | | / \ / | | / \__|___|__/ \__|___|__/ - \ /| _____ _____ _____ _____ _____ _____ \ / | | | | | | | | | | | | \ / | | | | | | | | | | | | / | | | | | | | | | | | | / \ ----- ----- ----- ----- ----- ----- / \ / \| - where the crossed arrows mean that the right edge of the last rectangle matches up with the left edge of the first rectangle *after a twist*. Imagine, if you can, driving on a planet whose surface was shaped like one of these Mobius strip or the other. The point is that the atlas that tells you everything is connected up on the first surface works just fine for the second surface. So from you experience living within the surface, you can't distinguish between a 1/2 twist and a 3/2 twist. Thus we make the following informal definition. We regard two surfaces as having the same topology if a SINGLE ATLAS WILL SERVE FOR BOTH. The atlas above was unnecessary complicated. For a mobius strip a single patch will do: -------------------------------- | | / \ | | | | \ / | | -------------------------------- Here the arrows tell you to match up the left edge and the right edge with a twist. (The top edge and bottom edge will then form a single continuous boundary.) We notate the surface we get, that is, the Mobius strip, as *x. The * indicates the ``circular'' boundary; the x indicates the twisted region within the boundary. We also use the name *x for the symmetry type of a plane pattern which serves as a map for such a surface. Here's an example, a bit different from the one I use at the blackboard (since my typewriter has no backward R): M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- Note that one has refection lines between each row of type and glide refection lines running along the middle of each row of type. Another surface that interests us is the Klein bottle, which also has a one patch atlas: \ -------------------------------- | / | / \ | | | | \ / | \ | -------------------------------- / Here we first match up the top edge and the bottom edge, without any twist, to form a long open cylinder. Then match up the circular edges at each end, not the way leads to a donut like surface, but with the opposite orientation. If you try to do this physically you will find that you must pass the surface through itself. WE JUST WON'T LET THIS BOTHER US since the physical realization of surfaces in 3-dimensional space plays no role in our main topic, symmetry of patterns on the plane. Incidentally, we wouldn't have any trouble if we had four dimensions of space to move around in. One can illustrate this by using color as a fourth dimension. Then we can arrange for a red region of the surface to pass through a blue region; the difference in color means that there's no collision as far as we're concerned. The Klein bottle gets the symbol xx because its actually two Mobius strips taped together along their boundaries. It's easy to see one of them, running along the center of the strip. \ -------------------------------- | / | / \- - - - - - - - - - - - - - -| | M o b i u s s t r i p | |- - - - - - - - - - - - - - -\ / | \ | -------------------------------- / What's left if you remove this Mobius strip is another Mobius strip. There are at least to ways to see this. First you could rearrange the picture, since the top and the bottom were suppose to get taped together anyway. Second, you could think of this Mobius strip as the front of the cylinder, so what's left is the back of the cylinder, but by symmetry, the front and the back must have the same topology. The following pattern could be a map of a Klein bottle: M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- W- M- The repetitions as you move up and down correspond to going around and around the cylinder. The glide reflections as you move either between lines of type or along the middle of a line of type correspond to moving along the spines of the two Mobius strips. Note that the Klein bottle has no boundary and correspondingly, the pattern has no reflections. The final surface in today's lecture is the projective plane. It's atlas looks like this: / -------------------------------- | \ | / \ | | | | \ / | \ | -------------------------------- / The projective plane is twisted up so severely that visualizing it in 3-dimensional space, though possible, isn't an easy matter, even if you don't mind having it pass through itself. The projective plane would have the symbol x , but it does not actually arise in connection with plane symmetries in its raw form. What we do see though is a 22x. Consider a rectangular pillow case with some decoration on the front (we'll use M-) and the same decoration on the back, except upside-down and backwards (so it comes out -W). So this pillow case has a symmetry -- it looks the same after you turn it inside out. That means that when you open it out, you get *two* copies of the template for the map; after all we've compelled the artist to base the decoration of the back on the decoration of the front. We would get a map that looks like: M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W- -M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W- -M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W- -M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W- -M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W- -M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W- -M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W-M-W Notice that we have two types of gyration points, the centers of the configurations W- and -M -M W- and also glide reflections running along middle of each row of type. How can we *begin* to visualize the surface? Since turning the pillowcase inside-out brings the right side to the left and the left side to the right, we can cut the pillowcase in half down the middle and discard the right half. Here's what's left: ___________ | /\ | | | | | | | | | | A| |B | | | | | | | | | | \/ ----------- Now turning the pillow case inside out carries the segment of the edge marked A to the position of the segment of the edge marked B, so we should tape (or sew) these segments together. The hard part is that we must tape the *top* of A to the *bottom* of B and visa-versa. You can start to do this with actual needle and thread, but you run into trouble before the job is done. The mathematician isn't concerned (in this instance) with physically doing it, though. For mathematicians its enough to say that we declare the points running down A in the picture to represent the same points in the surface as the points running up B. This means, for instance, that someone moving around in the surface could jump from the point marked * to the point marked o ___________ | /\ | | | | *| | | | | | | | | | | | | |o | | | | \/ ----------- Notice that if we cut out the horizontal middle third of the surface it makes a Mobius strip (explain this to yourself!). Then attaching the upper and lower horizontal thirds makes an open pillow case. So the projective plane can be thought of as a Mobius strip sewed together with an open pillowcase.