Math 629: History of Mathematics
Frank Sottile

• Opening Remarks: We begin with the story of non-Euclidean Geometry. This is a subject I have taught a few times (it is Math 467 in the TAMU undergraduate curriculum). The treatment in Stillwell is quite superficial (this is a flaw/feature in his book, and perhaps unavoidable as Mathematics has a very long history). One challenge in teaching and discussing non-Euclidean geometry is that there are no good models for it. Studying it in the abstract (e.g. via Saccheri's lines, or more fundamentally his quadrilaterals), one has obviously curved lines that are declared to be straight lines, and it does defy our intuition. Beltrami's model in Section 18.4 has straight lines (segments in the disc) and a reasonable notion of parallel, but distance is odd (how do you compute it?) and angles mean something other than what you see. Many geometers and all modern mathematicians find that Beltrami's conformal models in Section 18.5 to be the right place to work in hyperbolic geometry. Of particular note is the upper half plane model (I also like the two views of projecting to the disc and upper half plane). This makes sense in any dimension (the set of points in Rⁿ with last coordinate positive, where the metric is scaled by the inverse of the last coordinate, generalizing what you see on page 371 for n=2 and then page 372 for n=3.) Still, these models leave something to be desired, in my opinion and from my experience teaching.
  
      This is why I have developed the construction activity that you are to do. The model we are making, the hyperbolic soccer ball, is based on one of the tessellations of the hyperbolic plane, but not one shown in Section 18.6. I note that one of my collaborators (who is Romanian) is from Cluj (as was Bolyai), and he received one of his Ph.D.s there.
  
      Because there is a need to move on, I am also assigning Chapter 19 on group theory as part of your reading. It is worthwhile to consider the symmetry group of the tessellation of the hyperbolic plane that your model represents as you read the Sections 19.5, 19.6, and 19.7.
  
  
• Reading:
  1. Chapter 18 in Stillwell's book. This is the story of the discovery of non-Euclidean geometry.
  2. The handout, which I sent to you in a package. My general webpage on the hyperbolic football, as well as the description of the construction you are to do, and finally also the pictures from a construction of a smaller model with essentially the same pieces.
  3. Chapter 19 in Stillwell's book. Group theory is a staple of modern abstract algebra, and the story of its origins in the theory of equations and number theory as well as the tragedy of Galois make compelling reading.


• Assignment: Due Tuesday, 18 April 2017, (HW 13) To hand in: Email a .pdf to both Bennett Clayton bgclayton@math.tamu.edu and Frank Sottile fjsteachmath@gmail.com
  1. The main part of your homework is to build the hyperbolic soccer ball, and then do the activity I describe below for it. This will take a couple hours of your time. (I do this in 90 minutes to 2 hours with in-service teachers, but expect it you take longer as I am not in the room to make comments and answer questions.) If at any time, you are confused, go to Piazza with your question. This is the first time I have tried to run this activity remotely, the other 26 times I have done it have all been live and interactive. Feedback from you on this will be helpful for future students.
     1. First, follow the instructions to build a hyperbolic soccer ball. Do stop your construction for now where I say that you have enough real estate to do the mathematical activity. I will ask that you take a picture of your creation and email it with your homework (Try to keep it a reasonable size, say 800x600 pixels).
     2. Next, we will do the activity that is sketched on the instruction page I sent you. Read it, and follow the instructions below, which give a little more information.
        1. The first of Euclid's axioms is that any two points determine a line. Grasp, using thumb and forefinger of each hand in a pinching motion, two points on your model. Apply tension to straighten out the model between your fingers, and sight down them. You should manage to straighten out your model between your fingers, and can envision a line between them. Do this with several other pairs of points.
        2. Euclid also postulates that any line segment can be extended indefinitely. You can demonstrate this on your model by drawing a line. Note that you want to draw on the back of the model, where you put your tape. Notice that any polygon is flat (as it was cut from a single piece of paper), and any two polygons sharing an edge can be both flattened simultaneously (do this on a table). Using an index card draw a line segment across two flattened polygons, near (within a polygon or two) to, and `parallel' to, a boundary of your model. Avoid drawing towards a vertex, it is impossible to continue a line through a vertex. Move your index card along the line segment, and use it to guide you extending your line to the adjacent polygon. You can continue doing this until you get to the edge of the model. Now extend the line in the other direction. Pinch the endpoints of your line and sight down them. You should see a more-or-less straight line, as long as you managed to avoid a vertex. You can erase and redraw this line, if yours managed to pass through a vertex.
        3. Two lines are parallel if they do not meet. While this is hard to prove, as it involves points of the lines far, far away, in elementary geometry we show that if two lines share a common perpendicular, than they are parallel.
           Conveniently, your index card has four right-angled corners. Use one to draw a short (about 1.5 polygons long) perpendicular to your line. At the end of this perpendicular (or at a propitious point along it), measure another perpendicular, and proceed to extend this to a line. Sight down this line as you did the previous one to verify that it is straight. The second line is parallel to the first, and you should be able to see that if the model is extended, then the two lines will never meet. Now stand back and look at your parallel lines. Do they look parallel ?
        4. An equivalent form of Euclid's parallel postulate is Playfair's Axiom: Given any point P not on a line l, there is a unique line through P that is parallel to l. Let us test this as follows:
           On one of your lines l, and at a different point than before, erect a perpendicular and extend it towards towards the other line. Let P be the point where this second perpendicular meets the other line. Is it also perpendicular to the second line? What can you conclude about Playfair's Axiom on your model.
        5. The two perpendiculars and two lines form a Lambert quadrilateral, which has three right angles. Look this up.
        6. Now we turn to triangles. Perhaps starting on one of your lines, draw a triangle on your model. Make it reasonably large, and do not forget to sight down each edge of the triangle.
        7. Now we measure the angles as described on your handout. Read it and ask questions if you are stymied.) Make the semi circle about 2 cm in radius. If you have a protractor, you can get the angle measures as follows. Lay your semi circle on a blank sheet of paper, marking the centre point, as well as the four points (starting point and points where the edges leave the semi circle) along the semi circle that you measured. Now extend these to get four rays, and record the angles that you measured, both the individual angles, as well as their sum.
        8. If you can, make a couple of other triangles, both a smaller one, and one as big as possible. Measure their angle sums. For each triangle, count the number of vertices internal to the triangle.
     3. Send in a couple of pictures, your information about your triangles, and a couple of paragraphs on what you learned. Report separately in that email to me the information about your triangles, such as: First triangle 143 degrees and four vertices, etc. I will make a chart and post these data.
  2. After all that, I ask for you to do the somewhat more mundane homework from Stillwell, Exercises 19.5.1, 19.5.2, 19.5.3, 19.5.4, which investigate the rotation group of the cube.