Math 629: History of Mathematics
Frank Sottile

• Opening Remarks: We will begin with a couple of topics from Chapter 17.
  
      The feature this week is 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 of the hyperbolic plane. 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.
      You are invited to explore my webpage on the hyperbolic football, as well as the description of constructions with it. My time in Germany and the coronavirus crisis kept me from mailing you information for an activity based on this that I like to do with students and teachers. To Frank: Redo the construction page using lessons from Spring 2024: High-level description and uniformity of vertices, and emphasize using two pieces of tape for each edge.     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.
  
  
• Reading:
  1. Chapters 17 and 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 (This is from 2018. Note that I did not send the last two pages of hexagons listed there; I no longer have printouts of them), and finally also the pictures from a construction of a smaller model with essentially the same pieces.


• Assignment: Due Monday, 1 April 2025, (HW 11)
  Here is a .pdf and a LaTeX source of the assignment.
  Note that there is no concept quiz and no group portion of the assignment. The construction of the hyperbolic soccerball obviates that.
  To hand in: We are using Gradescope for homework submission.
  1. Please do exercises in Section 17.6 (there are typos here), numbers 17.7.1 and 17.7.2. Here, excess means in excess of the angle sum of a polygon. Having done that, use the idea of 17.7.2 to deduce that the angle sum of a polygon in the Euclidean plane with n sides is π(n-2). Along the same line of spherical geometry, do problem 18.2.1.
  2. Look up Gauß on the web and write about one mathematical accomplishment of his that is not explained in either book, or only explained in passing. This should be brief; such as only one paragraph.
  3. 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. 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 post it to Piazza.
     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, then 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. Make a couple of triangles, the larger, the better.
        7. Now we measure the angles as described on your handout. (Read it and ask questions if you are stymied.) Do not try to measure individual angles, but use the semicircle method I describe below; ask questions if this is not clear. 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. Placing your semi-circle on one of the (images of) proctractors that I sent, or on a sheet of paper if you have your favourite protractor, extend these to get four rays, and record the sum of the angles that you measured. (If you measure the angles individually and then sum them, versus just marking their sum on your semicircle and then measuring it, you will be less accurate. Measuring thrice and summing the numbers gives an error of about 1.7 (∼3^1/2) times the error of making one measurement.
        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 in the interior of the triangle. Please send to me your triangle data: For each triangle, its number of internal vertices and angle sum as a pair [number of vertices, angle sum], e.g. [1,172],[6,135],[12,80]. I will plot that and share it with you. This will illustrate a discrete version of the Gauss-Bonnet formula.
     3. To Hand in: First, write a couple of paragraphs (be brief) describing what you did with lines on your hyperbolic soccerball. Second, another couple of paragraphs about what you learned from this activity.
        These are part of the homework to post to Gradescope.
        
        The last part, which you share a picture or two on Piazza of your constructions; possibly both sides, and second, report on your triangles, say in a group post on Piazza (I think that you can edit the posts of others) information such as First triangle 143 degrees and four vertices, etc. I will make a chart and post these data along with your pictures. It is very interesting.

Last modified: Tue Mar 26 09:54:09 PDT 2024 by sottile