David W. Henderson, Department of Mathematics, Cornell University,
Ithaca, NY14853-7901, USA. e-mail: email@example.com
Daina Taimina, Faculty of Physics and Mathematics, University of Latvia,
Raina blvd 19, Riga LV1586, Latvia. e-mail: firstname.lastname@example.org
The workshop included hands-on small-group cooperative experiences of the geometry of the 450° cone, studying the intrinsic geometry (as a crawling ant would experience it) of surfaces. This was related to Differential Geometry and to our understandings of the possible global geometries of our physical universe.
If I draw a straight line on a flat piece of paper and then bend the paper, the line will (in general) no longer be straight from our extrinsic point of view -- as we look at the line on the bend paper it will be curved in 3-space. But from the intrinsic point of view of an ant on the surface the line has not changed and thus is said to be intrinsicly straight. In the workshop we studied the intrinsicly straight lines (in differential geometry these are called geodesics) on the 450° cone. See Figure 1.
Figure 1. How to make a 450° cone.
We discovered that paths AC, AB, BE, EF, FD, DC were all intrinsicly straight because a neighborhood of each could be flattened onto a plane in which the lines were straight. The path AF we considered straight in the sense that it has all the bilateral (reflection) symmetries of a straight line and thus it makes sense to say that the ant will experience no turning. The path AF can also be produced by folding the cone in the same way that we can use folding to produce straight lines on an ordinary plane sheet of paper. The path AD is extrinsicly straight but not intrinsicly straight because the two sides of the path are not the same and thus the ant will experience some turning. Path AE is the shortest path from A to E but it is not straight in the sense of symmetry.
Since Euclid defined a right angle to be the angle formed when two straight lines intersect such that the four angles formed are all congruent, we see that at the point of the 450° cone a right angle has 112.5°. Thus the 450° cone violates Euclid's Fourth Postulate which says: All right angles are equal.
Our experiences on the cone were then related to Differential Geometry which can be considered as the general study of the intrinsic geometry of surfaces and space. There is a theorem that says that: On a surface in which all right angles are equal (Euclid's Fourth Postulate) and every intrinsicly straight line can be continued indefinitely (Euclid's Second Postulate), then (1) every shortest path is straight, (2) every extrinsicly straight path is intrinsicly straight, (3) every straight path is the shortest path between nearby points.
These are ideas (particularly of intrinsic and extrinsic) that we should start teaching to students because they relate to experiences in our physical universe. In a few years it is expected that astronomers, physicists and mathematicians will be able to determine the shape of our physical 3-dimensional universe and the intrinsic geometry is likely to NOT be Euclidean geometry. How do we (and our students) start to think about this? By considering the intrinsic experiences of the ant on various surfaces we can start to get an idea of our intrinsic experience in 3-space -- in our experience of the universe we are like ants.
The notions of differential geometry are usually only studied in advanced university classes because the formalism is very very complicated. However, it can be made more accessible (even to school children) by emphasizing experiences such as these, which are explored further in the two texts listed below.
My philosophy of mathematics learning is summarized by the following quote from the Dalai Lama:
Do not pay attention to the words,
Instead pay attention to meanings behind the words;
But do not just pay attention to meanings behind the words,
Instead pay attention to your deep experience of those meanings.
D. Henderson, Experiencing Geometry on Plane and Sphere, Prentice-Hall, 1995.
D. Henderson, Differential Geometry: A Geometric Introduction, Prentice-Hall, 1997.
D. Henderson, I Learn Mathematics From My Students -- Multiculturalism in Action, For the Learning of Mathematics, 16, 34-40.
Jane-Jane Lo, Kelly Gaddis and David Henderson, Learning Mathematics Through Personal Experiences: A Geometry Course in Action, For the Learning of Mathematics, 16, n.2.