The topic for this session was on the general question: What does "straight" mean, on the plane and on the sphere and other spaces?

Students were asked to work in groups and write down characteristic properties of straight lines on the plane, with an eye towards using these properties to define straightness on the sphere. Then the answers were discussed.

Then some ideas were put forward as describing or defining straightness, especially ideas based on symmetry. These are described in the textbook in Chapter 1.

Then a series of experiments on the sphere were done by students in groups:

In each case, the path was a great circle. Circles that are not great circles are not straight.

Additional experiments that were performed or described:

• Fold a piece of sphere to get a folded edge.
• Place a hemisphere on a mirror to see that the reflection completes the sphere.
• Cut a perpendicular plane section

Symmetries on the sphere were described. It was observed that if a point P is on a circle, the image of the circle under point reflection is a different circle (forming sort of a figure 8) unless the original circle is a great circle. It was also observed that reflection in 3-space in a plane through the center of the sphere defined a "line reflection" on the sphere with the mirror line being a great circle.

Composition of point symmetries was posed as a question.

Students drew great circles on spheres with a "ruler" on the Lenart sphere models. It was concluded that there are no parallel great circles on the sphere. Any two distinct circles intersect in a pair of opposite points.

In particular, on a globe the parallels of latitude may be parallel by some definitions, but they are not straight, except for the equator, for they are not great circles.

The general term for a straight path is "geodesic". The word "line" is convenient, but is used in some contexts to mean a path or curve and only a "straight line" is a geodesic. So we should be careful when we use the word "line".