What it shows:
You can use a spherical blackboard for many things, including the teaching of geographical coordinates, as a model for a closed Universe, or simply as a mathematical shape.
In the non-Euclidean geometry of the sphere, a circle will have a circumference greater than 2πr and an area greater than πr2. A triangle’s angles will add to more than 180°, and two parallel lines, called Great Circles, will converge.
A Universe with a density parameter Ω greater than unity will have too much mass to overcome its own gravitational attraction. Such a Universe will cease to expand, and will re-collapse to (maybe) a singularity. The geometry of this closed Universe has a positive or spherical curvature, and it can be represented two-dimensionally by a spherical surface.
How it works:
We have two spherical blackboards, a small bench-top one (diameter 30cm) that is a custom made blackboard (origin unknown) and a larger (diameter 67cm) that was once a globe (see Comments) but was painted black.
Setting it up:
The size globe you want will depend upon the size hall you're teaching in. Use chunky multi-colored chalk for the big one, such as Prang™ fluorescent lecturer's chalk.
So, if you have an old globe lying about, paint it black and you have an instant spherical blackboard. That's exactly what they did at Harvard; they found an old globe, one of a pair dating from 1757, stripped the paper off the papier-mâché sphere and removed the six cabriole legs, painted it black and mounted it on a plywood box stand. Its twin, a celestial globe was restored and currently resides in the Harvard Collection of Historical Instruments. Originally priced at £35 for the pair, there is now little doubt which has the greater monetary value. However, which of the twins has the greater educational value is open to debate.
D. P. Wheatland, The Apparatus of Science at Harvard 1765-1800 (Harvard University Press 1968)