cosmology - geography - non-Euclidean geometry
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. Rating**
D. P. Wheatland, The Apparatus of Science at Harvard 1765-1800 (Harvard University Press 1968)