For the Museum of Mathematics
Well, Math Mondays might as well just go ahead and admit it. It is incurably addicted to great circles. Inspired by the latest reader feedback on the topic, Math Mondays decided to team up with Grace Whitney of Stony Brook, NY to polish off the outstanding great circle challenge: the 25-circle arrangement beloved of Buckminster Fuller. The ingredients of today’s approach are simple: a big transparent inflatable ball and 18.4 meters of elastic sewn into 25 loops just the right size to stretch around a great circle of the ball.
The great circle arrangement starts off in a by-now familiar way: use four elastics to create a spherical cuboctahedron. Note that the segment between every pair of adjacent crossings is identical in length. It’s well worth it at this stage to use a tape measure to be very careful to get them all uniform since all of the remaining placements will be guided by these initial four.
Next, add six medians of triangles to produce this lovely arrangement of 10 great circles:
Three diagonals of squares mesh harmoniously, for a running total of 13 great circles:
Finish up by stretching each of the remaining elastics as follows. Pick any vertex V of a square S. Stretch the elastic through V to the center of a triangle which is adjacent to S but does not include V. These rays are close to the angle bisectors of the existing 60-degree angles at the center of each of the triangles. Keep in mind that any place where three great circles almost hit the same point, they really do all hit the same point. In particular, the final elastics will in aggregate make an octagon around the center of each of the squares. Here’s the finished product, hopefully nice enough to do old Bucky proud:
[email protected] may now have reached saturation on great circles, but it would be happy to be convinced otherwise by a truly amazing pic…