Tuesday, November 17, 2009

Kicking the Can Down the Road

Kicking a can down the road: Ernest Hemingway, Iowa.

Hemingway joined the fight against fascism from the front lines in Spain, having sustained earlier injuries during WW1. His work on a film documentary about the attack against Spain by Hitler and Mussolini is mentioned in the movie Into the Fire.

1, 12, 42, 92, 162...

"Kicking a can" may be likened to generating successive terms in a number sequence, perhaps according to some rule. Sequences have various properties: some are convergent, others divergent, some oscillate, others are chaotic.

The sequence below is described by the growth of the above lattice, defined by closest packed balls, starting with 12 balls around a nuclear ball at the corners of a cuboctahedron.

The Python built-in function, next, is here bound to the name kick, while the generator is assigned the name can.

A generator returns control to the caller when it encounters the keyword yield, then waits for a triggering next, or in this case, kick, upon which it loops around to the yield expression again, providing the next term in the sequence.

The successive terms, in this case, are pairs (tuples), the first being the number of balls in the next concentric layer, the second being the cumulative number of balls in all layers so far, including the nuclear ball.

The balls define a matrix or lattice of equal-length edges known as the octet truss in architecture, because of the tetrahedral and octahedral volumes that define it. Alexander Graham Bell worked with this same truss in his famous kite designs.

Because a cuboctahedral layer of balls may be rearranged into an icosahedral layer without changing their number, the sequence above relates to the number of hubs, struts and windows in an n-frequency geodesic sphere and/or naturally occurring virus and/or carbon cage molecule.

The formula 10 * n * n + 2 was derived by R. Buckminster Fuller and featured on the front page of the New York Herald Tribune.

As Siobhan Roberts tells it in her King of Infinite Space:
When a reporter from LIFE magazine called in 1970, Coxeter gave Fuller a somewhat backhanded -- but then accidentally glowing compliment.... Coxeter sent back a letter saying that one equation would be 'a remarkable discovery, justifying Bucky's evident pride,' if only it weren't too good to be true. The next day, Coxeter called: 'On further reflection, I see that it is true.'
Here's a proof (by me), and a link to Coxeter's subsequent write-up Polyhedral Numbers (my thanks to CJ Fearnley of SNEC for bringing it to my attention).