My newest design on Shapeways is a model of the Lawson-Klein surface : a stereographic projection of

( cos(u)cos(2v), cos(u)sin(2v), sin(u)cos(v), sin(u)sin(v) )

I often have trouble giving meaningful concise names to variables in the programs I write, perhaps because, until I reach for the keyboard, my thinking is largely nonverbal. I suspect that it would be less of a problem for someone more exposed to the accumulated lore of programmer culture; though perhaps not in this case:

I’m thinking of breaking up this process into multiple rounds, to obtain increasing degrees of geometric continuity. The initial nodes would coincide with the input dots but have no defined theta (tangent angle) or kappa (curvature); the first round of replacements determines theta for G¹ continuity, the next round determines kappa (the first derivative of theta) for G² continuity — and subsequent rounds may seek higher degrees of continuity by matching further derivatives.

This means that the node object, instead of exactly two fields called theta and kappa, should have a list of theta and its known derivatives (one more than the current replacement-round needs), and I’m at a loss for a good name for this list.

This curve-fitting thingy is one of several projects on which I’ve made progress in rare fits over several years. It ran into two big snags. I haven’t found how to determine which gridpoints are within the pen-width of a blending arc; two methods that ought to work don’t. (What would help: tutoring in drawing pictures in a MacOS display, so that I might have a better idea where they go wrong. And a pony.)

The other snag is this: For each pair of arcs, there is an infinite family of blending arcs; how to choose the osculation points to minimize rapid changes in curvature, while meeting the gridpoint constraints? ( . . more . . )

I noticed that Shapeways had 13 models of the roundest of the fullerenes (one of the 1812 forms of C60), but none of the less regular forms; so I made some.

Each of the white pieces has mirror symmetry; the red pieces are chiral. Not shown (because it hasn’t been printed yet): the blue set, which is a reflection of the red set. The idea is that you buy both red and blue if and only if you count reflected chiral forms separately.

These figures have 12 pentagons and up to 8 hexagons. They include the two smallest forms with no nontrivial symmetries, and the two smallest with no ‘peaks’ where three pentagons meet.

I could have used this a week ago.

def fccstack():
	newlimit = 0
	while True:
		oldlimit = newlimit
		newlimit += 1
		# extend z
		for x in xrange(oldlimit):
			for y in xrange((x+oldlimit)&1, oldlimit, 2):
				yield (x,y,oldlimit)

		# extend y
		for x in xrange(oldlimit):
			for z in xrange((x+oldlimit)&1, newlimit, 2):
				yield (x,oldlimit,z)

		# extend x
		for y in xrange(newlimit):
			for z in xrange((y+oldlimit)&1, newlimit, 2):
				yield (oldlimit,y,z)

g = fccstack()
for dummy in xrange(512):
	p,q,r = g.next()
	print "%d %d %d\t%d" % (p,q,r, p+q+r)

This lists coordinates of sites in a face-centred cubic lattice, filling the smallest cube that contains the number of sites required.

My bundle of 19 fullerenes is arranged in a face-centred cubic lattice, each ball occupying one of the 24 even-numbered cells of a 4×4×3 array (and parts of the adjacent cells). The spacing of the grid planes is determined by the sums of the radii of the circumspheres of nearest-neighbor balls.

There appears to be much more ‘daylight’ between balls than necessary. That’s not a problem here, as the size of the bounding box is not near the limits of the process; but still it got me thinking about how to tighten the packing. ( . . more . . )

Shapeways has thirteen models of (the most symmetrical and famous form of) C60, but none of the thousands of smaller fullerenes. So I designed models of the 35 smallest, i.e. up to C36; this includes the smallest with no symmetries, and the smallest with no ‘peaks’ where three pentagons meet. One bundle has the 19 forms with mirror symmetry; two other bundles will have the 16 or 32 chiral forms (arbitrarily divided), so that the buyer can choose to have both or only one of a pair distinguished only by handedness.

I’ll make them available when I have them in my hands and am satisfied of their strength.