I got a notice from Zazzle that if I don’t post at least one new product in every fifteen months they’re going to charge me a Non-Contributing Account Fee. So here’s one. I can spin more variations on this theme until the cows come home.

Sometimes I search my blog for this picture and scratch my head in puzzlement that it’s not here, before remembering that I posted it on Google Plus back when I used that.

So here it is. Stephen Guerin (the shaven one) displays his canvas print of one of my designs.

The colors came out better than I hoped, in stark contrast to a couple of mugs with related designs that I got from the same shop.

I’ve had other designs made in steel but not this one. (The sintering leaves the steel highly porous, so liquid bronze is brought in by capillary action to fill it; the result is about three parts steel to two parts bronze – if I understand right. Hence the color.) (**Later:** I was mistaken: the steel powder is not sintered but glued; the bronze presumably burns away the glue.)

While it was on its way to me, I thought of some improvements. ( . . more . . )

I’ve been designing printable models of the Lawson-Klein surface

x = cos(u) sin(2v)

y = sin(u) cos(v)

z = sin(u) sin(v)

As you can plainly see, this figure lives in S3 (positively curved 3-space), so stereographic projection can bring it into E3 (Euclidean 3-space) without adding more self-intersections. (It crosses itself at u=nπ.)

To minimize the distortion of the projection, I want the projection center to be as far as possible from the surface. One thing I tried was pursuit: starting with an arbitrary point P in S3 and an arbitrary point L(u,v) in the surface, move (u,v) to bring L closer to P while simultaneously moving P away from L. This gets me nowhere so far: either it fails to converge or P converges to the antipodes of L, which is also in the surface (change u by π).

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) )

models of the 35 smallest fullerenes

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.

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.