The projective plane is definitely the weirdest object I've ever knitted. Of course, I cheated because you can't embed RP2 into R3, so it's really a projective plane with a (small) disc removed. I created it by making a Möbius band, and contracting the boundary as much as I could. Directions for knitters: follow my instructions for a Möbius band, but instead of casting off after a few rows, start to decrease steadily row by row. Eventually you'll get to the point that you really can't decrease any more, and then cast off.
I've often wondered what a projective plane really looks like; there are some nice polyhedral models that give one an idea of the structure, and some sculptures that hint at it... and now that I've made the darned thing, I still feel like I don't know what it looks like... even when I'm looking at it. But here are some pictures of the first (and ugliest) projective plane I made...
This is the way I most commonly try to view it. The pastel band around the outside is the original Möbius band (then I ran out of yarn and switched colors) and you can sort of see it twisting through the white part (which is where the boundary of the missing disk is).
(that's the other side, i.e. flipped over, not just rotated)
One can see
that most of a projective plane is really just a disk.
But flipping it over reveals the pathology! Opposing points on the boundary of the disk are identified. Or, well, they would be if this were a true projective plane. One can pull the bulk of the disk-y part through the boundary, which one would think would be enlightening, but is merely confusing.
Since the time of the above photos, I've made some much better projective planes.
Here they are:
One was a gift for Doug Shaw.
This blue one has one stripe, which represents the Mobius band I started with.
In the second picture, you can almost see the Mobius band passing through the
This green projective plane has one stripe, too---just 'above' the Mobius band
I started with. That's why it looks like it has two stripes; the stripe doesn't
go through the central S1.
Here you can see the stripe twisting through the twisted missing disk.