A knitted cross-cap
“The real projective plane, RP^2, is the space of all lines through the origin in R^3, with a distance defined by the (smaller) angle between two given lines. Topologically, this is the same thing as starting with a sphere, and identifying (i.e., gluing together) every pair of antipodal points. RP^2 is a non-orientable surface, and cannot be embedded in R^3.”
The word cross-cap refers to this particular image of RP^2 in R^3 (its construction is illustrated in one of the pictures). Like any image of RP^2 in R^3, the cross-cap is self-intersecting, but nevertheless you can knit one! Self-intersecting knitting is achieved through a magic loop technique which is described and illustrated in the pattern. The basic idea is to hold your needles in figure-8 shape.
The finished object measures approximately six inches by six inches by three inches. The pattern calls for aran/worsted-weight yarn, but really, any yarn knit at a tight gauge will work.
This pattern requires: knitting and purling; provisional cast-on; decreasing; and self-intersecting magic loop knitting (described in the pattern).