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Title: Cross-cap Real Pro jective Plane
Artist and Designer: Emily Peters
Materials: Noro Kureyon yarn and stuffing
Technique: Knitting
Artist’s Statement: Why is this RP²? Well, RP² is the object created by identifying antipodal points on a sphere. To see the shape created in this way, imagine identifying all points in the Northern hemisphere with their corresponding southern hemisphere points. Now we just need to identify antipodal points on the equator. If we pinch together twelve o’clock and six o’clock, we’re left with two intervals which must be identifed—but we want to identify the clockwise direction of one with the counterclockwise direction of the other. To accomplish this in R³, we have to introduce a self-intersection.
The bottom half of this object is a self-intersecting hemisphere, knit on circular needles held in a figure- eight shape. To identify its boundary with itself, we knit a hat-shaped disk as the top half. Because RP² is non-orientable, this disk has to change from stockinette stitch to reverse stockinette (i.e., from flat knitting to bumpy) to be consistent with the knitting of the self-intersecting disk.