The description of RP 2 that feels the closest to the artistic perspective is an example of what I think of as math by fiat. Those train tracks that seem to recede in the distance-how do you draw them? The seeds for projective geometry were planted when Renaissance artists started painting scenes in perspective using vanishing points, the spots on (or off) the canvas to which parallel lines would appear to converge.Ĭredit: Wolfram Gothe Wikimedia ( CC BY-SA 3.0) Despite its difficulty and abstraction, though, the projective plane had concrete origins. I’m calling it basic, even though it’s not one of the first mathematical objects you meet, because it’s one of the building blocks for other surfaces, as Emille Davie Lawrence mentioned when we talked to her for the My Favorite Theorem podcast. The projective plane is one of the hardest “basic” mathematical objects for me to imagine. This can be defined a few equivalent ways, and looking at the different definitions can help us understand the space better. There are four points such that no line contains more than two of them.Īnything that satisfies these rules is a projective plane, but when mathematicians refer to the projective plane, they generally mean a space more properly known as the real projective plane, or RP 2.
Every pair of lines intersects in exactly one point.Every pair of points is connected by exactly one line.The seven lines are the six line segments that connect 3 points each, along with the circle in the middle. The seven points are the seven large black circles.
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