When learning about something a bit min
When learning about something a bit mind-bending, like hyperbolic space, I sometimes find myself wishing that Douglas Hofstadter had written about it. He'd be sure to find some insightful analogy, a new way of brain-seeing, a fearless exploration of difficult concepts taken on our behalf, returning with shining treasure and a detailed map. Well, this time my wish has come true.
Originally shared by Allen KnutsonGeometry in the dual affine plane
I was reminded today (by +Jordan Peacock) how amazing Douglas Hofstadter is at coining names of things.
Many (20, I think) years ago I saw a talk of his about plane geometry. Mathematicians know that almost everything's easier in the projective plane than the affine plane. To make the projective plane from the affine plane, add a new point "at infinity" for each possible slope of a line. You're used to two (unequal) points always determining a unique line, but two (unequal) lines only sometimes determining a unique point of intersection, because if you're terribly, terribly unlucky the lines are parallel and don't meet. But now, if they're parallel, they have the same slope, so they meet at one of your new points!
There are many other reasons to like the projective plane, e.g. there's no longer a distinction between hyperbolae (which get two new points at infinity), parabolae (which get one new point at infinity), and ellipses (which get no new points at infinity) -- each one is now connected and looks like a loop.
But the phenomenon I'll focus on is that now there's a perfect symmetry between points and lines. Two points determine a line, two lines determine a point.
So Hofstadter asks the following provably worthless question: if
1) we can get the usual familiar affine plane (yuck!) by taking the projective plane (yay!) and ripping out one line, which only after the fact gets the pejorative name "the line at infinity" as if it were somehow different than the other lines (basically, it's not), and
2) there's a symmetry between points and lines, then
3=1+2) what if we take the projective plane and rip out a point instead?
The answer, of course, is that we get the usual theory of plane geometry, as familiar to Euclid. But everything looks different!
In the usual theory, you think of a line as the collection of points lying on it. Dually, you should think of a point as a sort of organizational principle describing the collection of lines through it.
The main things I remember were the dual concepts to "ray" and "distance". A usual ray starts with a point p on a line L, and then gives a bunch of points p' on L, between p and the intersection of L with the line at infinity. Dually, we again have a point on a line, now the point l on the line P, and we look at rotations of P still through l, until we rotate P too far and it goes through the single "point at infinity" we ripped out. Hofstadter called this collection of lines the spray.
What about distance? Now we have two lines P_1,P_2 through the same point l. Affinely you might think they divide the real plane into four regions, but projectively you can sail off one side and come back on the other, so it's only two. One of those two regions has the "point at infinity". So we look at the other region, and the set of lines in it through the point l. Hofstadter calls the angle so subtended the twistance.
The names kept coming, and I'm sorry I can't bring more of them to mind, 20 years on. But a little Googling turns up a paper, on his website!
I was reminded today (by +Jordan Peacock) how amazing Douglas Hofstadter is at coining names of things.
Many (20, I think) years ago I saw a talk of his about plane geometry. Mathematicians know that almost everything's easier in the projective plane than the affine plane. To make the projective plane from the affine plane, add a new point "at infinity" for each possible slope of a line. You're used to two (unequal) points always determining a unique line, but two (unequal) lines only sometimes determining a unique point of intersection, because if you're terribly, terribly unlucky the lines are parallel and don't meet. But now, if they're parallel, they have the same slope, so they meet at one of your new points!
There are many other reasons to like the projective plane, e.g. there's no longer a distinction between hyperbolae (which get two new points at infinity), parabolae (which get one new point at infinity), and ellipses (which get no new points at infinity) -- each one is now connected and looks like a loop.
But the phenomenon I'll focus on is that now there's a perfect symmetry between points and lines. Two points determine a line, two lines determine a point.
So Hofstadter asks the following provably worthless question: if
1) we can get the usual familiar affine plane (yuck!) by taking the projective plane (yay!) and ripping out one line, which only after the fact gets the pejorative name "the line at infinity" as if it were somehow different than the other lines (basically, it's not), and
2) there's a symmetry between points and lines, then
3=1+2) what if we take the projective plane and rip out a point instead?
The answer, of course, is that we get the usual theory of plane geometry, as familiar to Euclid. But everything looks different!
In the usual theory, you think of a line as the collection of points lying on it. Dually, you should think of a point as a sort of organizational principle describing the collection of lines through it.
The main things I remember were the dual concepts to "ray" and "distance". A usual ray starts with a point p on a line L, and then gives a bunch of points p' on L, between p and the intersection of L with the line at infinity. Dually, we again have a point on a line, now the point l on the line P, and we look at rotations of P still through l, until we rotate P too far and it goes through the single "point at infinity" we ripped out. Hofstadter called this collection of lines the spray.
What about distance? Now we have two lines P_1,P_2 through the same point l. Affinely you might think they divide the real plane into four regions, but projectively you can sail off one side and come back on the other, so it's only two. One of those two regions has the "point at infinity". So we look at the other region, and the set of lines in it through the point l. Hofstadter calls the angle so subtended the twistance.
The names kept coming, and I'm sorry I can't bring more of them to mind, 20 years on. But a little Googling turns up a paper, on his website!
www.cogsci.indiana.edu/pub/hof.geom.diamond.pdf
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