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stoutfellowIn my last post about mathematics, I was deliberately vague about my student's and my discoveries. I'm not going to be much more precise here, but I think it's fairest to be a little more clear.
Consider quadrilaterals. Two well-known (among students of Euclidean geometry) classes of quadrilaterals are the "equidiagonal" and "orthodiagonal" classes. In my classification, both are "rho-negative" classes. A quadrilateral is equidiagonal if its diagonals are equal in length; it is orthodiagonal if its diagonals are perpendicular.
If ABCD is a quadrilateral, its "bimedians" are the lines connecting the midpoints of opposite sides - the midpoint of AB to the midpoint of CD, or the midpoint of AD to the midpoint of BC. Here is the interesting phenomenon: a quadrilateral is equidiagonal if and only if its bimedians are perpendicular, and a quadrilateral is orthodiagonal if and only if its bimedians have the same length. In other words, each class has a characterization involving diagonals and one involving bimedians, and the diagonal characterization of each corresponds to the bimedian characterization of the other; and this is the phenomenon I was speaking of.
This sort of relationship happens all the time with first-order (never mind) rho-negative classes; what's unusual here is that it goes both ways. What stymied me before is that it doesn't go both ways most of the time; what pleases me now is that I know why it does go both ways for the four rho-negative classes of hexagons (two pairs), and I know when it will do so for evengons with more than six sides.
Still too vague, I know, but I hope it's a little clearer what I'm talking about.
Consider quadrilaterals. Two well-known (among students of Euclidean geometry) classes of quadrilaterals are the "equidiagonal" and "orthodiagonal" classes. In my classification, both are "rho-negative" classes. A quadrilateral is equidiagonal if its diagonals are equal in length; it is orthodiagonal if its diagonals are perpendicular.
If ABCD is a quadrilateral, its "bimedians" are the lines connecting the midpoints of opposite sides - the midpoint of AB to the midpoint of CD, or the midpoint of AD to the midpoint of BC. Here is the interesting phenomenon: a quadrilateral is equidiagonal if and only if its bimedians are perpendicular, and a quadrilateral is orthodiagonal if and only if its bimedians have the same length. In other words, each class has a characterization involving diagonals and one involving bimedians, and the diagonal characterization of each corresponds to the bimedian characterization of the other; and this is the phenomenon I was speaking of.
This sort of relationship happens all the time with first-order (never mind) rho-negative classes; what's unusual here is that it goes both ways. What stymied me before is that it doesn't go both ways most of the time; what pleases me now is that I know why it does go both ways for the four rho-negative classes of hexagons (two pairs), and I know when it will do so for evengons with more than six sides.
Still too vague, I know, but I hope it's a little clearer what I'm talking about.
