stoutfellow (![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png)
stoutfellow) wrote2005-01-30 12:04 pm
stoutfellow) wrote2005-01-30 12:04 pm
Entry tags:
"Dem Bones Gonna Rise Again"
As I've mentioned before, my primary research interest at present has to do with the properties of polygons - old-fashioned Euclidean geometry. Since there seems to be some interest, I thought I'd try to give some broad idea of the goals of my work.
Associated with a triangle ABC are various special points. For example, if you draw the lines connecting A to the midpoint of BC, B to the midpoint of AC, and C to the midpoint of AB, you'll find that they intersect in a single point, the centroid of ABC. If you draw the line through the midpoint of AB which is perpendicular to AB, and do likewise to AC and BC, again they intersect in a single point, the circumcenter. (There is exactly one circle passing through the vertices of ABC - the circumcircle - and the circumcenter is its center.) If from the point A you drop the perpendicular to BC, from B the perpendicular to AC, and from C the perpendicular to AB, the three lines (the altitudes) intersect in the orthocenter. And if you draw the lines which bisect the angles BAC, ABC, and ACB, they intersect in the incenter, which is the center of a circle - the incircle - inside ABC which just touches the sides.
These points were known to the ancient Greeks, and there the matter stood for centuries. In the early seventeenth century, Fermat (showing off the powers of the proto-calculus he was developing) identified the point F with the property that the sum of its distances from A, B, and C was a minimum; it is now known as the Fermat point, or the (first) isogonic center. ("Isogonic" comes from Greek roots meaning "equal angle"; it gets the name because the angles AFB, BFC, and CFA turn out to be equal.) This was the first new special point to be discovered, but more were to follow. There was a burst of activity in the nineteenth century, and another starting in the late 1970s, and the number of interesting points has mushroomed. (One catalog, constantly being updated, listed over two thousand of them, the last time I checked.)
It's not just special points; there are special lines - the Euler line, the Brocard axis, the symmedian trail, the Lemoine line - and a plethora of special curves - conics, cubics, and more complex ones as well. This is rather an embarrassment of riches, and there are ongoing efforts to bring some order to the chaos. This is one of the areas I'm working in.
Quadrilaterals and other sorts of polygons don't seem to have such a wide variety of special points and whatnot. For example, most quadrilaterals don't have a circumcircle, or an incircle either. This seems unfair to me, and a second area I'm working in involves locating interesting points and such for polygons with more than three sides. (My New Year's paper, among other things, points towards several of these for quadrilaterals.) The two areas actually tie together in intriguing ways.
I don't know how well what I'm doing in these areas can be explained to non-mathematicians, but I may try to sketch a little of it later, if there's any interest.
Associated with a triangle ABC are various special points. For example, if you draw the lines connecting A to the midpoint of BC, B to the midpoint of AC, and C to the midpoint of AB, you'll find that they intersect in a single point, the centroid of ABC. If you draw the line through the midpoint of AB which is perpendicular to AB, and do likewise to AC and BC, again they intersect in a single point, the circumcenter. (There is exactly one circle passing through the vertices of ABC - the circumcircle - and the circumcenter is its center.) If from the point A you drop the perpendicular to BC, from B the perpendicular to AC, and from C the perpendicular to AB, the three lines (the altitudes) intersect in the orthocenter. And if you draw the lines which bisect the angles BAC, ABC, and ACB, they intersect in the incenter, which is the center of a circle - the incircle - inside ABC which just touches the sides.
These points were known to the ancient Greeks, and there the matter stood for centuries. In the early seventeenth century, Fermat (showing off the powers of the proto-calculus he was developing) identified the point F with the property that the sum of its distances from A, B, and C was a minimum; it is now known as the Fermat point, or the (first) isogonic center. ("Isogonic" comes from Greek roots meaning "equal angle"; it gets the name because the angles AFB, BFC, and CFA turn out to be equal.) This was the first new special point to be discovered, but more were to follow. There was a burst of activity in the nineteenth century, and another starting in the late 1970s, and the number of interesting points has mushroomed. (One catalog, constantly being updated, listed over two thousand of them, the last time I checked.)
It's not just special points; there are special lines - the Euler line, the Brocard axis, the symmedian trail, the Lemoine line - and a plethora of special curves - conics, cubics, and more complex ones as well. This is rather an embarrassment of riches, and there are ongoing efforts to bring some order to the chaos. This is one of the areas I'm working in.
Quadrilaterals and other sorts of polygons don't seem to have such a wide variety of special points and whatnot. For example, most quadrilaterals don't have a circumcircle, or an incircle either. This seems unfair to me, and a second area I'm working in involves locating interesting points and such for polygons with more than three sides. (My New Year's paper, among other things, points towards several of these for quadrilaterals.) The two areas actually tie together in intriguing ways.
I don't know how well what I'm doing in these areas can be explained to non-mathematicians, but I may try to sketch a little of it later, if there's any interest.
no subject
Very cool!! Interesting, thanks!! :-)
(and yes, definitely interested in hearing more)
no subject
no subject