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stoutfellowI think maybe I'll have to write seven Taxonomy papers.
I've been thinking more about mixed classes, and I've discovered a few new things. This is a result of the realization I mentioned last time - not a logical implication of it, but a consequence of checking out specific examples. I can now identify no less than four special kinds of mixed class, and it may be worth devoting one paper to those before going on to the general attack.
First: if m divides n, then every n-gon has an m-gon associated with it, in a natural way. For instance, the midpoints of the long diagonals of a hexagon ABCDEF (AD, BE, and CF) are the vertices of a triangle. Then, for any class of m-gons (isoceles triangles, triangles whose vertices are collinear, triangles whose vertices are identical), there's a corresponding class of n-gons - the ones whose associated m-gons belong to that class. Some of those are pure classes, but most of them are mixed. These are the "derived classes".
Second: if n = 2m is even, there's a special line, the "centroidal line" of any n-gon. For quadrilaterals, the centroidal line is the "Newton line", the line connecting the midpoints of the diagonals, and it has all kinds of interesting properties, and it's connected to several classes of quadrilaterals. For example, in a trapezoid, the Newton line is parallel (actually, identical) to one of the bimedians. In the same way, there are classes of n-gons defined by the centroidal line being parallel or perpendicular to some other special line. These are the "centroidal classes".
Third: rho-negative classes. These only exist when n = 2m is even, and they are where I first noticed the Varignon transform. But it turns out that every rho-negative class of n-gons is associated with a rho-positive class of m-gons, in a natural way, and this interacts with the Varignon transform in a neat way. I noticed these a long time ago, but only recently recognized the Varignon connection. [For example, the class of hexagons ABCDEF for which the triangles ACE and BDF have the same area is one of these; more generally, you can compare the value of some rho-positive function on the m-gon formed by the even vertices to its value on the odd-vertex m-gon.]
Fourth: again, let n = 2m be even, but require m to be odd. Then every pure class of m-gons gives rise to a mixed class of n-gons, in the same way that rho-negative classes arise. This I didn't realize until just recently - it's part of what I realized on that dog-walk. Probably the fact that it only works when m is odd stalled my realization.
Each of these, or all of them together, form a small minority of mixed classes, but each of them is interesting in its own right, and I'm thinking of splitting Taxonomy V into two parts - this, and then the general theory of mixed classes.
Gotta get to work, this semester.
I've been thinking more about mixed classes, and I've discovered a few new things. This is a result of the realization I mentioned last time - not a logical implication of it, but a consequence of checking out specific examples. I can now identify no less than four special kinds of mixed class, and it may be worth devoting one paper to those before going on to the general attack.
First: if m divides n, then every n-gon has an m-gon associated with it, in a natural way. For instance, the midpoints of the long diagonals of a hexagon ABCDEF (AD, BE, and CF) are the vertices of a triangle. Then, for any class of m-gons (isoceles triangles, triangles whose vertices are collinear, triangles whose vertices are identical), there's a corresponding class of n-gons - the ones whose associated m-gons belong to that class. Some of those are pure classes, but most of them are mixed. These are the "derived classes".
Second: if n = 2m is even, there's a special line, the "centroidal line" of any n-gon. For quadrilaterals, the centroidal line is the "Newton line", the line connecting the midpoints of the diagonals, and it has all kinds of interesting properties, and it's connected to several classes of quadrilaterals. For example, in a trapezoid, the Newton line is parallel (actually, identical) to one of the bimedians. In the same way, there are classes of n-gons defined by the centroidal line being parallel or perpendicular to some other special line. These are the "centroidal classes".
Third: rho-negative classes. These only exist when n = 2m is even, and they are where I first noticed the Varignon transform. But it turns out that every rho-negative class of n-gons is associated with a rho-positive class of m-gons, in a natural way, and this interacts with the Varignon transform in a neat way. I noticed these a long time ago, but only recently recognized the Varignon connection. [For example, the class of hexagons ABCDEF for which the triangles ACE and BDF have the same area is one of these; more generally, you can compare the value of some rho-positive function on the m-gon formed by the even vertices to its value on the odd-vertex m-gon.]
Fourth: again, let n = 2m be even, but require m to be odd. Then every pure class of m-gons gives rise to a mixed class of n-gons, in the same way that rho-negative classes arise. This I didn't realize until just recently - it's part of what I realized on that dog-walk. Probably the fact that it only works when m is odd stalled my realization.
Each of these, or all of them together, form a small minority of mixed classes, but each of them is interesting in its own right, and I'm thinking of splitting Taxonomy V into two parts - this, and then the general theory of mixed classes.
Gotta get to work, this semester.
no subject
Date: 2019-01-12 03:11 am (UTC)no subject
Date: 2019-01-12 03:17 am (UTC)no subject
Date: 2019-01-12 04:06 am (UTC)I was wondering how n could possible not be even if it was defined as multiplying m-presumably-discrete vertices by two, and it's more about setting m.
Thanks!