Thursday, February 23, 2012

Visualizing map operations as vertex inflations

One way to visualize map operations is as an inflation of vertices, like blowing bubbles. For example, let’s use a triangle on the sphere as our base map.

We have drawn the triangle on the sphere, though it may look like we have drawn it on the plane. Given any topological map on the sphere, we can place all but perhaps one vertex on the front side of the sphere. We will draw spherical maps on the plane in this way, knowing that occasionally we will need to imagine what is happening at a singular point on the back side of the sphere.

OK, let’s dispense with drawing little black dots at the vertices, except where they are 2-valent and needed, and redraw the original vertices as little bubbles. This already gives us the truncate of the base map.

Inflate the vertices some more, and eventually they meet each other at the midpoint of the edges. That forms medial:

Continued inflation of the bubbles causes the frontier of meeting to enlarge from a point to a line segment, forming leapfrog:

Inflating still more causes the bubbles to meet at the center of the original faces. This is dual. (In the drawing, the dashed arrows indicate that the meeting of three of the bubbles takes place at the singular point on the rear of the sphere.)

We have shown that the dual of the triangle on the sphere is the theta graph on the sphere, a graph with 2 vertices and 3 edges.

We may choose to now consider Du(t) to be the base map. Inflating its two vertices to form its truncate, we discover that we are just running the same film in reverse. We have just discovered the identity:

Tr(Du()) = Le().

Playing the rest of the film back we also discover these map operation identities:

Me(Du()) = Me(),

Le(Du()) = Tr(),

Du(Du()) = Id(),

where Id() is the identity operation, the operation that does not change the base map.

Altogether, there are 5 map operations (Id, Tr, Me, Le, and Du) that can be understood as inflations of the original vertices. This chart summarizes the sequence. The middle column shows polar views of the action of the operation on the map of a monogon on the sphere. The right column shows the action of the map operation on the quadrilateral associated with each edge of the base map. Look first at the diagrams for Id, the identity operation, to get oriented.

The map operation subdivide, Su(), inserts a new vertex at the midpoint of each edge in the base map. So Su(t) is:

Starting from Su(t), we now have two classes of vertices, old and new, that we can inflate differently. If we choose not to inflate the new vertices, we get a rather uninteresting sequence that reprises Id-Tr-Me-Le-Du with an additional vertex in all but Me (where it is redundant.)

Inflating only the new vertices first gives a truncation of only the new vertices that we will call lens, Ln():

Inflating the new vertices further causes them to meet at the old vertices, which gives parallel:

Inflating still further gives chamfer:

Inflating still more closes up the non-inflated faces completely, giving radial:

The following chart summarizes the map operations produced by inflating the new, mid-edge vertices:

Finally, if both original vertices and mid-edge vertices are inflated (neither being allowed to contract to a point) we get these map operations:

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