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After all that talk about which patterns have which types, I really wanted to see an index of patterns by type, so I put together this subessay. To begin with, here's a table of counts by section and type, with totals … basically the joint and marginal distributions.
And here are the details for each type.
A. Type 2
Checkerboards are the only patterns of these types (that I like). I said in the last subessay that centers can't move in any of these ways, but actually we can make that statement a bit stronger: the only ways that centers can move are specified by the other four of the seven types.
D. Type 22
There are three simple patterns of type 22: dots, a checkerboard, and crosses. If you add any two of them together you get the third, so together with the identity pattern they form the group Z22. There's also that one pattern based on the edge-corner swap.
E. Type 4
There are two patterns of type 4: crosses and stripes.
F. Type 33
I've talked about six points of variation in these patterns: centers, corners, two sets of inner edges, and two sets of outer edges. Since the points can be varied independently, the space of possible patterns looks like a six-dimensional cube. However, I'm mainly interested in patterns with centers and no corners, and when one set of inner or outer edges is active I don't care which set it is. That simplifies things a lot. Just as we can split a cube into two squares, we can split the six-dimensional cube into four four-dimensional cubes and focus on the one with centers and no corners. And, just as we can sum the coordinates of a square by projecting at a 45-degree angle, we can project that four-dimensional cube into a table with axes that are counts of inner and outer edges. And here it is!
It only takes one conjugated primitive per step to move around in the table!
We can project the other four-dimensional cubes into other tables, but it's kind of pointless since I only like one pattern in each cube.
Just for the record, the space of all possible patterns of type 33 is even larger than the space I was just talking about. There are two other obvious points of variation (two sets of three corners), and there are also some other less obvious ones. For example, we can flip the inner edges as well as move them around. I don't like any of the resulting patterns, though.
G. Type 2AA
I'm also not going to discuss the space of all possible patterns of type 2AA. The part I do want to discuss looks like a pair of tetrahedra stuck together. At one end we have the sandwich, which if you're into origami is pretty much the bird base of type 2AA patterns. At the other end we have crosses, where all four edges are swapped between the upper and lower layers. In between we have three patterns where two edges are swapped. I've laid the whole thing out below as the first row of a table. I know it's a poor substitute for a 3-D diagram, sorry.
There are three kinds of primitives we can use to move around in the diagram. Edge swaps conjugated by RεB connect the sandwich to the present to crosses, and also connect the basket to the unnamed pattern. Vertical edge swaps conjugated by U±1 connect the sandwich to the basket to crosses, and also connect the present to the unnamed pattern. Finally, awkward swaps like (B2R2)3 conjugated by FU2F-1U-1 complete the symmetry.
If you look at the middle layer of any of those patterns, you'll see that there are two edges that can be flipped without messing up the type. Doing that drops you into the second row of the table, which has the same internal structure as the first.
In addition to the patterns in the table, there are also columns of type 2AA.
In this last category we have the two heart patterns of type 5 and the four patterns that don't fit into the classification: flowers, stripes, and two kinds of columns.
@ January (2013)