Information — where it appears
Named by 3 essays across one field — each of them below, with the objects they name alongside it.
The field is empty where it would say nothing
The interchange format for crease patterns has a field for the layer order and nothing ever fills it in. Filling it in where the folded states can be listed — four of the eight printed patterns, and Miura patches to twelve panels — finds that the preliminary base and both twists have exactly one folded state, so every one of the field's signs follows from the crease pattern and the field would record nothing a reader could not compute. The fold-and-cut triangle has two states. The Miura is different: every patch with three or more columns has several — three, six and eleven on the three-by-two, three-by-three and four-by-three — so on the pattern that gets built the field carries information from six panels up, and the field's size had been measured as log₂ of the panels' orderings, which on the preliminary base is fifteen bits for an object that has zero.
One choice with eleven answers
A folded state was proposed as a short list of free choices — which way a flap lies, where a rim panel sits — with the layer-order field's signs following from them. Listed exhaustively on every Miura patch small enough, the choices are never independent: every sign that varies is tied to every other through a shared panel, so the states are one choice with many answers. And there are more answers than the record said. The overlap test had a blind spot a third of a panel wide, and with it corrected the three-by-three Miura has six folded states, not one, and the four-by-three eleven, not five.
A line in the columns, a binomial in the rows
A Miura strip two rows high was found to have 2c − 3 folded states for c columns, from three columns to seven, and the search that found it ran out at eight. Counted instead from the pairs of panels that actually overlap, the count runs to twenty columns and nine rows, and 2c − 3 turns out to be one row of a formula: a patch c columns by r rows has 1 + (c − 2)(B − 1) folded states, with B the central binomial coefficient of r + 1. The line in the columns has a reason that can be drawn. The binomial in the rows is measured and not explained, and it holds only while the folded columns two apart still overlap, which is the same as the pile being more than two deep.
Named alongside it
The objects these essays reach for when they reach for this one.