# Herparian agar

By definition, an agar is a pattern filling the plane which is periodic in both space and time. A wick is a similar pattern except that it is supposed to be one - rather than two - dimensional. Although these definitions are easy to interpret, they are not always enforced with strict rigor. For example, the wires in the chicken wire agar have 2 as a uniform height but the cells of the mesh lying between them have a length which could be uniform. But that is only a special case; worse than being periodic, the distribution could well be variable - even - irrational, and still be called an agar. A discernible regularity, rather than periodicity, is what is really required.

A pattern which is finite with a border which renders it stable, but which could readily be enlarged is also called an agar. Little more than exhibiting a unit cell, or a small cluster of them, is usually sufficient to describe an agar; it could hardly be shown in its entirety.

Wick is a similarly flexible concept; one-dimensional isn't necessarily a straight line. It is a frieze. Also, one might think of Hilbert's space filling curve as an extreme example of a curve so crinkly that its dimension has to be calculated with care. Wicks can also be semi infinite, or even finite; again it is the regularity underlying their formation rather than their particular extent which matters.

Niemiec still life 12.105
Niemiec still life 12.14
Niemiec still life 12.15

With these generalities in mind, consider what can be done by stringing out dominoes to form ponds, lakes, and so forth. A snake is a figure similar to a domino, with the same proportions but thicker because of its internal structure, and selfstanding where a domino is not. But snakes can be used to create an assortment of patterns similar to the lakes by laying them out corner to corner; one such arrangement has already been called snake bridge snake. The figures at the right show all three; they are called herparian (from Greek: ἑρπετόν, herpeton, "creeping animal") agars.

Two of the bridges leave the snakes perpendicular; with the other they are parallel. Since a snake is symmetric by 180 degree rotation, this pairwise joining can be carried out at both ends, resulting in a chain. If indefitely prolonged a wick results; unlike chaining dominoes, finite, even highly convoluted chains are possible. In both cases, chains can close to form loops.

Rather than choosing to connect just one snake to the next, they can be joined two at a time, filling space with an ordinary lattice rather a highly convoluted curve. Either way, given enough regularity an agar can be constructed.

 several copies of the 6×6 unit cell of the smallest herparian agar Lake style snake agar

The figure at the left above contains a design which is readily extended to a square lattice, thus constituting an agar. Were horizontal bridging used, or the two combined, the figure at the right illustrates further possibilities.