And among such false means largeness of scale in the dwelling-house was of course one of the easiest and most direct. All persons, however senseless or dull, could appreciate size: it required some exertion of intelligence to enter into the spirit of the quaint carving of the Gothic times, but none to perceive that one heap of stones was higher than another. And therefore, while in the execution and manner of work the Renaissance builders zealously vindicated for themselves the attribute of cold and superior learning, they appealed for such approbation as they needed from the multitude, to the lowest possible standard of taste; and while the older workman lavished his labor on the minute niche and narrow casement, on the doorways no higher than the head, and the contracted angles of the turreted chamber, the Renaissance builder spared such cost and toil in his detail, that he might spend it in bringing larger stones from a distance; and restricted himself to rustication and five orders, that he might load the ground with colossal piers, and raise an ambitious barrenness of architecture, as inanimate as it was gigantic, above the feasts and follies of the powerful or the rich.
A ri is a unit of measure, it’s about how far a person can walk in an hour at a reasonable pace. It clocks out at roughly 3.93 kilometers.
Remnants of the ri system are scattered along the old roads of Japan. During the Edo period, ri were marked recurrently by hulking earthen mounds that flanked the road — ichi-ri zuka, “one-ri mounds.” There are only a handful of “originals” left. When you pass one with an old cypress or oak growing from its center it becomes a tiny moment of celebration.
Wang tiles (Hao Wang, 1961) are a class of formal systems. They are modelled visually by square tiles with a color on each side. A set of such tiles is selected, and copies of the tiles are arranged side by side with matching colors, without rotating or reflecting them.
The basic question about a set of Wang tiles is whether it can tile the plane or not, i.e., whether an entire infinite plane can be filled this way. The next question is whether this can be done in a periodic pattern.
In 1966, Wang's student Robert Berger solved the problem in the negative. He proved that no algorithm for the problem can exist, by showing how to translate any Turing machine into a set of Wang tiles that tiles the plane if and only if the Turing machine does not halt. The undecidability of the halting problem then implies the undecidability of Wang's tiling problem.