In the year 2026, after a storm of unusual violence, I was cast upon a shore that no chart had ever recorded. Koch island as I named it, was small enough that, standing on a low hill, I could see its entire extent. Its meadows and woods occupied a finite space — a man might, with diligence, paint every inch of its surface before nightfall. Yet when I set out to walk its circumference, the path refused to end. Each stretch of beach I thought familiar dissolved, upon closer approach, into a jagged promontory: the middle third of every shore replaced by two new sides of a smaller triangle, thrusting seaward. What had been one length became four, each a third as long. I walked for hours. The island remained no larger, yet the distance before me grew without bound. No matter how carefully I followed the water’s edge, I could not complete the circuit in any finite time. A native, watching me from the trees, explained in a language of gestures that the coastline lived between dimensions. Magnify any portion by three and four identical copies appeared. Its measure, he indicated, was the logarithm of four divided by the logarithm of three — greater than a line, less than a surface. Continuous underfoot, yet nowhere could a straight staff be laid truly tangent to the shore. I had nearly abandoned the attempt when, from the interior of the island, a low horn sounded — long, pure, and unending. The native smiled and pointed toward a narrow, tapering valley that ran farther than the eye could follow. “Gabriel’s,” he said. “You may fill it with a finite measure of wine, yet no amount of paint will ever coat its walls.” I listened a long while, and did not try to walk around either wonder again.