Gosper Island

Start with a hexagon and replace each side with three shorter lines in a zigzag: turn 60 degrees one way, then 60 degrees back. Each line is $1/\sqrt{7}$ of the side, about 38%. Then repeat on every line.¹

Three lines of 38% make about 1.13 times the old length, so the coastline grows without limit.

The area does not change at all. The zigzag looks the same if you spin it half a turn about its middle, so it adds exactly as much on one side of the old line as it removes on the other, and every island has the area of the hexagon it started from.²

That same half turn is why islands fit together. Two neighboring hexagons share a side and read it in opposite directions, and because the zigzag looks the same from both ends, both islands get the identical coastline along that side. A grid of hexagons becomes a grid of islands, with no gaps or overlaps.³

Now put six hexagons around one. The outer edge of those seven has 18 sides, and it is exactly a larger hexagon with each of its six sides zigzagged once.

As you raise the round, the outline stays right on the seven islands, because zigzagging every line of two matching outlines leaves them matching. So seven islands make one island, $\sqrt{7}$ times as wide and seven times the area.

This is why the Gosper curve fills this shape: its route also joins seven smaller copies of itself. Set Show to Whole plane in Exploration controls to see the seven tints repeat.

Further exploring

• Gosper Island — MathWorld
• Gosper curve — Wikipedia

Sources and notes

1. Flowsnake. The coastline's dimension, $2\ln 3/\ln 7$, comes from these numbers: A113211. ↩
2. Calculation. This page's islands were also measured up to five rounds. The area matched the hexagon's to nine decimal places. ↩
3. Flowsnake Earth. ↩