Abstract or Additional Information
The Ancient Greeks proved that the circle is the least-perimeter way to enclose given area. Similarly the round soap bubble provides the least-perimeter way to enclose a given volume of air, although that was not proved until 1884 by Schwarz. Similarly the double bubble that forms when two soap bubbles come together is the least-perimeter way to enclose and separate two given volumes of air, although that wasn't proved until 2000 by Hutchings, Morgan, Ritoré, and Ros. Lord Kelvin sought the least-perimeter way to divide all of space into unit volumes, and his conjecture stood for 100 years, until Weaire and Phelan found a better way in 1994. Whether their new candidate is best remains open today. Even the least- perimeter way to divide the plane into unit areas, using the bees' hexagonal honeycomb tiling, though conjectured by the Ancient Greeks, was not proven until 1999 by Hales. The most efficient tiling by pentagons remains open. In many simple non-Euclidean possible universes, even the ideal shape for a single soap bubble remains open.