Tailors and cartographers have long realized an ironclad mathematical truth: Objects that are curved in most ways cannot be seamlessly covered or directly represented by flat materials unless those materials are stretched or torn. For that reason, tailors need cuts and seams to make a shirt from flat pieces of cloth, and mapmakers representing Earth on a flat surface can accurately depict relative areas or local shapes but not both. A well-known consequence is the wildly out-of-proportion representation of Greenland and Antarctica in Gerardus Mercator’s well-known projection. Indeed, distorted spherical projections are known from antiquity, dating back at least as far as Ptolemy’s Planisphaerium.
Clothing a rounded form or mapping shapes and distances from a globe onto a flat sheet requires artifice or compromise, if not both. Curved and flat spaces are generically incompatible. But what if we could transcend that limitation? Imagine the possibilities for device design if...