A FLEXIBLE but inextensible thin spherical shell may be bent inwards through an angle θ on and only on a circle whose radius of curvature (expressed in angular measure on the sphere) is ½θ. This is easily proved with a geometrical construction involving two equal intersecting spheres, and can be demonstrated on a punctured ping-pong ball. This affords a simple explanation of the shape of island arcs and related arcuate geographical structures. The interpretation1 of island arcs, or rather the ocean trenches which lie outside them, as the places where oceanic crust, spreading from mid-oceanic ridges, bends downwards to return into the mantle is now widely accepted (compare various contributions to ref. 2). The dip of this return path is revealed by the location of earthquake foci on a surface which slopes downwards to a depth of 700 km at 45° (ref. 3) (if one discards an analysis in terms of two successive slopes, one shallower and the other steeper, which the uncertainty of focal depth determination does not justify). The Kamchatka-Kurile arc, the Alaska Peninsula-Aleutian arc and the Sumatra-Java arc all have radii of about 20°, or about 22° measured out to their associated trenches, in consonance with dips of 45°. The arcuate curvatures of the Himalaya and the Andes, though rather more difficult to specify precisely, are of about the same magnitude.