The purpose of this note is to draw attention to two infinite families of antiprisms that differ slightly from the usual Keplerian kind. These figures have surely been noticed earlier by geometers and philomorphs, but I was unable to track down a reference to them. My attention was drawn to these figures by a Christmas gift that my family received recently. The gift was a cake in a box of a rather unusual shape (see Figure 1). On examining the box, I saw that it could be characterized variously as a deformed antiprism, an unfolded pyramid or a truncated prism. The box had two square faces parallel to each other, with the line joining the centres of the squares being perpendicular to the plane of either square. The squares were rotated 45° relative to each other, exactly as in a square antiprism, but one square had only half the area of the other. The separation of the squares was such that, of the eight triangles linking them, four were identical equilateral triangles while the rest were identical isosceles triangles.