Variations of the Gringorten square equal-area map projection

Gringorten's square equal-area map projection has been forgotten since its appearance in 1972. I describe a modern implementation, including details of how to arrange, in different ways, the fundamental Gringorten projection of a sexadecant (one sixteenth of the surface of the sphere) onto a triangle. The Gringorten Mark I projection is an arrangement in which one hemisphere forms a square, with the other hemisphere disposed around it so that the whole sphere projects as a diamond, which may then be rotated to appear as a square. I introduce an alternative arrangement, the Gringorten Mark II, which is twice as high as it is wide, with one hemisphere on top of the other. These variants are compared with some other square map projections. Maps that fill a rectangular space completely can be very useful where, as on computer screens, space is limited and must be used efficiently.