[xoxoxoBruce]

The AuthaGraph company makes world maps which are rectangular but still accurate in area proportions.

AuthaGraph map is able to transform an entire sphere to a rectangle as Mercator projection does while it substantially keeps sizes and shapes of continents as Dymaxion map does.

This rectangular world map called AuthaGraph World Map is made by equally dividing a spherical surface into 96 triangles, transferring it to a tetrahedron while maintaining areas proportions and unfolding it to be a rectangle.
The world map can be tiled in any directions without visible seams. From this map-tiling, a new world map with triangular, rectangular or parallelogram’s outline can be framed out with various regions at its center.

AuthaGraphic Projection
An original method for maintaining areas proportions is called "iso-area-mapping."
And an original mapping process by combining different projection methods via intermediate objects is called "multilayer-mapping." These ideas for mapping aims to reduce errors during projecting a sphere to a tetrahedron because a simple optical projection from a sphere to a tetrahedron causes huge distortions therefore useless.

link