Myriahedral projections are a new class of methods for mapping the earth. The globe is projected on a myriahedron, a polyhedron with a very large number of faces. Next, this polyhedron is cut open and unfolded. The resulting maps have a large number of interrupts, but are (almost) conformal and conserve areas. A general approach is presented to decide where to cut the globe, followed by three different types of solution. These follow from the use of meshes based on the standard graticule, the use of recursively subdivided polyhedra and meshes derived from the geography of the earth. A number of examples are presented, including maps for tutorial purposes, optimal foldouts of Platonic solids, and a map of the coastline of the earth.