Miroslav Vicher found many . Here is one I found, with fans, pyramids, corks, and a few triangles. I liked that pattern, but I felt it was possible to use fewer shapes. Here is a Mitre System tiling that just uses two shapes, and an occasional third. To me, it seems like there is still much to know about tessellations.
Much beauty remains to be discovered.
This new shape, in purple, seemed promising. The below pattern uses only pyramids and the new shape, dubbed the Millstone.