This is a truncated 4-dimensional cube, drawn in a curved style by Jos Leys. You can take an ordinary 3-dimensional cube, cut off its corners and get a truncated cube. Similarly, you can take a 4-dimensional cube, cut off its corners, and get a 4-dimensional uniform polytope with $2 \times 4 = 8$ truncated cubes as faces and $2^4 = 16$ tetrahedral faces! It’s called the truncated 4-cube.
A deltoid is a curve formed by rolling a circle inside a circle whose radius is 3 times larger. Similarly, an astroid is a curve formed by rolling a circle inside a circle whose radius is 4 times larger. The picture here, drawn by Greg Egan, shows a deltoid rolling inside an astroid. It fits in a perfectly snug way!