what is the shape that can fit without spaces?

I think all platonic solids would work. No proof to offer you, if you have some D4 or D8 lying around it would be kind of simple to test.

Platonic solids are only regular shapes, so if you can use non-regulars shapes then the number of tessellateable 3D shapes is a bounded infinity

Platonic solids are also the only fair dice (each side has an equal chance of showing given sufficiently random initial orientation, velocity, and rolling surface. (Human dice rolls are usually considered to be a strong random)

You are probably familiar with tessellation of a 2D shape to fill a plane.

Tessellation can be extended to three or more dimensions. Certain polyhedra can be stacked in a regular crystal pattern to fill three-dimensional space, including the cube (the only regular polyhedron to do so); the rhombic dodecahedron and the truncated octahedron. Some crystals including Andradite and Fluorite can take the form of rhombic dodecahedra.

Tessellations in three or more dimensions are called honeycombs. In three dimensions there is just one regular honeycomb, which has eight cubes at each polyhedron vertex. Similarly, in three dimensions there is just one quasi-regular honeycomb, which has eight tetrahedra and six octahedra at each polyhedron vertex. However there are many possible semi-regular honeycombs in three dimensions