Kinds of Blocks

1D

• The only blocks are line segments of various lengths.

2D

• Squares.
• Circles, semicircles, wedges with various angles.
• Rectangles of various sizes (1x2, 2x2, 1x4, whatever block or Lego sizes you like).
• Regular polygons (triangle, pentagon, hexagon, octagon).
• Rectangles cut in half diagonally to make right triangles of various sizes.
• Maybe that traditional block shape where you take a 2x4 rectangle and cut out a 1x2 semicircle to make an arch.

3D

Now things get interesting. We can construct 3D blocks by taking products of 2D and 1D ones. (Here that just means extend 2D shapes orthogonally into the third dimension, but in 4D it becomes more powerful.) In addition to prisms / cylinders we can also make pyramids / cones with whatever bases. And, in addition to that we have the following full 3D shapes.

• Spheres, half-spheres, quadrants, octants.
• Regular polyhedra (cube we already have, but also tetrahedron, octahedron, etc.).

4D

4=3+1 : We can extend any 3D block into 4D as a prism or as a pyramid. For example, just as a circle extended into 3D makes a wheel that can roll in one direction (but not two) or lie flat on a circular face, a sphere extended into 4D can roll in two directions (but not all three) or lie flat on a spherical face.

4=2+2 : We can also take products of two 2D blocks, for example circle x circle, one I'm very curious to see because I don't understand it. Products of regular polygons with different numbers of sides can be understood, though, and they approximate circle x circle.

There are also full 4D shapes.

• 3-spheres, etc.
• Pentachoron (like tetrahedron).
• Hexadecachoron (like octahedron).
• 24-cell (unique 4D shape).
• 120-cell and 600-cell (probably too big to be useful).

That's all the regular polychora, but there are also all kinds of truncations and less regular shapes, just as in 3D.