Construct the 4D analog of a tetrahedron
Arrange four red vertices symmetrically around a central violet vertex. The color shows that the red vertices sit in the negative direction in the fourth dimension (w < 0) while the violet vertex sits in the positive direction in the fourth dimension (w > 0). The hypertetrahedron's five 0D vertices and ten 1D edges will be obvious. Can you also find its five 3D hyperfaces (“cells”)? More challenging still, can you find all ten of its 2D faces?
Construct the 4D analog of an octahedron
Hint: An ordinary 3D octahedron may be drawn with its vertices at (±1, 0, 0), (0, ±1, 0) and (0, 0, ±1). Similarly, a 4D “hyperoctahedron” may be drawn with its vertices at (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0) and (0, 0, 0, ±1).
Your hyperoctahedron's 0D vertices and 1D edges will be obvious. Can you also find its 2D faces and 3D cells?
Construct the 4D analog of a cube
You'll need eight red vertices and eight violet vertices. Each red vertex sits at exactly the same 3D location as the corresponding violet vertex (same x, y and z coordinates). Which pairs of vertices do you need to connect with edges? Don't forget to connect each red vertex to it corresponding violet vertex.
Can you find the hypercube's 3D cells? How many 3D cells does it have?
Can you find the hypercube's sixteen 0D vertices, its thirty-two 1D edges and its twenty-four 2D faces?
What sort of 4D polyhedron do alternating vertices of a hypercube give you?
To get warmed up, start with an ordinary 3D cube (make it all green in 4D Draw). Delete every other vertex in alternation, so only four of the cube's original eight vertices remain. Connect those four vertices with edges. What sort of polyhedron do you get?
Now try the same construction starting with a 4D hypercube. Delete every other vertex in alternation, so only eight of the hypercube's original sixteen vertices remain. Where should you put the edges, to make a convex 4D hyperpolyhedron?
Can you see this 4D hyperpolyhedron's 3D cells? How many are there? What is the shape of each one?
This 4D hyperpolyhedron's 0D vertices and 1D edges will be obvious. Can you also find its 2D faces?
Compare this 4D hyperpolyhedron to the 4D analog of an octahedron (see above). How are they similar? How are they different?
What does a hypercube's “diagonal slice” look like?
First consider a 2‑dimensional person's drawing of an ordinary 3D cube, if the cube sits with a long diagonal perpendicular to that person's 2D world:
If the 2D world passes through the cube's center, the cross-sectional slice will be a green hexagon:
The question now becomes, if we 3‑dimensional people draw a 4D hypercube with its long diagonal perpendicular to our 3D space, and with our 3D space passing through the hypercube's center, what will the 3D cross-sectional slice be?
Hint: Start by putting the “lowest” corner at (0, 0, 0, red). Add the next four corners at (‑2, ‑2, ‑2, yellow), (‑2, 2, 2, yellow), (2, ‑2, 2, yellow) and (2, 2, ‑2, yellow). Then keep adding more corners, until you've built the complete hypercube. Finally, connect the vertices at the “green” level to see what the hypercube's diagonal slice looks like. If you want, you can then delete all the hypercube's non-green vertices, so you can see the diagonal slice by itself, without distraction (but save a copy first!).
What does a 4D fused checkerboard tiling look like?
To get warmed up, start with a 2D checkerboard
split every other square into four triangles
and then fuse each triangle with the nearest whole square
The result is again a checkerboard, but the new squares are tilted 45° and have twice the area of the old squares.
Now build yourself a 3D checkerboard (a cubic tiling, all green) and imitate the same procedure: split every other cube into six pyramids, and then fuse each pyramid with the nearest whole cube to get a new tiling of 3D space. In this new tiling, what is the shape of each 3D cell? How many 2D faces does each 3D cell have? Are those 2D faces regular or irregular (that is, are they perfect squares or merely rhombi)?
Finally, build yourself a 4D checkerboard and imitate the same procedure: split every other hypercube into eight hyperpyramids, and then fuse each hyperpyramid with the nearest whole hypercube to get a new tiling of 4D space. In this new tiling, what is the shape of each 4D hypercell? How many 3D hyperfaces (“cells”) does each 4D hypercell have? Are those 3D cells regular or irregular?