> 4D Blocks

> Version 6

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> Toys and Puzzles
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Toys and Puzzles

Before I get started, let me point out that many of the previous examples can usefully be thought of as puzzles. For example, you can take any of the "geom-arch" examples, scatter the pieces, and then put the arch back together again. This is surprisingly satisfying. I think the arch-ring-sphere-3sphere sets of examples work bestóboth the original ones in Geometry Examples and the compact versions in More Geometry Examples. The examples "geom-nav3c" and "geom-nav4c" are also fun to mess around with, although there it's less clear what the goal is.


If you just want to play with blocks, these are the examples for you. The scenes start out empty, so you should hit "insert" a few times to create some blocks. You can also hit "shift-insert" and enter a quantity, but I quite enjoy seeing the blocks pop up one by one.

In the "a" variants, the blocks are all 1x1 blocks, just with different colors. These are the easiest to play with and build things with.

One good exercise to do in 4D is to get an overhead view, get into align mode, and create blocks one at a time while paying attention to their positions on the 3D mat. Once you've created enough of them, you'll have something like a hedge maze that you can fly down and navigate through. (And rearrange!)

The "b" variants are based on the wooden blocks I had when I was little. The shapes and colors are the same except for the following differences.

  • There were no 1x1 blocks.
  • The 2x2 blocks should be darker blue. (But then they're too hard to see on the screen.)
  • The 2x2 ramps were green, not orange. (Some of the 2x4 blocks were orange, though.)
  • The pillars were circular, not octagonal.
  • There was one yellow 2x6 block, great for making bridges.
  • There were some 2x4 blocks with 1x2 semicircles carved out to make arches. The arches and semicircles were both green, I think. Unfortunately the blocks program can only handle convex shapes right now.
  • The corners and edges were slightly rounded.
  • And, of course, none of the blocks were four-dimensional!

The "c" variants include every standard block, which is usually more kinds of block than you want, but not always. Note that you can delete blocks to get rid of them.


Speaking of hedge mazes, here are a couple, with some pleasing scenery to boot. No cheating please! That means no deleting the walls and no flying up out of the maze until after you've solved it. Also, note how at any point you can look up and see the sun and sky. It should help to keep in mind that the cyan sky is forward.


I'm really proud of these four-dimensional dice, even though the design would be obvious to any four-dimensional person.

Three-dimensional dice are numbered 1-6 and are arranged so that opposite sides add up to 7. In the same way, four-dimensional dice are numbered 1-8 and are arranged so that opposite sides add up to 9. However, in 4D there are eight corner positions on the faces, which together with the center position makes exactly nine. So, opposite sides not only add up to 9, they can and do have spots in exactly inverted positions! It's also nice that there's no need to use any edge positions like the 6 does in 3D.

On three-dimensional dice, the spot pattern for an odd number is always just the spot pattern for the preceding even number plus a center spot. If we require the same thing for four-dimensional dice and also apply the inversion property, that uniquely determines the spot patterns for all numbers except 2 and 4! But, those are easy enough to work out. The 2 should obviously have diagonally opposite spots. We can justify that by saying that it's the most symmetrical choice, but then it turns out that there's a symmetrical choice for the 4 as well: the four vertices of an inscribed tetrahedron! Then, as a bonus, the backward analogue of both the 2 and the 4 in 4D is the 2 in 3D.

The other numbers have nice spot patterns too. When we construct 5 by adding a center spot to 4 we get Ö a vertex-first view of a pentachoron! The 6 looks like a mess at first sight, but if you find the right angle you can see that the spots form a hexagon; and then the 7 is a hexagon plus a central point. And, of course the 8 is a cube.

There are three variant files here, but the only difference is in the number of dice.


Now we come to some things that I'd actually classify as puzzles. These are large blocks that have been painted different colors on different faces and then sliced up into 2x2x2 (x2) and 3x3x3 (x3) smaller blocks. The game is to scatter the pieces and then reassemble them so that the colors match up. Note that opposite faces have been painted similar colors. The inside parts are green, which as you may have noticed is the natural color of most shapes. The colors on the edges of the blocks are hard to predict, so I prefer to use the "A" key here to switch the boundary texture to white.

The cubes of size 3 are also fun to play with in their unscrambled state. You can dig into the center by moving or deleting cubes and make either a little cubbyhole or a tunnel that leads through to the other side. Right-angle bends in the middle are pleasing too.


The idea here is similar. We have a unit cube that's been painted red on the outside and then sliced up in various ways. The 3D versions (to me) are engaging but not challenging, but the 4D versions are mind-boggling because of all the slanted faces and diagonals. I won't spoil the puzzles by telling you what the shapes are, but you can always peek in the scene files if you're curious.

I haven't made a 4D version of "redcube3d" yet, but maybe someday I will. It requires an unusual shape, an irregular 24-sided gizmo with 8 cuboctahedral faces and 16 tetrahedral faces.


If that's not enough for you, here are the same puzzles with the faces painted different colors.


Finally, I made a couple of traditional jigsaw puzzles. The first ones are simple 2x2 (x2) puzzles. The "a" variants are printed on big blocks. I used to have some puzzle blocks like that where the outside surface was clear plastic. The "b" variants are almost as flat as real jigsaw puzzles. Maybe we should think of them as wooden puzzles, those are usually pretty thick.

The puzzles, even the "b" variants, are shown standing up so that you can see the pictures from the starting position, but after you scramble the pieces it's more natural to work with a top-down view. Before you scramble the "b" pieces you might want to fly over the top and see how thin they are.

In the 4D case, it's well worth your time to stop and think about what you're seeing. Just as in 3D the puzzle pieces have square faces, so in 4D the puzzle pieces have cubical faces. The pieces fit together so snugly that you can't rotate them in place, but if you pick them up, you can turn them around and see the flat edges (in the "b" case) and the large blank back face. When you have a piece that's right-side up, you can use the spin keys (shifted versions of the turn keys, by default) to reorient the piece without turning it over.

The "x" files are there for technical reasons, please just skip over them. If you're paging through the files, they'll look like empty scenes.


For the larger puzzles I made my best attempt at a traditional child's drawing of a house with a sun and a tree and some grass. Since I couldn't scribble all over the place with a blue crayon, I made some blue polygons for the sky instead. Please hit "A" to switch the boundary texture to white so that there's no confusion between the puzzle piece boundaries and the green colors in the drawing.

The 4D picture is exactly analogous. In fact, if you switch out of align mode and spin things around a little, you can see that the 3D picture is a projection of (parts of) the 4D one. Good luck!