4D Hypercube fun facts

[Renslay]

Hi all!

Ever since I've got Melinda's physical 2x2x2x2, I fell in love with hypercubes. I already solved my first 3x3 hypercube in 2007, it's just my interest towards higher dimensional puzzles are reignited.

So, just out of randomness, here are a couple of fun facts about Rubik's hypercubes, mostly comparing 3d and 4d 2x2 cubes. If you don't know much about the topic, these can be quite mind blowing!

#0
The shape of a 3d cube is called... well, cube. Or regular hexahedron; but who calls it that?
The shape of a 4d cube is called a tesseract. Or 8-cell. Hypercube is a general term for any n-dimensional cube.

#1
You create a 3d 2x2 Rubik's Cube by splitting a solid 3d cube with three 2d planes, halving the 3d cube across each dimension.
You create a 4d 2x2 Rubik's Cube by splitting a solid 4d cube with four 3d spaces, halving the 4d cube across each dimension.

#2
The 3d cube has 6 colors: two colors for each axis.
The 4d cube has 8 colors for the same reason. The extras are usually purple and pink.

#3
The sides of a 3d cube is U,D,F,B,R,L.
The sides of a 4d cube is U,D,F,B,R,L,I,O. I stands for Inner, O is for Outer (as we don’t have better words for direction along the 4th axis). Sometimes they are called Top/Bottom, but that can be confused with U/D.

#4
There are 8 corner pieces on a 3d 2x2.
There are 16 corner pieces on a 4d 2x2.
It's not much of a surprise if you know their full name is 2x2x2 (or 2^3), and 2x2x2x2 (or 2^4).

#5
A corner piece of a 3d cube has 3 square (2d) stickers.
A corner piece of a 4d cube has 4 cube (3d) stickers.

#6
Every corner piece on a 3d cube has 3 neighbours; differing in exactly one color.
Every corner piece on a 4d cube has 4 neighbours; differing in exactly one color.

#7
A 3d corner can have 3 different orientations.
A 4d corner can have 12 different orientations.

#8
You can orient a 3d cube in 24 different ways (for example, red on top and green on front; or white on top and blue on front, etc).
You can orient a 4d cube in 192 different ways.

#9
On a 3d cube, including no-turn, there are 4 different ways you can apply a turn on a side (for example, R, R', R2, nothing).
On a 4d cube, including no-turn, there are 24 different ways you can apply a turn on a side. The turning notation is a bit more complicated, but not that bad.
[Disclamer: using a certain turn metric, like HTM.]

#10
The number 24 in #8 and #9 are the same for the same reason; and the rule goes further in higher dimensions. For example, on a 5d cube you can turn a single side in 192 different ways; and you can orient a 5d cube in 1920 ways. On a 6d cube you can turn a single side in 1920 ways, and so on. The pattern for these numbers are 4*6*8*10*...

#11
When you do a turn on a 3d 2x2, you move half of the pieces.
Actually, this is true in any n-dimensional 2x2.

#12
If you randomly assemble a 3d 2x2, there is a 1/3 chance it is solvable.
If you randomly assemble a 4d 2x2, there is a 1/6 chance it is solvable.

#13
You cannot twist a single corner on a 3d 2x2 (without affecting the other pieces).
You can twist a single corner on a 4d 2x2 (without affecting the other pieces), but out of the 12 possible orientations only 4 is achievable (including the no-twist solved state).

#14
To reverse the roles of #13:
You can swap two corners on a 3d 2x2 (while leaving the other pieces).
You cannot swap two corners on a 4d 2x2 (while leaving the other pieces).

#15
Fun fact #13 and #14 are the reason for #12.

#16
The number of possible solvable states of a 3d 2x2 is 3.6 million: 3,674,160.
The number of possible solvable states of a 4d 2x2 is 3.3 octillion: 3,357,894,533,384,932,272,635,904,000. That’s about a billion times larger than the number of states of a 3d 3x3 (which is 43.2 quintillion, or 43,252,003,274,489,856,000).

#17
There are physical, playable analogues of many of the 4d cubes, including the 4d 2x2 and 4d 3x3. They are like 3d "unfolded" versions of their 4d representations, similarly how you can unfold a 3d cube into its 2d net of squares (not an exact analogue, but close enough).
Unfortunately, not all of the possible turns are equally easy to apply on a physical 4d cube. Think about a 3d cube where the only turns you can do is U2, R, x and y. You can still do an F turn as y’ R y, but it requires more substeps than a simple R turn.
Still, they are extremely fun to play around with!

Which fun fact is your favorite, or most surprising? Do you know more?

 

[abunickabhi]

My favourite fact is #7

 

[Renslay]

My favourite fact is #7

Ah, yes, the number of orientations. That comes from the following argument:
In 3d:
Imagine you want to put a specific corner, for example the white-red-green to a specific place, like UFR, but you do it by sticker by sticker. How many ways you can do that?
You can chose from 3 stickers which goes to U, then you can chose from 2 remaining stickers which goes to F, and the remaining sticker goes to R.
So, there are 3*2*1 way you can "put" this corner into that place. But half of them are not valid: they would produce a mirrored, wrong version of the corner! So in 3d there are 3*2*1/2 = 3 orientations.

Same argument works in 4d. You want to put a specific corner, for example the white-red-green-purple to a place, UFRO for example, sticker by sticker. You can chose from 4 colors what goes to U, then from 3 colors what goes go F, from 2 colors what goes to R, and the remaining color goes to O. That's 4*3*2*1 way to sticker-up that corner, but again, half of them are invalid mirrored cases.
So the number of orientations is 4*3*2*1/2 = 12.

Can you gess the number of corner orientations on a 5d 2x2?

 

[xyzzy]

#12
If you randomly assemble a 3d 2x2, there is a 1/3 chance it is solvable.
If you randomly assemble a 4d 2x2, there is a 1/6 chance it is solvable.

The funny thing is that 3D and 4D are the exceptional cases, because in all higher dimensions the orientation constraints are different.

1. 3D is the only dimension where parity doesn't matter. In all higher dimensions, every move is an even permutation of the pieces, so there's a 50% chance the permutation alone isn't even solvable. (Not 100% sure of this but it feels like it should be true.)

2. The global orientation constraints can be expressed in terms of the abelianisation of the alternating group A_n. (This incidentally also answers the above question; the individual piece orientations lie in the alternating group A_n, so there are n!/2 of them. Non-orientable variants (e.g. on Klein bottle surface) would instead have n! piece orientations.) A_3 is already abelian, so its abelianisation is itself, with order 3. A_4 is not abelian, with the Klein 4-group as its commutator subgroup, and has an abelianisation of order 3. All larger alternating groups are simple and non-abelian, so have trivial abelianisation. The probability of a random element of (A_n)^ab being the identity element is thus 1/3 for n=3 or 4; and 1 for n≥5.

 

[OtaMota]

every move is an even permutation of the pieces, so there's a 50% chance the permutation alone isn't even solvable. (Not 100% sure of this but it feels like it should be true.)

I believe this is true. I have tried to apply a pure 2-swap on 2^4 before; I don't believe it is possible.

 

[ruffleduck]

My favorite fun fact is #15 because it's a meta fun fact (or maybe a fun meta-fact?)

 

[Renslay]

I believe this is true. I have tried to apply a pure 2-swap on 2^4 before; I don't believe it is possible.

Yes, it is impossible to do a pure 2-swap on a 2^4, and it's easy to prove.

Without loss of generality, inspect a single RO turn (also known as Rx). The 4 elements closer to the Inner side does a 4-cycle a -> b -> c -> d -> a around the x axis, which is an odd permutation, but the 4 elements closer to the Outer side also does a similar 4-cycle around the x-axis, independently of the others. So two odd permutations happen at the same time, which is an even permutation overall.

A pure 2-swap is an odd permutation, but a series of even permutations always lead to a state of even permutation. Q.e.d.

By the way this is the same reason why you cannot do a pure 2-swap on a 3^3 regular Rubik's Cube. There with quarter turn you do a 4-cycle of 4 corners and a 4-cycle of 4 edges; here you have two 4-cycles of 8 corners. But the principle is the same.