2x2x2x2, part 7

August 30th, 2016

Once you start playing with 4D puzzles, it’s difficult to stop. They are highly addicting.

After thinking about the 2x2x2x2 puzzle for a while, I started wondering how many ways there were to build a kind of 4D Soma puzzle. That is, little pieces all fitting together perfectly to create the 2x2x2x2 hypercube.

There are endless variations on such a puzzle, so I narrowed it down a bit. Consider that there are a total of 16 little hypercubes in a 2x2x2x2 hypercube. Suppose we only consider puzzles where each “piece” consists of exactly 4 little hypercubes.

It will take four such pieces to build the 2x2x2x2 hypercube (since 4×4 = 16). Suppose we restrict things even further: All four of those pieces need to be the same shape.

How many different 4D puzzles can we make, if we follow those rules? More tomorrow.

Relativity

August 29th, 2016

In some 4D world
Someone’s solving a puzzle
In five dimensions

2x2x2x2, part 6

August 28th, 2016

Now that you’ve seen what it looks like to rotate a 2×2 block inside a three dimensional cube, let’s see what happens when we rotate a 2×2 block inside a four dimensional hypercube. It turns out that there are at least six different ways you can rotate the same 2×2 block within a 2x2x2x2 hypercube:


0 1
2 3


4 5
6 7


8 9
a b


c d
e f


0 1
2 3


4 5
6 7


8 9
a b


c d
e f


0 1
2 3


4 5
6 7


8 9
a b


c d
e f


0 1
2 3


4 5
6 7


8 9
a b


c d
e f


0 1
2 3


4 5
6 7


8 9
a b


c d
e f


0 1
2 3


4 5
6 7


8 9
a b


c d
e f

All six of the above cases represent the same 2×2 shape embedded in a 2x2x2x2 hypercube. The only difference is how they are rotated differently (in four dimensions, of course). The good news is that all those rotations provide lots of opportunities to make really interesting puzzles.

If I had a Holo…, part 2

August 27th, 2016

One problem with trying to answer the question “What would I create if I really had Holodeck?” is that the question is woefully broad. The Holodeck is not a genre. It may not even be a medium.

Which means that there isn’t one thing or another that it particularly suggests — it suggests everything. It’s sort of like asking the question “What should a book be about?”

In fact, the analogy with a book is fairly good. The possibilities attainable in written language are vastly greater than the possibilities attainable in cinema. If you doubt that, just go back and reread Gogol’s The Nose.

Similarly, the very power of the Holodeck can make it difficult to narrow down creative choices. But I can think of various interesting directions.

Storytelling, games, construction toys, music (anything from enjoying a musical performance to creating original music, to creating original musical instruments), visits to exotic places, both possible and impossible, these are just a few of the possibilities.

Then there are the serious uses: Architecture, medicine, art, science, literary analysis, financial models, the list goes on and on.

In each case, the Holodeck manifests itself differently. To compare one Holodeck experience to another might be as futile as comparing a romantic farce to a Shakespearean tragedy.

I suspect that the Holodeck, if we ever manage to make it a reality, will not turn out to be a medium at all. Like the computer, it will more likely be akin to the computer — a “meta-medium”. That is, a vastly protean substrate through which new media are continually discovered and developed.

If I had a Holo…

August 26th, 2016

Let us, for the moment, take our leave of the sunny shores of the fourth dimension. We now pay a visit to the twentyfourth century.

It is the year 2364. Captain Picard has just kindly taken you on a tour of the Holodeck. You are aboard the U.S.S. Enterprise because StarFleet Command has decided that you have a high aptitude for storytelling. Should you win, your prize is to author a Holodeck experience to be staged on the real thing.

But what single experience will you choose for this? After all, on the Holodeck anything is possible.

In particular, suppose you could describe only one scenario? What would it be?

2x2x2x2, part 5

August 25th, 2016

One problem with “flattening” an object into two dimensions is that it can be hard to understand what happens when you rotate the object. Let’s go back to our three dimensional case.

Suppose we turn every little cube red if it is on the right side of the big cube. Here is what this looks like as a 3D object:

In the image below I’m flattening this, so that the left part shows negative z (the little cubes in the back), and the right part shows positive z (the little cubes in the front):


0 1
2 3
4 5
6 7

If we rotate the cube in different ways in the 3D view, it’s always easy for us to see where the highlighted red cubes go. After all, we’ve been looking at rotating 3D objects all our lives.

But if we look at the flattened view, the same object rotated in differently ways can look a little strange:




0 1
2 3
4 5
6 7
 
0 1
2 3
4 5
6 7

These all represent the same shape rotated in various ways, but they don’t really look the same. I don’t have any answers for what to do about this. I think you just need to get used to the strangeness of it.

Because when we start rotating things in four dimensions, it’s going to get even stranger.

String theory

August 24th, 2016

There are times a string
Of only twelve syllables
Beats four dimensions

2x2x2x2, part 4

August 23rd, 2016

So we’ve seen that you can represent a 2x2x2 cube on the page by placing its third dimension next to its first two dimensions. You can also represent a 2x2x2x2 hypercube on the page using a similar trick.

It takes sixteen little cubes to make a 2x2x2x2 hypercube. We can visually represent those sixteen little cubes as follows:











0 1
2 3


4 5
6 7


8 9
a b


c d
e f


I’ve represented each hypercube as a tiny square, and all the hypercubes are labeled. The labels are in base sixteen. In base sixteen, after you run out of numbers, you continue on with letters, using a, b, c, d, e, f to represent 10, 11, 12, 13, 14, 15.

There are four dimensions: x, y, z and w. Each tiny 2×2 square represents x and y (columns for x, rows for y). Each large collection of squares represents a value of z and w (columns for z, rows for w).

So you can see that every one of the 16 locations represents a unique combination of x,y,z and w. For example, the top left corner represents 0,0,0,0. The top right corner represents 1,0,1,0 (in other words, x and z both have value 1, while y and w both have value 0).

It’s important to keep in mind what we are doing here: We are discussig a 4D object, but we sre visually flattening it out, so that we can show it on the page.

More tomorrow.

2x2x2x2, part 3

August 22nd, 2016

It’s not easy to talk about a four dimensional puzzle on a blog, because the page you are reading right now is basically only two dimensional. So I’m going to resort to an old trick: Rather than try to show you four dimensions directly, I’m going to show you four dimensions flattened out to two dimensions.

To understand the principle, imagine you were a creature in Flatland, and a friend from the 3D universe wanted to talk to you about the eight little subcubes that make up a bigger cube. Your 3D friend knows you can’t actually see a 3D cube, so she flattens it out for you, showing you the top 2×2 cubes side right next to the bottom 2×2 cubes:


0 1
2 3
4 5
6 7

In your mind you’re supposed to think of the two images above as being stacked up, one on top of the other. But if you live in Flatland, you don’t know which way is up, so looking at them side by side let’s you see the eight little sub-cubes in a way that you can understand.

More tomorrow.

Long dead days

August 21st, 2016

True tale, long told, from some dead bard,
Time past when life ’twas very hard:

Each year ’twas writ, this time long past,
Some rule, some game, till days flew fast.

That year they said each word must hold
Some self same size (true tale, long told).

Four, nary more, this pact thus made,
True lore, we’re sure, from long dead days.