Penn Station

I am sitting in Penn Station, NY, NY, waiting for a train that will take me to the airport. And I am realizing that there is something about Penn Station which is both fascinating and disturbing.

Nobody is here because they want to be — everyone here is just passing through on their way to somewhere else. So there is an odd sort of energy, a kind of pervasive restless feeling.

It’s different from, say, Grand Central Station, which is an actual destination in its own right. Over there, people stop and look around, draw in a deep breath, spend some time taking in the magnificent vaulted ceiling, luxuriate in the sense of spacious grandeur.

Here it hasn’t been that way since 1963. Once one of the most magnificent buildings in the world, this place is now cramped and soulless, a place nobody wants to be.

But it’s a place that lots of people need to pass through, while waiting to get to somewhere else. So if nothing else, it’s a really awesome place for people watching.

Amazing coincidence

I was telling a colleague yesterday about the oddest coincidence. You see, I have lots of friends who have had experience with contractors. Maybe they needed their roof fixed, or their kitchen redone. I also have some other friends who are contractors, who have had various experiences with clients.

As you probably know, before a contractor starts to work, a contract is written up. It says exactly what work will be done, with an estimate of how much it will cost. Just so there will be no surprises.

Yet it happens that every person I know who has worked with a contractor has horror stories to tell. The contractor didn’t actually do the work, or did the wrong work, or ended up needing to fix something that shouldn’t have needed fixing, and everything ended up costing vastly more than the estimate.

It also happens that every person I know who has worked as a contractor has their own horror stories to tell. The customer changed their minds in the middle, or complained about some phantom problem with the work, or refused to pay what they had agreed to pay.

So here is the amazing coincidence: Every person I know who has worked with a contractor turns out to be an honest person who was taken in by an unscrupulous contractor. Yet every person I know who is a contractor turns out to be an honest person who was taken in by an unscrupulous customer.

I ask you, what are the odds?

2x2x2x2, part 7

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.

2x2x2x2, part 6

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

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…

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

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.