Author: admin

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.

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

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.

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.

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.

Today I started watching the second season of Grace and Frankie. I didn’t get very far before the first shot of Sam Waterston got me thinking about his earlier work, and that’s when I started going down the rabbit hole.

First I looked on Youtube to see if I could find the video recording of the 1973 Joseph Papp production of Much Ado about Nothing, starring Waterston as Benedick and Kathleen Widdoes as Beatrice. Turns out it’s not on Youtube. Fortunately, used copies of the videotape are available at Amazon for purchase.

Next I set my sights on Kathleen Widdoes. I wasn’t interested to learn more about what she’s been up to in recent years (she has mainly been a mainstay of the Soaps), but rather how she got her start.

I searched on Youtube and quickly got a hit on a 1961 TV show called Way Out, which I had never heard of. Widdoes, who at the time was only 22 years old, guest stars in one episode called “Dissolve to Black”. I watched it.

From that one episode, I think I got a good sense of the show. Like similar TV offerings of the era, such as One Step Beyond, and in the spirit of the much missed EC Comics, it mixed several surefire ingredients, including a love of the macabre, familiar from Alfred Hitchcock Presents, and a disturbing supernatural twist, in the style of The Twilight Zone.

It was all in shadowy black and white, with a very low production budget, like F. Murnau trying to shoot a film in his garage. The show is clearly aiming to scare you, but this episode at least did not succeed.

By the way, I’ve read on-line from still-traumatized baby boomers that episode 12, “Side Show”, is truly terrifying. I look forward to seeing it.

But the most surprising thing about the show is that its narrator each week is Roald Dahl! I had never actually seen the man speak before, and it was quite a revelation.

He is clearly aiming to imitate the style and diction of Alfred Hitchcock in Alfred Hitchcock Presents, a show that had started six years earlier, and in 1961 was still going strong. But far more disturbing than that, perhaps scariest thing of all, far more disturbing than anything about the show itself, is Roald Dahl himself.

For in a twist right out of a Freddy Krueger nightmare, the man looks and sounds exactly like William F. Buckley Jr. I may not be able to sleep for weeks.

I have never met Sam Biederman of the New York City Parks Department. But today he is my hero.

You could look it up. 🙂

I dreamed that I wrote
This as a haiku, and so
I woke up, and did

Two times in the last few days people close to me have done things that seemed to me, in the moment, really destructive, or aggressive, or self-destructive, or potentially self-sabotaging, or some combination of all of the above. In both cases I responded by feeling anger, but pretty much backing away from the situation in the moment, until I could find time to process my feelings.

It’s not so much that I am afraid of expressing anger. It’s more that I don’t want to say things that I don’t really mean, based on some unformed mental model of what’s going on. After all, it’s a lot easier to make situations worse than it is to make them better.

In both cases I eventually got to a place in my own head where I could separate myself from what had happened. One conclusion I’ve reached from this is that I often become angry at people when they act out of some subconscious directive that seems destructive or self-destructive. And that’s a mistake on my part.

It’s a mistake because I am generally assuming, in that moment, that they know what’s going on. “If it’s obvious even to me what game they are playing here,” I tell myself, “then surely it is obvious to them!”

But no, it is not obvious to them. They don’t know what game they are playing. They don’t even know they are playing a game. That’s the whole point with this subconscious thing.

We all sometimes do or say the most absurd things, and the people around us may be mortified, yet we yourselves may have no idea, in that moment, that we have crossed any boundaries. Because it’s our subconscious doing it, acting out of some important emotional need that our conscious mind cannot access.

Our subconscious, of course, really sucks as a long term planner. So when it decides to take action, it will often do so in ways that will come across to other people as very agressive or self-destructive, or at the very least unsettling.

On the other hand, these subconscious minds of ours are pretty darned awesome. For one thing, without them we probably wouldn’t fall in love all that much, and the human race would have died out long ago. So when they end up popping up in the darnedest places, I guess we should cut them some slack.

have you ever tried
to catch that exact moment
when you fall asleep?