Donald Knuth: this is a turn

When you enter my house ... um ... My wife and I wanted to have some works of art in the house that were personal and also related to what we do. And so you can see two things: one of them is the dragon curve, I will tell you more about it in a minute. And the other is this work. This is a line from “The Road to Wisdom” by Pete Hain:

Should We,
To
Get to Eternal Wisdom ,
That
It
Tempts Us So Temptingly,
Again
And Again,
And Again
Mistaken,
But Less
And Less,
And Less.

This is one of my favorite phrases. A kind of story of my whole life is making mistakes and trying to learn something from them. This phrase was designed in the form of a superellipse, which is one of the great discoveries of Pete Hine. This ellipse is more complete and it has been used in many architectural products.




So what is the idea of ​​a dragon curve? This is what was born thanks to a couple of physicists in the late sixties.



Take a piece of paper and bend it, bend it again, and again and you do it n times. N is nine in the case of a dragon curve on the wall to the left.



When you look at this, you see that some folds are bent up and some inwards. If you imagine that you are moving along this curve, then you turn left here, then turn left here, turn left here again, but then you turn right and then another turn right, and again turn left and so on.



This is a pattern consisting of left and right. The most amazing thing is that if you continue this curve, do it 9 times, and then open all these folds at an angle of ninety degrees, you will get a track that begins to fill the space and creates a beautiful practical pattern. Let's start here, from this tile, for example.



If you bring the camera a little closer, you will see that my wife and I wrote our names here, but we can talk about this story later. This curve begins, turns left, left again, then right, then left, then left again, and so on. This is exactly the pattern you get when folding paper. Since I folded the sheet 9 times, 512 steps are shown on the wall. Well, probably 511. The curve ends here and it continues, but in the middle it becomes very confused.

The wonderful thing, from the point of view of geek art, is that you only need three types of tiles.



You need a completely empty tile or tile on which there is only one loop, as here, or a tile on which two loops, like this one. And, of course, I can take a tile with two loops and turn it, say, in this direction. Then it will not connect here, but it forms an isolated circle here.



I had to calculate how many empty tiles I would need, how many tiles with one loop, and so on. We made a breadboard model of paper and cardboard and stored it in the apartment in which we lived, while our house was being designed. And we decided that it was a cool design and it would be cool to have something like that in our house when it was finally built.



We both liked the tiles from HIT Ceramics in Sausalito, California, and we had a mock up of the design, so we went to look at the ceramics and talk to the people working there. And we were wondering if they would be able to make the order for us a certain number of empty tiles, tiles with one loop, etc. They replied that yes, they can do it, but they make tiles with a 3.5 inch square.

And for my design, 2 inch squares were required. The calculations could not be changed in any way, even if I changed 9 to 8. It would be the same interesting pattern, but at the same time it would be huge and would not look normal on such a scale.

This kind of pattern is very attractive in that size, and not in a larger size. We apparently looked very upset and disappointed because the person we were talking to saw that we were upset, since we traveled all this way with this great idea. And he said: “Wait a minute, in June we are going on vacation, we are closing our factory, but we have a guard here.

You can come and use our equipment, we will let you use our clay and make tiles. If you are going to cut them to a size of 2x2 inches, then we will cover the tiles with our icing and burn the tile for you. So we did. We went there and spent the whole day there, making all this tile, including several spare tiles. And we placed our initials on this tile, at the starting point. Then this tile was fired. Some marks from the fire can still be seen sometimes, the tile in some places cracked a little during firing.



We still had 9. After eight steps, the line is here, after 16 here, at step 32 I am here.

Years passed, I finally posted a chapter in my book, Fun and Games, which talks about it. And I read all the articles on this topic, I wrote about the dragon curve and came to the shocking discovery that this * points to the wall * is wrong.



You get this curve with 9 folds, but I made the 9 fold in the wrong direction. Instead of going this way, I went the other way.



The last half of this curve actually has a different fold. From the point of view of mathematics, I lived with this curve for 30 years and did not realize that I had a mathematical error in design.

- What do you feel about this?
- I felt the meaning of the phrase:

To
get to the wisdom of the Eternal,
That it is
so tempting for us It
squeals,
Again
And again,
And again To make a
mistake,
But less
And less,
And less.

---

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