Do you remember free time as a kid that you wasted with connecting dots? If you actually liked it, here’s some good news: There are dot-to-dot books for grown-ups! Most notable, there are the books by Thomas Pavitte with great pictures with 1000 dots.
So, these are some of the books
and here is some video:
Actually, it takes some time to connect 1000 dots; I need ten minutes or so, depending a bit an the picture.
For the fun of it, I coded some lines in MATLAB to connect the dots automatically. And since I am a lazy programmer, I did not bother to connect the dots in the manner that was prescribed by the artist but more efficiently:
1. Greedy paths
For the greedy path, we start at some randomly chosen dot and connect the dot where we are with the closest possible dot where we haven’t been already.
Here’s how this looks for one of Thomas’ pictures:
2. Shortest paths
The greedy path sometimes makes very large jumps (when it runs into some corner, using all the dots in the vicinity). This leads to some spurious large jumps in the picture. In used some simple heuristics to find some “locally shortest paths” through the thousand dots. (And “locally shortest” means that there are no two edges for which swapping improves the total lengths of the paths.) Actually, I started out with the goal to solve the travelling salesman problem over the thousand dots, i.e., to find the shortest path of all. Then it turned out that
- Solving the travelling salesman problem is not that simple to solve – well it’s one of the best investigated NP-hard optimization problems and there is no doubt that it would take my laptop only little time to solve it with 1000 dots if fed with the right code.
- The locally shortest path already looks quite appealing and I am not sure how the shortest path would look any different.
Here is the locally shortest path:
Oh, by the way: the image is a pug dog, the one that you can party see on this cover
Here are some more pictures (not by Thomas Pavitte). Middle is the greedy, right the locally shortest path: