The Harriss spiral is constructed from rectangles in the ratio of the plastic number (1.3247…), in a similar way to how a Fibonacci spiral is created from rectangles in the related golden ratio (1.6180…). These plastic rectangles can be split into two smaller plastic rectangles, leaving a square. Recursively splitting the rectangles, and drawing curves in the squares gives this fractal spiral.
Archive for the 'Math' Category
A teaser. For the setup, go to the link.
In each case the rolling coin has made one complete rotation. But the red arc at the top is half the length of the red line at the bottom. Why?
I have a more physics-y than a formal math-y explanation of why, which I will post soon.
OK, here’s my answer.
In the rolling case, all you have is rotation. On rotation gives you 2*pi, so it rolls one circumference.
But in the other case you have rotation and revolution (spin and also orbital motion). Going halfway around the coin gives you an equal contribution of each, so the amount of spin only requires pi rotation, and it rolls half of the circumference. If the coin’s point of contact never changed, it would still do a rotation over the course of its revolution. If the orientation stayed fixed, the point of contact would make a complete trip around the coin.
A related example of this is the moon. If viewed from an external inertial frame (where the distant stars appear to be fixed), the moon rotates around the earth every ~4 weeks. But since it’s tidally locked and always has the same part facing the earth, it also rotates once about its axis.
I was a tad confused at first, because to me “kinetic sculpture” refers to those systems where balls are continually moving about on various tracks in a closed system. But these are rather nice, too.
I haven’t done much fist-shaking from the porch recently, but here’s to changing that. Today’s curmudgeonly two-fer have one thing in common: overselling the product, in a way. The thing is, I don’t think they need the false advertising, whether it’s on purpose or owing to some comprehension gap. Ignore the rant if you wish, and just enjoy the technology/math-based artistry of these pieces.
These 3d-printed zoetrope sculptures were designed by John Edmark, and they only animate when filmed under a strobe light or with the help of a camera with an extremely short shutter speed.
… just like any other object would. Maybe it’s just me, but this sounds like the author is implying this is special to this particular class of structures — it’s not. That’s just how the strobe effect works.
These are wonderful. But having a few gears doesn’t turn it in to a Rube Goldberg device; it’s not just a matter of being slightly more complex than it needs to be — in this case, mostly by adding one layer of complexity. There are no chain reactions and no diversity of mechanism, two hallmarks of such devices.
The [2D] swings combine with each other to create swirling designs called Lissajous figures.
The patterns are so stunning that machines like Blackburn’s Y-shaped pendulum were made commercially in the Victorian era. They became known as “harmonographs”, since the variation in images results from the variation in harmonies between the different swings.
I arrived at a conclusion that I wasn’t really expecting or prepared for: Lionel Messi is impossible.
It’s not possible to shoot more efficiently from outside the penalty area than many players shoot inside it. It’s not possible to lead the world in weak-kick goals and long-range goals. It’s not possible to score on unassisted plays as well as the best players in the world score on assisted ones. It’s not possible to lead the world’s forwards both in taking on defenders and in dishing the ball to others. And it’s certainly not possible to do most of these things by insanely wide margins.
But Messi does all of this and more.
Algorithms are a fascinating use case for visualization. To visualize an algorithm, we don’t merely fit data to a chart; there is no primary dataset. Instead there are logical rules that describe behavior. This may be why algorithm visualizations are so unusual, as designers experiment with novel forms to better communicate. This is reason enough to study them.
But algorithms are also a reminder that visualization is more than a tool for finding patterns in data. Visualization leverages the human visual system to augment human intellect: we can use it to better understand these important abstract processes, and perhaps other things, too.
The transformation of a maze into a decision tree near the end is awesome.
It can be pretty awesome when these two get together.
No, not fracking. Space-filling curves, which are fractal in nature.
[S]uch a curve is quite unusual, and won’t quite look like anything encountered before. In fact, the complete curve is impossible to visualize, since it literally fills the square and, in the process, takes an infinite number of twists and turns along the way. However, we can get a feel for its behavior through an iterative process that generates curves of increasing complexity that approach the true space-filling curve in the limit of infinite iterations.
I had one of those awesome experiences this week where a student thinks of a better question.