Most tilings we see around us are periodic tesselations. They are tesselations because there are no gaps, and periodic because they repeat like wallpaper. In fact, until the 1960s, people thought that any finite set of tile shapes that could tesselate could be used to tesselate periodically. Then researchers discovered tiles that only tesselate non-periodically. Such tiles are called aperiodic.
From 1998 to 2008, a team of researchers worked to unravel the secrets of a document first written in Byzantine Greek over 1000 years ago (check out this TED talk by the lead researcher, William Noel). Over the centuries, monks periodically cleaned off the markings on the document, and wrote fresh text, thus
This was an early exercise with p5. It stops drawing after 1000 frames. Refresh the browser to see it redraw.
I didn't formally studied computer graphics, so like many other areas of computer science, I get introduced to well-known ideas only when they appear in my line-of-sight. Years ago, when my son mentioned a college CS assignment using Bresenham's line algorithm, I'd never heard of it. I was attracted to the simplicity of the algorithm, and the fact that it addresses one of the earliest ad simplest computer graphics problems.
This isn't about any particular card game, but rather about the process of how I opted to represent cards in p5 and React.
To artists and mathematicians, knots can be beautiful, interesting structures. Until recently, I hadn't explored them much from either perspective. Then my mother-in-law gave me a copy of George Bain's
We've been captivated by the Cipra Loops tiles, so we added another Barry Cipra invention to our lineup. Cipra again found inspiration in the work of Sol Lewitt, who often explored the possible combinations of a graphic idea. For these tiles, that graphic idea is vertical, horizontal, and diagonal lines on a square, and Cipra created a tile for each unique combination. The tiles are oriented. In the image below, the orientation is indicated by the blue arrow. All blue arrows will point in the same direction when composing the 16 tiles. Can you convince yourself that all possible combinations of lines are represented?
clock arithmetic
Most things that we observe vary. We don't pay attention to things that don't change. When we measure those variables, we often use statistics like average or median to summarize. But those statistics don't describe the variability of the distribution. Standard deviation works well if the distribution is a bell curve, aka normally distributed. But many things we observe are not normally distributed. Yet we want a concise summary. One way to summarize is using quantiles, which involves dividing our observations into more-or-less equal size buckets of non-overlapping ranges of values. Common examples of quantiles are quartiles, with four buckets, and deciles, with ten buckets.
This was something we played around with at Cherry Arbor Design, and used it to create wall art, earrings, and when they were still a thing, fidget spinners.
rotating links
it's not Quicksort…
At Cherry Arbor Design, my wife and I created a lot of tilings. Some allow periodic tilings. Others are aperiodic, and can't repeat like wallpaper. Part of the appeal of aperiodic tiles is that on first encounter, it is remarkably difficult to tesselate without hitting a dead end, making them a kind of puzzle. We've made aperiodic Penrose tiles, Fractal Penrose tiles, the Golden B, and Ammann-Beenker tiles.
Many of the things we play around involve geometric symmetry. There are several types. This post is about an unusual symmetry. But first, a quick recap of some more well-known symmetries…