How small can a set of aperiodic tiles be? The first aperiodic set had over 20000 tiles. Subsequent research lowered that number, to sets of size 92, then 6, and then 2 in the form of the famous Penrose tiles.
Penrose’s work dates back to 1974. Since then, others have constructed sets of size 2, but nobody could find an “einstein”: a single shape that tiles the plane aperiodically. Could such a shape even exist?
Taylor and Socolar came close with their hexagonal tile. But that shape requires additional markings or modifications to tile aperiodically, which can’t be encoded purely in its outline.
In a new paper, David Smith, Joseph Myers, Chaim Goodman-Strauss and I prove that a polykite that we call “the hat” is an aperiodic monotile, AKA an einstein. We finally got down to 1!
Some sites, including Facebook, add parameters to the web address for tracking purposes. These parameters have no functionality that is relevant to the user, but sites rely on them to track users across pages and properties.
Mozilla introduced support for URL stripping in Firefox 102, which it launched in June 2022. Firefox removes tracking parameters from web addresses automatically, but only in private browsing mode or when the browser’s Tracking Protection feature is set to strict. Firefox users may enable URL stripping in all Firefox modes, but this requires manual configuration. Brave Browser strips known tracking parameters from web addresses as well.
Facebook has responded by encrypting the entire URL into a single ciphertext blob.
Since it is no longer possible to identify the tracking part of the web address, it is no longer possible to remove it from the address automatically. In other words: Facebook has the upper hand in regards to URL-based tracking at the time, and there is little that can be done about it short of finding a way to decrypt the information.
There’s ahead of its time, then there’s Traité des couleurs servant à la peinture à l’eau — or, in its original Dutch title, Klaer Lightende Spiegel der Verfkonst, a 900-page book of paint colors made before any such things were common tools of the artist’s, scientist’s, and industrial designer’s trade. Author and artist A. Boogert created one, and only one, copy of his extraordinary manual on color mixing in 1692. Appearing on the threshold of modern color theory, and featuring over 700 pages of color swatches, the book draws on Aristotle’s system of color rather than the new understanding of the color spectrum, fully elaborated by Newton in his Opticks over a decade later.
It would be another hundred years before a flood tide of color books began to make the theory more practical: from Goethe’s 1810 Theory of Colors and Werner’s 1814 Nomenclature of Colour to the dream of color standardization realized: the Pantone company, launched in 1963.
But if A. Boogert had much influence on the theory or practical application of color in his day, there doesn’t seem to be much evidence for it. Of course most of the Dutch masters had died when the book was completed, and it seems unlikely that those still working in 1692 would have been familiar with its single copy.
Instead, the book was meant to educate watercolorists, hence its French title, which refers to “water-based paint.” (A literal translation of the Dutch runs something like “clearly lighting mirror of the painting art.”) Medieval historian Erik Kwakkel found the book in a French database, “and it turns out to be quite special,” he writes, “because it provides an unusual peek into the workshop of 17th-century painters and illustrators.
In over 700 pages of handwritten Dutch, the author, who identifies himself as A. Boogert, describes how to make watercolour paints. He explains how to mix the colours and how to change their tone by adding ‘one, two or three portions of water.’… In the 17th Century, an age known as the Golden Age of Dutch Painting, this manual would have hit the right spot.”
The book is currently housed at Bibliothèque Méjanes in Aix-en-Provence, where you’ll find full-page, zoomable, hi-resolution scans. “Beyond being informational, the images from the book are stunning and addictive flip through,” notes Refinery29. “They resemble page after page of Pantone color chips, except without the household name.” One wonders if “A. Boogert” would have become a household name had his book been printed and distributed. But his color system was already passing away in the Newtonian age of color spectrums and wheels, until paint chips finally came back in style. Visit the color manual online here.
This is the story of one strange moment in animation history, two reluctantly warring teams of animators, and three movies, only one of which you can see today. Photo: Buena Vista Pictures/Everett Col
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