math
On beauty bare
Wang tiles
Trees and graphs
A tree is a kind of graph, but a graph can be considerably more complex than a tree.
I have reason to believe, which for brevity’s sake I will treat elsewhere, that the most complex class of processes and structures we humans can consciously prescribe, reduces mathematically to a tree. A tree has a top, bottom, left and right. Its branches fan out from the trunk and they don’t intersect with one another. They are discrete, contiguous, identifiable objects which persist across time. Trees are Things.
Software and websites, however, reduce to arbitrarily more complex structures: they are graphs. A graph has no meaningful orientation whatsoever. No sequence, no obvious start or end—at least none that we can intuit. It is better considered not as one Thing, but as a federation of Things, like the brain or a fungus network, or perhaps a composite artifact left behind from an ongoing process, like an ant colony or human city.
Trees and semilattices
The tree of my title is not a green tree with leaves. It is the name of an abstract structure. I shall contrast it with another, more complex abstract structure called a semilattice.
Both the tree and semilattice are ways of thinking about how a large collection of many small systems goes to make up a large and complex system.
A collection of sets forms a semilattice if, and only if, when two overlapping sets belong to the collection, the set of elements common to both also belongs to the collection. That is, if [234] and [345] belong to the collection, then [34] belongs to the collection.
A collection of sets forms a tree if, and only if, for any two sets that belong to the collection either one is wholly contained in the other, or they are wholly disjoint. Every tree is trivially a simple semilattice.
We are concerned with the difference between structures in which no overlap occurs, and those structures in which overlap does occur.
The semilattice is potentially a much more complex and subtle structure than a tree. It is this lack of structural complexity, characteristic of trees, which is crippling our conceptions of the city.
A City Is Not a Tree
An Essay by Christopher Alexander- Strands of life
- Impending destruction
- The right overlap
- The difficulty of designing complexity
- Political chains of influence
Notes on the Synthesis of Form
A Book by Christopher AlexanderVisualizing Data
A Book by William S. ClevelandExploratory Data Analysis
A Book by John TukeyPlus Equals #4
An Article by Rob WeychertOne of the seeds for Plus Equals was planted a few years ago with Incomplete Open Cubes Revisited, my extension of a Sol LeWitt work. I learned a lot about isometric projection from that project, but my affection for the concept didn’t begin there. Whether I’m looking at a Chris Ware illustration or an exploded-view technical drawing of a complex machine, an isometric rendering always stirs something in me.
A brief foray into vectorial semantics
An Article by James SomersOne of the best (and easiest) ways to start making sense of a document is to highlight its “important” words, or the words that appear within that document more often than chance would predict. That’s the idea behind Amazon.com’s “Statistically Improbable Phrases”:
Amazon.com’s Statistically Improbable Phrases, or “SIPs”, are the most distinctive phrases in the text of books in the Search Inside!™ program. To identify SIPs, our computers scan the text of all books in the Search Inside! program. If they find a phrase that occurs a large number of times in a particular book relative to all Search Inside! books, that phrase is a SIP in that book.
tixy.land
A Websitesin(t * x) * cos(t * y)
Creative code golfing.
Rafael Araujo's Golden Ratio
A GalleryBlue Morpho Double Helix & Icosahedron
The Tiling Patterns of Sebastien Truchet and the Topology of Structural Hierarchy
A Research Paper by Cyril Stanley SmithA pattern of tiles illustrated by Douat in 1722.
A translation is given of Truchet's 1704 paper showing that an infinity of patterns can be generated by the assembly of a single half—colored tile in various orientations.
Everything and More
A Book by David Foster WallaceInfoCrystal
A Research PaperThis paper introduces a novel representation, called the InfoCrystal, that can be used as a visualization tool as well as a visual query language to help users search for information. The InfoCrystal visualizes all the possible relationships among N concepts.
You're Probably Using the Wrong Dictionary
As if a word were no more than coordinates
The New Oxford American dictionary, by the way, is not like singularly bad. Google’s dictionary, the modern Merriam-Webster, the dictionary at dictionary.com: they’re all like this. They’re all a chore to read. There’s no play, no delight in the language. The definitions are these desiccated little husks of technocratic meaningese, as if a word were no more than its coordinates in semantic space.
Another mind as alive as yours
In 1807, Webster started writing a dictionary, which he called, boldly, An American Dictionary of the English Language. He wanted it to be comprehensive, authoritative. Think of that: a man sits down, aiming to capture his language whole.
Dictionaries today are not written this way. In fact it’d be strange even to say that they’re written. They are built by a large team, less a work of art than of engineering. When you read an entry you don’t get the sense that a person labored at his desk, alone, trying to put the essence of that word into words. That is, you don’t get a sense, the way you do from a good novel, that there was another mind as alive as yours on the other side of the page.
Webster’s dictionary took him 26 years to finish. It ended up having 70,000 words. He wrote it all himself, including the etymologies, which required that he learn 28 languages, including Old English, Gothic, German, Greek, Latin, Italian, Spanish, Dutch, Welsh, Russian, Aramaic, Persian, Arabic, and Sanskrit. He was plagued by debt to fund the project; he had to mortgage his home.
A soft and fitful luster
Who decided that the American public couldn’t handle “a soft and fitful luster”? I can’t help but think something has been lost. “A soft sparkle from a wet or oily surface” doesn’t just sound worse, it actually describes the phenomenon with less precision. In particular it misses the shimmeriness, the micro movement and action, “the fitful luster,” of, for example, an eye full of tears — which is by the way far more intense and interesting an image than “a wet sidewalk.”
It’s as if someone decided that dictionaries these days had to sound like they were written by a Xerox machine, not a person, certainly not a person with a poet’s ear, a man capable of high and mighty English, who set out to write the secular American equivalent of the King James Bible and pulled it off.
Pathos
With its blunt authority the New Oxford definition of “pathos” — “a quality that evokes pity or sadness” — shuts down the conversation, it shuts down your thinking about the word, while the Webster’s version gets your wheels turning: it seems so much more provisional — “that which awakens tender emotions, such as pity, sorrow, and the like; contagious warmth of feeling, action, or expression; pathetic quality; as, the pathos of a picture, of a poem, or of a cry” — and therefore alive.
Most important, it describes a word worth using: a mere six letters that have come to stand for something huge, for a complex meta-emotion with mythic roots. Such is the power of actual English.
An affection for words
There’s an amazing thing that happens when you start using the right dictionary. Knowing that it’s there for you, you start looking up more words, including words you already know. And you develop an affection for even those, the plainest most everyday words, because you see them treated with the same respect awarded to the rare ones, the high-sounding ones.
Which is to say you get a feeling about English that Calvin once got with his pet tiger on a day of fresh-fallen snow: “It’s a magical world, Hobbes. Let’s go exploring!”