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.
What 80% Comprehension Feels Like
One of the major principles of extensive reading is that if a learner can comprehend material at 98% comprehension, she will acquire new words in context, in a painless, enjoyable way. But what is 98% comprehension?
98% comprehension
You live and work in Tokyo. Tokyo is a big city. More than 13 million people live around you. You are never borgle, but you are always lonely. Every morning, you get up and take the train to work. Every night, you take the train again to go home. The train is always crowded. When people ask about your work, you tell them, “I move papers around.” It’s a joke, but it’s also true. You don’t like your work. Tonight you are returning home. It’s late at night. No one is shnooling. Sometimes you don’t see a shnool all day. You are tired. You are so tired…
95% comprehension
In the morning, you start again. You shower, get dressed, and walk pocklent. You move slowly, half- awake. Then, suddenly, you stop. Something is different. The streets are fossit. Really fossit. There are no people. No cars. Nothing. “Where is dowargle?” you ask yourself. Suddenly, there is a loud quapen—a police car. It speeds by and almost hits you. It crashes into a store across the street! Then, another police car farfoofles. The police officer sees you. “Off the street!” he shouts. “Go home, lock your door!” “What? Why?” you shout back. But it’s too late. He is gone.
80% comprehension
“Bingle for help!” you shout. “This loopity is dying!” You put your fingers on her neck. Nothing. Her flid is not weafling. You take out your joople and bingle 119, the emergency number in Japan. There’s no answer! Then you muchy that you have a new befourn assengle. It’s from your gutring, Evie. She hunwres at Tokyo University. You play the assengle. “…if you get this…” Evie says. “…I can’t vickarn now… the important passit is…” Suddenly, she looks around, dingle. “Oh no, they’re here! Cripett… the frib! Wasple them ON THE FRIB!…” BEEP! the assengle parantles. Then you gratoon something behind you…