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.
The Mezzanine
White cloth
I used to be very interested in the fact that anything, no matter how rough, rusted, diffy, or otherwise discredited it was, looked good if you set it down on a stretch of white cloth. Because anytime you set some detail of the world off that way, it was able to take on its true stature as an object of attention.
Colors in nature
Twice every summer we discussed whether colors in nature could clash.
A blue glow
The neurons that do expire are the ones that made imitation possible. When you are capable of skillful imitation, the sweep of choices before you is too large; but when your brain loses its spare capacity, and along with it some agility, some joy in winging it, and the ambition to do things that don't suit it, then you finally have to settle down to do well the few things that your brain really can do well - the rest no longer seems pressing and distracting, because it is now permanently out of reach. The feeling that you are stupider than you were is what finally interests you in the really complex subjects of life: in change, in experience, in the ways other people have adjusted to disappointment and narrowed ability. You realize that you are no prodigy, your shoulders relax, and you begin to look around you, seeing local color unrivaled by blue glows of algebra and abstraction.
Each fascinating crisis
The problems themselves, though they once obsessed you, and kept you working late night after night, and made you talk in your sleep, turn out to have been hollow: two weeks after your last day they already have contracted into inert pellets one-fiftieth of their former size; you find yourself unable to recreate the sense of what was really at stake, for it seems to have been the Hungarian 5/2 rhythm of the lived workweek alone that kept each fascinating crisis inflated to its full interdepartmental complexity.
My skate blade's gorge
If you made a negative of that image of my skate blade’s gorge, you would arrive at the magnified record groove.
What is this static modernism?
Why can't office buildings use doorknobs that are truly knob-like in shape? What is this static modernism that architects of the second tier have imposed on us: steel half-U handles or lathed objects shaped like superdomes, instead of brass, porcelain, or glass knobs? The upstairs doorknobs in the house I grew up in were made of faceted glass. As you extended your fingers to open a door, a cloud of flesh-color would diffuse into the glass from the opposite direction. The knobs were loosely seated in their latch mechanism, and heavy, and the combination of solidity and laxness made for a multiply staged experience as you turned the knob: a smoothness that held intermediary tumbleral fallings-into-position. Few American products recently have been able to capture that same knuckly, orthopedic quality.
You can taste it with your eyes
It was one of those good rides, where the motion of the train is soothing, and the interior temperature pleasantly warm but not hot. I imagined the subway car as a rapidly moving load of bread. The motto "You can taste it with your eyes" occurred to me.