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.
Suburban Nation
A system for living
Unlike the traditional neighborhood model, which evolved organically as a response to human needs, suburban sprawl is an idealized artificial system. It is not without a certain beauty: it is rational, consistent, and comprehensive. Its performance is largely predictable. It is an outgrowth of modern problem solving: a system for living. Unfortunately, this system is already showing itself to be unsustainable.
The five components of sprawl
The dominant characteristic of sprawl is that each component is strictly segregated from the others.
- Housing subdivisions, also called clusters and pods
- Shopping centers, also called strip centers, shopping malls, and big-box retail
- Office parks and business parks
- Civic institutions
- Roadways
Subdivisions
Subdivisions can be identified as such by their contrived names, which tend toward the romantic—Pheasant Mill Crossing—and often pay tribute to the natural or historic resource they have displaced.
An unmade omelet
The successes of turn-of-the-century planning, represented in America by the City Beautiful movement, became the foundation of a new profession, and ever since, planners have repeatedly attempted to relive that moment of glory by separating everything from everything else. This segregation, once applied only to incompatible uses, is now applied to every use. A typical contemporary zoning code has several dozen land-use designations; not only is housing separated from industry but low-density housing is separated from medium-density housing, which is separated from high-density housing. Medical offices are separated from general offices, which are in turn separated from restaurants and shopping.
As a result, the new American city has been likened to an unmade omelet: eggs, cheese, vegetables, a pinch of salt, but each consumed in turn, raw.
Beauty and function
In truth, a lot of sprawl—primarily affluent areas—could be considered beautiful. This raises a fundamental point: the problem with suburbia is not that it is ugly. The problem with suburbia is that, in spite of all its regulatory controls, it is not functional: it simply does not efficiently serve society or preserve the environment.
Six qualities of traditional neighborhoods
- The center
- The five-minute walk
- The street network
- Narrow, versatile streets
- Mixed use
- Special sites for special buildings
Market segments
The segregation of housing by “market segment” is a phenomenon that was invented by developers who, lacking a meaningful way to distinguish their mass-produced merchandise, began selling the concept of exclusivity: If you live within these gates, you can consider yourself a success.
Cookie cutter
One term that gets a lot of play these days is “cookie cutter.” Developers are mortified about the way this term is used to describe their subdivisions, and they expend a good deal of energy—and money—avoiding it. As much as 20 percent of their construction budget goes toward the application of superficial variety—different shapes, colors, window types, different styles of tack-on ornament, French Provincial next door to California Contemporary. But these efforts are in vain, because beneath the surface articulation is a relentless repetition of the same building. The best way to create real variety is to vary not the architectural style but the building type. Indeed, in places like Georgetown, styles vary only slightly, but one never hears the term cookie cutter, thanks to the wide range of building types.
Worthwhile destinations
Pedestrian life cannot exist in the absence of worthwhile destinations that are easily accessible on foot. This is a condition that modern suburbia fails to satisfy, since it strives to keep all commercial activity well separated from housing.
The twenty-minute house
Despite the way that it sounds, the “twenty-minute house” is not a derogatory label. Quite the opposite—it refers to the fact that a house has only twenty minutes to win the affection of a potential buyer, since that is the average length of a realtor visit. The building industry has responded to this phenomenon by creating a product that is at its best for the first twenty minutes that one is in it.
Like trying to cure obesity by loosening your belt
“Trying to cure traffic congestion by adding more capacity is like trying to cure obesity by loosening your belt.”
Homebuilders
The term homebuilder describes the house as a product that exists independent of its context. This approach would be appropriate if houses floated freely in space, or in some other environment where actual interaction between neighbors was neither possible nor desired. But houses are not meant to exist in isolation, so to think of the individual house as the ultimate outcome of the builder’s craft robs that craft of its broader significance.
The cul-de-sac kid
In this environment where all activities are segregated and distances are measured on the odometer, a child’s personal mobility extends no farther than the edge of the subdivision. Even the local softball field often exists beyond the child’s independent reach.
The result is a new phenomenon: the “cul-de-sac kid,” the child who lives as a prisoner of a thoroughly safe and unchallenging environment.
This “isolation and boredom” is the outcome of an environment that fails to provide teenagers with the ordinary challenges of maturing, developing useful skills, and gaining a sense of self.
On-site parking
Most cities require new and renovated buildings to provide their own parking on site. This is probably the single greatest killer of urbanism in the United States today. It prevents the renovation of old buildings, since there is inadequate room on their sites for new parking; it encourages the construction of anti-pedestrian building types in which the building sits behind or hovers above a parking lot; it eliminates street life, since everyone parks immediately adjacent to their destination and has no reason to use the sidewalk; finally, it results in a low density of development that can keep a downtown from achieving critical mass.
Architectural mysticism
In response to their growing sense of insignificance, some architects have tried to regain a sense of power through what can best be described as mysticism. By importing arcane ideas from unrelated disciplines—such as contemporary French literary theory (now outdated) —by developing illegible techniques of representation, and by shrouding their work in inscrutable jargon, designers are creating increasingly smaller realms of communication, in order that they might inhabit a domain in which they possess some degree of control. Nowhere is this crisis more evident than in the most prestigious architecture schools.
Globally, locally, regionally
Think globally, act locally, but plan regionally.