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.
A Burglar's Guide to the City
To commune with the space
...having realized long ago that the best way to commune with an architectural space was by breaking into it.
Every building is infinite
For the burglar, every building is infinite.
Putting the streets to use
Tad Friend writes, if you build “nine hundred miles of sinuous highway and twenty-one thousand miles of tangled surface streets” in one city alone, then you’re going to find at least a few people who want to put those streets to use. This suggests that every city blooms with the kinds of crime most appropriate to its form.
Topology by other means
The burglar is a three-dimensional actor amid the two-dimensional surfaces and objects of the city. This means operating with a fundamentally different spatial sense of how architecture should work, and how one room could be connected to another. It means seeing how a building can be stented: engineering short-circuits where mere civilians, altogether less aggressive users of the city, would never expect to find them. Burglary is topology pursued by other means: a new science of the city, proceeding by way of shortcuts, splices, and wormholes.
Burglary's White Whale
If all cities already contain the crimes that will occur there, then, taken to its logical conclusion, this suggests there might be a kind of Moby-Dick of crime, a White Whale of urban burglary: a town or city so badly designed that the entire place can be robbed in one go.
Tarzan of the concrete jungle
Weissmuller was most famous for playing Tarzan, and swinging into his apartment from a ledge outside had a wild irony, like some new Tarzan of the concrete jungle updating the character for an urban age.
The source code for SimCity
Local Code was Sorkin’s attempt to design a whole city from scratch—with one big twist. The whole thing had been written as if it were the byzantine, nearly impossible to follow codes and regulations for an entire, hypothetical metropolis. The effect is like stumbling upon the source code for SimCity. Sorkin’s exhaustively made point was that, if you know everything about a given metropolis, from its plumbing standards to its parking requirements, its sewer capacity to the borders of its school districts, then you could more or less accurately imagine the future form of that city from the ground up.
Architectural dark matter
Every building had its rhythms. These service corridors were the internal hinterlands—the architectural dark matter—so beloved by Bill Mason.
Vamburglars
Burglary was originally only possible in a household or dwelling; the very word contains an etymological variant on the Latin burgus, for “castle” or “fortified home” (from which other words, such as burgher and even borough, also derive). Common law definitions of burglary also originally required the person to break into a house or dwelling at night. Giving historical burglary an oddly vampiric dimension, you could not, legally speaking, be a burglar while the sun was still out.
The close
Think of it as an invisible geometric shape perceptible only to lawyers—a conceptual pane of glass that might not have kept the rain out but could, for legal purposes, be used to define the original limits of the car’s interior. This is the close, and defining it is ultimately just a form of connecting the dots: drawing an imaginary line from the corner of an open window to the edge of a nearby wall to the front gate of a home garden, and so on.
Breaking the close thus constitutes entry into a “protected interior” or “specified enclosure".
To deter crime
“To deter crime,” Cisneros explains, “spaces should convey to would-be intruders a strong sense that if they enter they are very likely to be observed, to be identified as intruders, and to have difficulty escaping.”
Architectural sequences
Noted designer and architectural theorist Bernard Tschumi would call the predictable repetition of events inside an architectural space a sequence: a linear series of actions and behaviors that are at least partially determined by the design of the space itself.
Architectural screenplays
Tschumi began to explore this notion through what he called screenplays: each “screenplay” was a black-and-white diagram breaking down a range of events that might occur inside an architectural space. Tschumi drew them in a way that resembled dance notation or the spatial analysis of a film scene.
The City of Light
Streetlights were one of many new patrol tools implemented by Louis XIV’s lieutenant general of police, Gabriel Nicolas de la Reynie. De la Reynie’s plan ordered that lanterns be hung over the streets every sixty feet—with the unintended side effect that Paris soon gained its popular moniker, the City of Light. The world’s most romantic city takes its nickname from a police operation.
Spatial expectations
Loya, who served seven years in prison for multiple bank heists before becoming a writer, explained to me that it was during the getaway that he often had the best chance of thwarting people’s spatial expectations. In his case, this meant that what he did immediately after leaving the bank was often the most important decision of all.
The getaway to end all getaways
Any attempt to track down the perfect getaway is made all the more complex because almost everything we know about burglary—including how they did (or did not) get away—comes from the burglars we’ve caught. As sociologist R. I. Mawby pithily phrases this dilemma, “Known burglars are unrepresentative of burglars in general.” Great methodological despair is hidden in such a comment. Studying burglary is thus a strangely Heisenbergian undertaking, riddled with uncertainty and distorted by moving data points. The getaway to end all getaways—the one that leaves us all scratching our heads—to no small extent remains impossible to study.
Every heist is a counterdesign
Heists obsess people because of what they reveal about architecture’s peculiar power: the design of new ways of moving through the world. Every heist is thus just a counterdesign—a response to the original architect.
All the things we want to do
This is precisely where “burglary” becomes a myth, a symbol, a metaphor: it stands in for all the things people really want to do with the built environment, what they really want to do to sidestep the obstacles of their lives.