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.
Two Cycles
Gorgeous artwork by Minori Asada.
Among the trees
To accommodate the spaces between the trees, I built three walls in a radial pattern. Filling out the spaces on both sides of these three spline-like walls, I came up with a structure that appears to be slipped in among the trees. This design allowed us to proceed without cutting down any of the woods.
Small economies
I refer to small money-earning business that consist of the work of a visible individual, or have evolved from a personal hobby or skill, as "small economies". We can include in this category newer forms of at-home work—side businesses, telecommuting and the like. The amount of income is unimportant; meager profits are compensated for by the motivation of the owner. A small economy may or may not be someone's main form of livelihood, but it is always a spontaneously conceived and continuing activity.
An extremely closed structure
Nearly all housing in Japan today consists of exclusively residential units for salaried workers and their nuclear families. Such residences have, by definition, no reason to interface with their surroundings.
Salaried workers commute to workplaces outside, and often a considerable distance from, their homes. Residences built for these workers do not contain a place of livelihood—in the broader sense, a place for exchange. This "residence-only housing" is only a place for the nuclear family to eat and sleep, with no occasions for interaction with the outside world, and no need to foster a sense of belonging to the community at large. Thus the only organizational principle is the maintenance of privacy. Both in external appearance and in lifestyle, it is an extremely closed structure.
Ecological cycles
This house exists in the midst of a year-long cycle of natural phenomena. One might say that this cycle entails the periodic "rise and fall" of the ground surface. In winter it sinks below a snow cover that grows head-high or more; as spring approaches, this height gradually decreases until we can see the actual ground surface, not yet covered with undergrowth. With summer the vegetation grows higher and higher until the plaza seems once again to be lower than its surroundings. With the falling of the leaves, autumn restores our ability to penetrate these surroundings at eye level, at least until the snow begins to fall again... Through the four seasons, we experience the sensation of the ground rising and falling, like the ebb and flow of the tide.
I call this cycle of natural phenomena an ecological cycle.
Doing community
There is a Japanese catchphrase, community suru, literally "making" or "doing" community. I will never forget the queasy feeling that came over me when I first heard that term, phrased as if community were a kind of event.
Hold an event, bring people together, get people who might otherwise never meet to interact. It's a wonderful thought. I have nothing against events per se. However, if they are not spontaneous and voluntary, they will not last. That is my objection to the keep-it-lively concept of community. The perception of community as event stems, I think, from a yearning for the festivals and rituals that once flourished in rural communities in Japan. But those events occurred precisely because a community existed, not the other way around.
What are those borders made of?
Functionalist modern architecture has prioritized the functionality of interiors and treated surfaces and external appearances as an outcome of that priority. Diagrams illustrating functional layouts generally frame them with thick borders. Updating conventional program theory entails questioning what those thick borders are actually made of, and how they should be designed. A dynamic program theory should be one that turns these thick borders into more organic interfaces that will foster exchanges and interactions.
An ecological cycle
In the design of his own residence / workplace, Toshiharu Naka created a small ecological cycle. Rows of green planters in front of the wall protect the house from the sun and help cool it in summer. Rainwater is collected via catch-basins from the roof, and used to water the planters.
In the water buckets is a micro-cycle — fish live in the buckets, eating mosquitos from the planters, eliminating the need for pesticides.