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.
In Praise of Shadows
- Things that shine and glitter
- A naked bulb
- The Japanese toilet
- Empty dreams
- Most important of all are the pauses
Things that shine and glitter
We find it hard to be really at home with things that shine and glitter. The Westerner uses silver and steel and nickel tableware, and polishes it to a fine brilliance, but we object to the practice. On the contrary we begin to enjoy it only when the luster has worn off, when it has begun to take on a dark, smoky, patina.
A naked bulb
The sight of a naked bulb beneath an ordinary milk glass shade seems simpler and more natural than any gratuitous attempt to hide it.
The Japanese toilet
The parlor may have its charms, but the Japanese toilet truly is a place of spiritual repose.
Empty dreams
But I know as well as anyone that these are empty dreams, and that having come this far, we cannot turn back.
Most important of all are the pauses
Japanese music is above all a music of reticence, of atmosphere. When recorded, or amplified by a loudspeaker, the greater part of its charm is lost. In conversation, too, we prefer the soft voice, the understatement. Most important of all are the pauses. Yet the phonograph and radio render these moments of silence utterly lifeless. And so we distort the arts themselves to curry favor for them with the machines.
The glow of grime
Of course this 'sheen of antiquity' of which we hear so much is in fact the glow of grime. In both Chinese and Japanese the words denoting this glow describe a polish that comes of being touched over and over again, a sheen produced by the oils that naturally permeate an object over long years of handling—which is to say grime. If indeed 'elegance is frigid', it can as well be described as filthy.
Lacquerware
There are good reasons why lacquer soup bowls are still used, qualities which ceramic bowls simply do not possess. Remove the lid from a ceramic bowl, and there lies the soup, every nuance of its substance and color revealed. With lacquerware there is a beauty in that moment between removing the lid and lifting the bowl to the mouth when one gazes at the still, silent liquid in the dark depths of the bowl, its color hardly different from that of the bowl itself.
To throw a shadow on the earth
In making for ourselves a place to live, we first spread a parasol to throw a shadow on the earth, and in the pale light of the shadow we put together a house.
The world of shadows
The 'mysterious Orient' of which Westerners speak probably refers to the uncanny silence of these dark places. And even we as children would feel an inexpressible chill as we peered into the depth of an alcove to which the sunlight never penetrated.
This was the genius of our ancestors, that by cutting off the light from this empty space they imparted to the world of shadows that formed there a quality of mystery and depth superior to that of any wall painting or ornament.
That one thing against another creates
Such is our way of thinking—we find beauty not in the thing itself but in the patterns of shadows, the light and the darkness, that one thing against another creates.
Wasting light
Yamamoto Sanehiko, president of the Kaizo publishing house, told me of something that happened when he escorted Dr. Einstein on a trip to Kyoto. As the train neared Ishiyama, Einstein looked out the window and remarked, "Now that is terribly wasteful." When asked what he meant, Einstein pointed to an electric lamp burning in broad daylight.
And the truth of the matter is that Japan wastes more electric light than any Western country except America.
- Poured
The eaves deep and the walls dark
I would call back at least for literature this world of shadows we are losing. In the mansion called literature I would have the eaves deep and the walls dark, I would push back into the shadows the things that come forward too clearly, I would strip away the the useless decoration. I do not ask that this be done everywhere, but perhaps we may be allowed at least one mansion where we can turn off the electric lights and see what it is like without them.
Follow the brush
One of the oldest and most deeply ingrained of Japanese attitudes to literary style holds that obvious structure is contrivance, that too orderly an exposition falsifies the ruminations of the heart, that the truest representation of the searching mind is just to 'follow the brush.'
I could never live in a house like that
Mrs. Tanizaki tells a story of when her late husband decided, as he frequently did, to build a new house. The architect arrived and announced with pride, "I've read your In Praise of Shadows, Mr. Tanizaki, and know exactly what you want."
To which Tanizaki replied, "But no, I could never live in a house like that."
There is perhaps as much resignation as humor in his answer.