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 Poetics of Space
Poetic drugs
In the final chapters Bachelard lets slip (a confession really) how if he "were a psychiatrist," he would recommend a poem by Baudelaire to treat "anguish." His squabble then is not with the purpose but rather the approach of a still-young profession. And of course, why not treat the power of great poems as something akin to "virtual 'drugs'"?
The world itself dreams
For Plato and many medieval philosophers, imagination was construed primarily as a mimetic act of mirroring, representing, copying. This approach was often associated with deceit and illusion, with confounding original realities with secondary substitutes. By contrast, for Kant and the romantics—including German idealists and existentialists like Sartre—imagination was hailed as a productive force in its own right, the source of all true meaning and value.
Bachelard resisted both extremes. For him, imagination was at once receptive and creative—an acoustic of listening and an art of participation. The two functions, passive and active, were inseparable. The world itself dreams, he said, and we help give it voice.
The past of his image upon me
The poet does not confer the past of his image upon me, and yet his image immediately takes root in me. The communicability of an unusual image is a fact of great ontological significance.
In the world of sunlight
And here we come back to that forgotten, outcast word, the soul.
Indeed, the soul possesses an inner light, the light that an inner vision knows and expresses in the world of brilliant colors, in the world of sunlight.
Refuges
Of course, thanks to the house, a great many of our memories are housed, and if the house is a bit elaborate, if it has a cellar and a garret, nooks and corridors, our memories have refuges that are all the more clearly delineated. All our lives we come back to them in our daydreams.
Deprived of all thickness
Here space is everything, for time ceases to quicken memory. Memory—what a strange thing it is!—does not record concrete duration, in the Bergsonian sense of the word. We are unable to relive duration that has been destroyed. We can only think of it, in the line of an abstract time that is deprived of all thickness. The finest specimens of fossilized duration concretized as a result of long sojourn, are to be found in and through space.
Memories are motionless, and the more securely they are fixed in space, the sounder they are.
The odor of raisins
What would be the use, for instance, in giving the plan of the room that was really my room, in describing the little room at the end of the garret, in saying that from the window, across the indentations of the roofs, one could see the hill. I alone, in my memories of another century, can open the deep cupboard that still retains for me alone that unique odor, the odor of raisins drying on a wicker tray. The odor of raisins! It is an odor that is beyond description, one that it takes a lot of imagination to smell. But I've already said too much. If I said more, the reader, back in his own room, would not open that unique wardrobe, with its unique smell, which is the signature of intimacy.
Oneiric topography
If I were the architect of an oneiric house, I should hesitate between a three-story house and one with four. A three-story house, which is the simplest as regards essential height, has a cellar, a ground floor, and an attic; while a four-story house puts a floor between the ground floor and the attic. One floor more, and our dreams become blurred. In the oneiric house, topoanalysis only knows how to count to three or four.
Winter is by far the oldest of the seasons
...and we feel warm because it is cold out-of-doors.
I am the space where I am
Je suis l'espace où je suis.
This is a great line. But nowhere can it be better appreciated than in a corner.
My house is diaphanous
My house is diaphanous, but it is not made of glass. It is more of the nature of vapor. Its walls contract and expand as I desire.