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.
Wittgenstein's Mistress
- I think very well of him indeed
- A perfect circle
- The Eiffel Tower
- Ceci n'est pas une pipe
- Erased de Kooning Drawing
I think very well of him indeed
When I was still doubtful as to his ability, I asked G.E. Moore for his opinion. Moore replied, ‘I think very well of him indeed.’ When I enquired the reason for his opinion, he said that it was because Wittgenstein was the only man who looked puzzled at his lectures. — Bertrand Russell
A perfect circle
Once, being asked to submit a sample of his work, what Giotto submitted was a circle.
Well, the point being that it was a perfect circle.
And that Giotto had painted it freehand.The Eiffel Tower
When the sun had gotten to the angle from which Phidias had taken his perspective, the Parthenon almost seemed to glow.
Actually, the best time to see that is generally also at four o’clock.
Doubtless the taverns from which one could see that did better business than the taverns from which one could not, in fact, even though they were all in the same street.
Unless of course the latter were patronized by people who had lived in Athens long enough to have gotten tired of seeing it.
Such things can happen. As in the case of Guy de Maupassant, who ate his lunch every day at the Eiffel Tower.
Well, the point being that this was the only place in Paris from which he did not have to look at it.Ceci n'est pas une pipe
There is nobody at the window in the painting of the house, by the way.
I have now concluded that what I believed to be a person is a shadow.
If it is not a shadow, it is perhaps a curtain.
As a matter of fact it could actually be nothing more than an attempt to imply depths, within the room.
Although in a manner of speaking all that is really in the window is burnt sienna pigment. And some yellow ochre.
In fact there is no window either, in that same manner of speaking, but only shape.
So that any few speculations I have made about the person at the window would therefore now appear to be rendered meaningless, obviously.
Unless of course I subsequently become convinced that there is somebody at the window all over again.
I have put that badly.Erased de Kooning Drawing
Once, Robert Rauschenberg erased most of a drawing by Willem de Kooning, and then named it Erased de Kooning Drawing.
I am in no way certain what this is connected to either, but I suspect it is connected to more than I once believed it to be connected to.
The cat
The cat I began to think about instead was the cat outside of the broken window in the room next to this one, at which the tape frequently scratches when there is a breeze.
Which is to say that I was not actually thinking about a cat either, there being no cat except insofar as the sound of the scratching reminds me of one.As there was, or is, no person at the window in the painting of this house.
It is still a house
Perhaps even the very house which I burned to the ground contained such examples, even though it would obviously not contain them any longer, no longer being a house.
Well, it is still a house.
Even if there is not remarkably much left of it, I am still prone to think of it as a house when I pass it in taking my walks.
There is the house I burned to the ground, I might think. Or, soon I will be coming to the house that I burned to the ground.The image of reality
Leonardo wrote in his notebooks backwards, from right to left, so that they had to be held up to a mirror to be read.
In a manner of speaking, the image of Leonardo’s notebooks would be more real than the notebooks themselves.Equidistance
Once, in the Rijksmuseum, I brought in new speakers for my phonograph. What the directions told me to do was make certain that the two speakers were equidistant from each other.
One certainly had to wonder what the person who wrote the instructions could have believed he meant by that.A strange calligraphy
Along the sand there will be frisky shadows, that will dance and fall away.
Or, if there is snow, the flames will write a strange calligraphy against the whiteness.Van Gogh
One of the things people generally admired about Van Gogh, even though they were not always aware of it, was the way he could make even a chair seem to have anxiety in it. Or a pair of boots.
The meaning of music
Once, somebody asked Robert Schumann to explain the meaning of a certain piece of music he had just played on the piano.
What Robert Schumann did was sit back down at the piano and play the piece of music again.
Good morning, Vincent
Perhaps I shall name the cat that scratches at my broken window Van Gogh.
Or Vincent.
One does not name a piece of tape, however.
There is the piece of tape, scratching at my window. There is Vincent, scratching at my window.Good morning, Vincent.
The illusion of anxiety
Or because of hormones.
And so which would not really have been anxiety at all, but only an illusion.
Even if one would certainly be hard put to explain the difference between an illusion of anxiety and anxiety itself.Somebody is living on this beach
Once, I had a dream of fame.
Generally, even then I was lonely.
To the castle, a sign must have said.
Somebody is living on this beach.