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.
Contrasting Concepts of Harmony in Architecture
The realm of feeling
Alexander: If you say: "Well, look, you're a feeling type, and I'm a thinking type, so let's not discuss that because we are always going to be on different sides", then it removes from this discussion what I feel to be the absolute heart and soul of the matter when it comes to buildings. Now I don't want to deny at all what you are saying about personalities. But I really cannot conceive of a properly formed attitude towards buildings, as an artist or a builder, or in any way, if it doesn't ultimately confront the fact that buildings work in the realm of feeling.
Panic
Alexander: It never occurred to me that someone could so explicitly reject the core experience of something like Chartres. It's very interesting to have this conversation. If this weren't a public situation, I'd be tempted to get into this on a psychiatric level. I'm actually quite serious about this. What I'm saying is that I understand how one could be very panicked by these kinds of feelings. Actually, it's been my impression that a large part of the history of modern architecture has been a kind of panicked withdrawal from these kinds of feelings, which have governed the formation of buildings over the last 2000 years or so.
Why that panicked withdrawal occurred, I'm still trying to find out. It's not clear to me. But I've never heard somebody say, until a few moments ago, someone say explicitly: "Yes, I find that stuff freaky. I don't like to deal with feelings. I like to deal with ideas.”
The pitched roof
Eisenman: I would argue that the pitched roof is – as Gaston Bachelard points out – one of the essential characteristics of "houseness". It was the extension of the vertebrate structure which sheltered and enclosed man. Michel Foucault has said that when man began to study man in the 19th century, there was a displacement of man from the center. The representation of the fact that man was no longer the center of the world, no longer the arbiter, and, therefore, no longer controlling artifacts, was reflected in a change from the vertebrate-center type of structure to the center-as-void.
The trick of little machines
Alexander: Up until about 1600, most of the world views that existed in different cultures did see man and the universe as more or less intertwined and inseparable ... either through the medium of what they called God or in some other way. But all that was understood. The particular intellectual game that led us to discover all the wonders of science forced us to abandon temporarily that idea. In other words, in order to do physics, to do biology, we were actually taught to pretend that things were like little machines because only then could you tinker with them and find out what makes them tick. That's all fine. It was a tremendous endeavor, and it paid off.
But it may have been factually wrong. That is, the constitution of the universe may be such that the human self and the substance that things made out of, the spatial matter or whatever you call it, are much more inextricably related than we realized. Now, I am not talking about some kind of aboriginal primitivism. I am saying that it may actually be a matter of fact that those things are more related than we realize. And that we have been trained to play a trick on ourselves for the last 300 years in order to discover certain things. Now, if that's true – there are plenty of people in the world who are beginning to say it is, by the way, certainly in physics and other related subjects – then my own contribution to that line of thought has to do with these structures of sameness that I have been talking about.
In other words, the order I was sketching out last night is ultimately, fundamentally an order produced by centers or wholes which are reinforcing each other and creating each other. Now, if all of that is so, then the pitched roof would simply come about as a consequence of all that – not as an antecedent. It would turn out that, in circumstances where one is putting a roof on a building, in the absence of other very strong forces that are forcing you to do something different, that is the most natural and simple roof to do. And, therefore, that kind of order would tend to reappear – of course, in a completely different, modern technological style – simply because that is the nature of order, not because of a romantic harkening back to past years.
Merely a building
Eisenman: Le Corbusier once defined architecture as having to do with a window which is either too large or too small, but never the right size. Once it was the right size it was no longer functioning. When it is the right size, that building is merely a building. The only way in the presence of architecture that is that feeling, that need for something other, when the window was either too large or too small.
Fucking up the world
City Hall by Rafael Moneo, Logroño, La Rioja, Spain.
Alexander : At least my experience tells me, that when a group of different people set out to try and find out what is harmonious, what feels most comfortable in such and such a situation, their opinions about it will tend to converge, if they are mocking up full-scale, real stuff. Of course, if they're making sketches or throwing out ideas, they won't agree. But if you start making the real thing, one tends to reach agreement. My only concern is to produce that kind of harmony.
The thing that strikes me about your friend's building – if I understood you correctly – is that somehow in some intentional way it is not harmonious. That is, Moneo intentionally wants to produce an effect of disharmony. Maybe even of incongruity.
Eisenman: That is correct.
Eisenman: I find that incomprehensible. I find it very irresponsible. I find it nutty. I feel sorry for the man. I also feel incredibly angry because he is fucking up the world.
A good table
Eisenman: Tolstoy wrote about the man who had so many modern conveniences in Russia that when he was adjusting the chair and the furniture, etc., that he was so comfortable and so nice and so pleasant that he didn't know – he lost all control of his physical and mental reality. There was nothing. What I'm suggesting is that if we make people so comfortable in these nice little structures of yours, that we might lull them into thinking that everything's all right, Jack, which it isn't. And so the role of art or architecture might be just to remind people that everything wasn't all right.
Alexander: I can't, as a maker of things, I just can't understand it. I do not have a concept of things in which I can even talk about making something in the frame of mind you are describing. I mean, to take a simple example, when I make a table I say to myself: "All right, I'm going to make a table, and I'm going to try to make a good table." And of course, then from there on I go to the ultimate resources I have and what I know, how well I can make it. But for me to then introduce some kind of little edge, which starts trying to be a literary comment, and then somehow the table is supposed to be at the same time a good table, but it also is supposed to be I don't know what; a comment on nuclear warfare, making a little joke, doing various other things... I'm practically naive; it doesn't make sense to me.