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.
Towards a New Architecture
The house is a machine for living in
But men live in old houses
It is not right that we should produce bad things because of a bad tool; nor is it right that we should waste our energy, our health and our courage because of a bad tool; it must be thrown away and replaced.
But men live in old houses and they have not yet thought of building houses adapted to themselves.
Primitive resources
There is no such thing as primitive man; there are primitive resources.
Employs nothing at all
The man of today planes to perfection a board with a planing machine in a few seconds. The man of yesterday planed a board reasonably well with a plane. Very primitive man squared a board very badly with a flint or a knife. Very primitive man employed a unit of measurement and regulating lines in order to make his task easier. The Greek, the Egyptian, Michaelangelo or Blondel employed regulating lines in order to correct their work and for the satisfaction of their artist’s sense and of their mathematical thought. The man of today employs nothing at all and the result is the boulevard Raspail.
All the work of an epoch
Style is a unity of principle animating all the work of an epoch, the result of a state of mind which has its own special character.
Our own epoch is determining, day by day, its own style.
Our eyes, unhappily, are unable yet to discern it.A taste for fresh and clear daylight
Tail pieces and garlands, exquisite ovals where triangular doves preen themselves or one another, boudoirs embellished with “poufs” in gold and black velvet, are now no more than the intolerable witnesses to a dead spirit. These sanctuaries stifling with elegance, or on the other hand with the follies of “Peasant Art,” are an offense.
We have acquired a taste for fresh and and clear daylight.
Eyes which do not see
Our epoch is fixing its own style day by day. It is there under our eyes—Eyes which do not see.
The problem of the house has not yet been stated
The lesson of the airplane is not primarily in the forms it has created, and above all we must learn to see in an airplane not a bird or a dragon-fly, but a machine for flying; the lesson of the airplane lies in the logic which governed the enunciation of the problem and which led to its successful realization. When a problem is properly stated, in our epoch, it inevitably finds its solution.
The problem of the house has not yet been stated.
At the Green Mosque
In Broussa in Asia Minor, at the Green Mosque, you enter by a little doorway of normal human height; a quite small vestibule produces in you the necessary change of scale so that you may appreciate, as against the dimensions of the street and the spot you come from, the dimensions with which is is intended to impress you. Then you can feel the noble size of the mosque and your eyes can take its measure. You are in a great white marble space filled with light. Beyond you can see a second similar space of the same dimensions, but in half-light and raised on several steps (repetition in a minor key); on each side still a smaller space in subdued light; turning round, you have two very small spaces in shade.
From full light to shade, a rhythm. Tiny doors and enormous bays. You are captured, you have lost the sense of the common scale. You are enthralled by a sensorial rhythm (light and volume) and by an able use of scale and measure, into a world of its own which tells you what it set out to tell you.
Poems of an Indian summer
To build one's house is very much like making one’s will. When the time does arrive for building this house, it is not the mason’s nor the craftsman’s moment, but that moment in which every man makes one poem, at any rate, in his life. And so, in our towns and their outskirts, we have had during the last forty years not so much houses as poems, poems of an Indian summer, for a house is the crowning of a career.
A grave and noble beauty
An architecture of our own age is slowly but surely shaping itself; its main lines become more and more evident. The use of steel and reinforced concrete construction; of large areas of plate glass; of standardized units (as, for example, in metal windows); of the flat roof; of new synthetic materials and new surface treatments of metals that machinery made possible; of hints taken from the airplane, the motor-car or the steamship where it was never possible, from the beginning, to attack the problem from an academic standpoint—all these things are helping, at any rate, to produce a twentieth-century architecture whose lineaments are already clearly traceable. A certain squareness of mass and outline, a criss-cross or “grid-iron” treatment with an emphasis on the horizontals, an extreme bareness of wall surface, a pervading austerity and economy and a minimum of ornament; these are among its characteristics. There is evolving, we may begin to suppose, a grave and classical architecture whose fully developed expression should be of a noble beauty.