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.
A Visual Inventory
Amassing the archive
I once sent a camera to a client, with a request that she keep a visual diary of her newly completed house. For a number of months she duly sent me one photograph a day, of whatever caught her attention, and it was fascinating seeing the spaces from her point of view.
In part it's simply about amassing the archive, but it's also about understanding the implications of every design decision and bringing this knowledge to bear on new projects. You have to keep pushing the learning process.
The spaces between things
It's easy to underestimate the significance of the spaces between things...as soon as you frame a section of the view with architecture, the eye has a place to rest and previously invisible details come into focus.
An absence at its centre
At first glance, the rocky outcrop reflected so sharply in the still surface of the water looks like the ghostly image of a house. Interestingly, once read this way, the image always seems to have an absence at its centre.
Drawing a frame
The panels of tessellating hexagons have been laid to stabilize a path running through what remains of the nave of Rievaulx Abbey. They demonstrate the impact of drawing a frame around anything, even if that frame is nothing more than a plastic cell and the subject an area of grass. I like the way the path simply peters out to either side, with no sharply defined boundaries.
Economy of line
With a composition as disciplined as this, everywhere you point the lens feels like a natural frame. The visible architecture comprises simply three walls, two benches, and the top of a flight of steps. It is a perfect expression of economy of line, with the dark green backdrop of the trees acting as a foil for the light grey concrete and granite.
The smallness of human life
The smallness of human life is graphically expressed in this graveyard, in the low stubs of the headstones dwarfed by the towering tree trunks. Perhaps unexpectedly, the effect of this monumental contrast of scales is a feeling of comfort — the secure tranquility of a final resting place overseen by these massive forms, whose benign nature seems to be underlined by the little wooden nesting box on the central tree.
Attenuation and repetition
The distortion here is produced by the movement of a car, on a road near La Ina in Andalusia. My eye is always drawn to attenuation and repetition and the stratified view here exhibits both characteristics to such a degree that the image appears stretched. The extended parallels of the power lines are layered above the repeating arches of the viaduct and the low mass of the roadside barrier.
The aesthetic potential of flaws
The archaeological quality of this section of exposed wall provides an example of the aesthetic potential of that which is flawed or broken.
Cantilevers of bronze
Set close to the surface of the water, the visible structure is made of only two materials — vertical cantilevers of bronze set between horizontal treads of dark grey granite.
A single material
There is something very appealing about a form constructed in a single material.
Of the plainest variety
It is unusual to find such mismatched elements on a single facade as this fine stonework coexisting with these stained and rotting shutters, on a house in the fortified town of Feltre, in the northern Italian province of Belluno.
Where considerable labour lies behind the cutting and fitting of the stone, the timber planks have been left in their raw state, with no paint or carved decoration. Even the iron hinges are of the plainest variety.
Stacking the rails
Stacking the rails in an interlocking zigzag configuration creates a self-supporting structure that is easy to repair and to take apart. This traditional construction method also has the advantage of requiring few tools, since no holes have to be dug for posts and there is no requirement for nails.
White walls
When people say that white walls are cold and characterless, I wonder whether they have ever stopped to look at one. It's not just about the drama of light and shadow, although I love the fragment of the ghost chair in this picture, but the way the smallest nuances of texture and tone come alive in certain conditions.
The precise construction of relationships
This flight of steps runs up the outside of a Modernist house in Switzerland. What is striking here is the precise construction of relationships. The gaps between steps allow crisp lines of light to fall on the darkly shadowed wall, reinforcing the subtlety of the dialogue between granite and the concrete, which has been bush-hammers to expose the stone aggregates.
In the lee of the sills
The first thing you register here is the dramatic inconsistency in the coloration of the timber cladding the house in Haldenstein: the natural hues of the wood survive only in the lee of the sills, like re-growth along the parties of a head of dyed hair. But a second glance takes in the precision of the cuts made to accommodate the window and the fact that the pine is used in seamless lengths.
An Escher-like quality
There is an Escher-like quality to these flights of steps, but it is the intricate net of shadows created by the roof structure of this sky-lit sculpture gallery, falling across a succession of vertical planes and reflecting back on the surface of the glass, which commands attention. Slender metal bars set crosswise between the rafters add their own animating rhythm. It all makes for a very complex visual arena in which to view art.
Monumental structures
These disused gas cylinders occupy a site on the outskirts of Stockholm. For the first ten years after moving to London, the view west across the train tracks was of a similar pair of monumental structures, transfigured by every sunset. One has since been dismantled to make way for the expanding national and international railway stations.
The character of a light box
In certain conditions, the white walls at home take on the character of a light box. In traditional Japanese architecture the intensity of atmosphere has a lot to do with the way natural light is filtered through the shoji paper panels, suffusing the interior spaces with subdued light, calming the spirit and sharpening the senses.