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.
Back to the Drawing Board
The lost art of drawing for engineers and architects.
- You can almost tell which software they were designed in
- Conversational drawing
- The effort heuristic
- Tablets have caught up
You can almost tell which software they were designed in
Tatiana von Preussen, cofounder of London practice vPPR Architects, says that certain software comes with constraints that encourage a particular style:
“Something I’ve noticed with new buildings is that you can almost tell which software they were designed in. For instance, if you take Revit, it’s very hard to freely create non-orthogonal, non-linear geometries, and it’s very easy to create repetitive elements, so it lends itself to a particular way of building.”
Conversational drawing
A skilled draughtsman guides design conversations by selecting and emphasizing details in a way that computer programmes cannot. Ron Slade, author of Sketching for Engineers and Architects and a structural director at WSP in London, calls it “conversational drawing”. He notes how botany field guides are always based on detailed drawings rather than photographs — as much for what they leave out as for what they show. “Extraneous material that might exist in a photograph is eliminated. It may be important to pick out and illustrate particular areas and leave other parts in sketchy or broad outline.”
The effort heuristic
Psychologists have noted that people tend to place greater artistic value on images when they can see the work that has gone into them — a tendency known as the “effort heuristic”. They are also more likely to connect emotionally with the work if they can detect the human hand, says Goldsmiths’ Chamberlain. “There’s an argument that if we see a brush stroke, we almost recreate it, and that’s part of the connection we feel with the artist — you can feel the intention.”
Perhaps to capitalize on this, some architects now show presentation drawings that look hand-drawn but are actually generated entirely by computer. “It’s totally fake,” says Brillhart. “They just take a computer image into Photoshop and put filters over it to make it look like it’s drawn by hand. It’s kind of amusing — instead of just sitting down and drawing for an hour, they spend eight hours making it look like a hand drawing.”
Tablets have caught up
But in the past couple of years, Brennan believes that tablets have caught up, with apps such as Apple’s ProCreate and Morpholio’s Trace becoming far more responsive to the user’s marks. “Tablets didn’t used to have that immediate response, from brain to eye to hand to pen to paper. A half-second delay has a huge impact on how you think — it causes you to stumble. But now that lag’s gone, it’s almost the same as drawing with pen on paper. You don’t need to engage with the airbrushes or other features — just use it in its purest form.” The stylus, too, far more convincingly apes a manual pen: “You’re able to tune it to almost replicate your favourite pen — and it doesn’t run out of ink.
Chetwood is an iPad devotee, using it to produce fantastical urban artworks as well as architecture. Far from hindering the drawing process, he believes tablets will give rise to a new era of creative drawing. “A lot of people say technology and computers are taking away the true art of drawing. That’s rubbish, it releases sketching. You can move so much quicker and change things much more quickly, and it keeps a record of what you’re doing. The control is just brilliant.” The polished glass surface is the only flaw, but textured acetates applied to the screen can make it feel more like paper.