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.
Why Sketch?
What you have observed closely
Drawing requires that you pay attention to every detail—even the seemingly unimportant ones. In creating an image (no matter how skillfully), the lines and tones on the paper provide ongoing feedback as to what you have observed closely and what you have not.
A single image
Scientific illustrations can achieve certain things that photographs cannot. A good illustration can portray difficult-to-photograph or rarely witnessed events. It can incorporate everything that’s important into one single image or show a special view of a subject.
It would be next to impossible to observe, in nature, a dozen different aquatic species in their natural habitat, posing perfectly together and all in focus at one time—but such a scene can easily come to life in an illustration.
Parallel refinement
My studies of live animals usually begin with a series of quick drawings, all on a single page. When the subject changes position—which it pretty much always does—I abandon the first sketch and start a new one. Continuing on in this way, the page fills with mostly unfinished squiggles until the animal eventually resumes one of its earlier poses. If that happens, I add as much detail or refinement to one of the easier sketches as I can. While this is all going on, I also jot down written notes that help explain what I observed and what seemed significant.
Color reproduction
In-person, live observation of color is a practice for which I feel there is no adequate substitute. Photographs are often imprecise in reproducing color.
The negative spaces
Focus on the negative spaces surrounding the object to give yourself a fresh perspective on the form.
Hues subdued
Colors in nature are, in general, more subdued than what comes directly from the pencil or the tube of paint. Greens, especially, tend to be a lot browner than we expect.
Unfinished
Leave the drawing unfinished. Record as much information as you need, but don’t draw any forms, details, or colors that are merely repetitive. The back and front of a representative flower on a plant, for example, or half of a bilaterally symmetrical animal may be all that’s necessary.
A red sea fan
Ink and watercolor drawing of a red sea fan.
Sketches
Sketches and written notes.