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.
What Good Means
The center of the way
The advice I’ve received from those who are close to the center of this timeless way of building is to start small. Like with a piece of tile, or a tea tray. And to then imagine along with Christopher Alexander:
What it would be like
to live in a mental world
where one’s reasons
for making something
functionally
and one’s reasons
for making something
a certain shape,
or in a certain
ornamental way
are coming
from precisely
the same place
in you
.Seduction
“The classic pervasive seduction to designers is finding a solution instead of the truth.” — Richard Saul Wurman
What the material wants to be
Part of how Lou Kahn made things be good was to ask the material what it wanted to do and be. He asked brick what it liked, and would get a different answer depending on the context for the building. In Dacca, the capital of Bangladesh, brick said it liked an arch. For the Korman House in Philadelphia, brick said it liked two giant fireplaces with a lintel between them for a doorway beneath and a balcony above.
Asking yourself some questions
All of the moves that we make in space will tend toward being in accord with this phenomenon of wholeness / beauty / life if we’re willing to bring the requisite level of care to the doing of our work.
Alexander says that each of us possess the means for accessing this order within ourselves and — here’s where he loses most other architects and many in the so-called sciences in academia — he contends that what we’re connecting with inside of ourselves is an objective criterion for what good means.
Applying the criterion is easy: you ask yourself some questions:
With any action you might take with regard to placement, and with regard to the situatedness of things in space you ask yourself: does this move increase wholeness / beauty / life?
Does the intervention you’re taking intensify the feelings of wholeness in you as the maker when you are performing the work?
How does your work on this one part enhance what’s going on among wholes at the system level?
Losing meaning
The people who’ve proven that they can make very good individual products with the radical focus of a spotlight seem to be pushed ever further from making good ecosystems.
Products are being made “consistent” with the application of so-called “design patterns,” and rather than bringing coherence to these various touch-points, the painting-on of interface standards and interaction patterns did something far less valuable.
Rote consistency, in the way many seem to be going about it (Material Design being just one example), is at odds with making things be good. It simplifies what needs to remain complex.
Always, when simplification is underway, meaning is being lost.
Two coffee trays
We speculate that the shop owners designed and built an initial quantity of these remarkable coffee trays, replete with what Alexander considers to be the fifteen geometric properties that correlate with wholeness / beauty / life.
Then they got busy. And then they got successful. They needed more coffee trays, and our hypothesis is that somebody decided to simplify the trays to ensure they could be produced in the quantities and at the price that worked for their budget, within an urgent food-service timeline.
The simplified tray fulfills every function the more complex tray does, with less fuss in manufacturing on account of having standardized its geometry. The simplified tray works, but isn’t alive. It lacks the gradients, local symmetries, levels of scale, contrast, and boundaries that are all present and accounted for in the tray that’s got wholeness / beauty / life. The tray with wholeness isn’t necessarily better than the simpler one. But it is good.