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's Wrong With This Model?
What's wrong with the rational model
- We Don’t Really Know the Goal When We Start
- We Usually Don’t Know the Decision Tree – We Discover It as We Go
- The Nodes Are Really Not Design Decisions, but Tentative Complete Designs
- The Goodness Function Cannot be Evaluated Incrementally
- The Desiderata and Their Weightings Keep Changing
- The Constraints Keep Changing
Deciding what to design
We Don’t Really Know the Goal When We Start
The most serious model shortcoming is that the designer often has a vague, incompletely specified goal, or primary objective. In such cases, the hardest part of design is deciding what to design.
I came to realize that the most useful service I was performing for my client was helping him decide what he really wanted.
Today, we recognize that rapid prototyping is an essential tool for formulating precise requirements. Not only is the design process iterative; the design-goal-setting process is itself iterative. Knowing complete product requirements up front is a quite rare exception, not the norm. Therefore, goal iteration must be considered an inherent part of the design process.
Evaluating goodness
The Goodness Function Cannot be Evaluated Incrementally
The Rational Model assumes that design involves a search of the decision tree, and that at every node, one can evaluate the goodness function of several downward branches. In fact, one cannot in general do this without exploring all the downward branches to all their leaves, which is possible in principle, but leads to a combinatorial explosion of alternatives in practice.
Changing constraints
The Constraints Keep Changing
The explicit listing of known constraints in the design program helps here. The designer can periodically scan the list, asking, “Can this constraint now be removed because the world has changed? Can it be entirely circumvented by working outside the design space?”
They just don't work that way
Perhaps the most devastating critique of the Rational Model, although perhaps the hardest to prove, is that most experienced designers just don’t work that way.
“Conventional wisdom about problem-solving seems often to be contradicted by the behavior of expert designers. Empirical studies of design activity have frequently found ‘intuitive’ features of design ability to be the most effective and relevant to the intrinsic nature of design. Some aspects of design theory, however, have tried to develop counter-intuitive models and prescriptions for design behavior.” — Nigel Cross
We must outgrow it
Why all this fuss about the process model? Does the model we and others use to think about our design process really affect our designing itself? I believe it does. I believe our inadequate model and following it slavishly lead to fat, cumbersome, over-features products and also to schedule, budget, and performance disasters.
The Rational Model, in any of its forms, leads us to demand up-front statements of design requirements. It leads us to believe that such can be formulated. It leads us to make contracts with one another on the basis of enshrined ignorance. A more realistic process model would make design work more efficient, obviating many arguments with clients and much rework.
The Waterfall Model is wrong and harmful; we must outgrow it.