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.
Design Thinking
Ducks and decorated sheds
A duck is a building whose confirmation is a complete symbol or icon. A decorated shed is a building to which symbols, often commonplace signs, have been attached.
Clinging to ideas
Another aspect of design thinking that was evident in the foregoing case studies is the tenacity with which designers will cling to major design ideas and themes in the face of what at times might seem insurmountable odds. Often the concept the designer has in mind can only come to fruition if a large number of apparently countervailing conditions can be surmounted.
Even when severe problems are encountered, a considerable effort is made to make the initial idea work, rather than to stand back and adopt a fresh point of departure.
A concept of style
It is a concept based not on the classification of various physical features of architecture and urban design but on the problem-solving process itself. We have seen that the final outcome of a design process is strongly determined by at least three aspects of that process:
- the subject matter of the organizing principles which are adopted,
- the manner in which these principles are interpreted and reinterpreted in the context of the problem at hand, and
- the sequent of applying such organizing principles.
Consistency in style among the output of designers can thus be understood as a habitual way of doing things, of solving problems.
Form and figure
Form applies to “a configuration with natural meaning or none at all,” whereas figure applies to “a configuration whose meaning is given by culture."
Design skirmishes
it is apparent that the unfolding of the design process assumed a distinctly episodic structure, which we might characterize as a series of related skirmishes with various aspects of the problem at hand.
As the scope of the problem became more determined and finite for the designer, the episodic character of the process seems to have become less pronounced. During this period a systematic working out of issues and conditions took hold within the framework that had been established. This phenomenon is not at all surprising when we consider the fundamental difference between moments of problem solving when matters are poorly defined and those with clarity and sufficiency of structure.
Within the episodic structure of the process, the problem, as perceived by the designer, tends to fluctuate from being rather nebulous to being more specific and well-defined. Furthermore, moments of "blinding" followed by periods of backtracking take place, where blinding refers to conditions in which obvious connections between various considerations of importance go unrecognized by a designer.
Such plans were deemed efficient
The terrain of cities was subdivided along the lines of distinct and discrete patterns of use, with very little opportunity for mixing (separation and concentration of functions). After all, the home environment should be just that…while places of work should be aggregated and serviced with their appropriate supporting functions.
Such plans were deemed efficient.
Constrained by the medium
The inevitable reciprocation that occurs between the act of drawing and the thinking associated with it. The hand moves, the mind becomes engaged, and vice versa. We might ask: How much does the medium of expression actually constrain a design process?
A medium has a way of constraining our choices, and this influence may not involve conscious choice at all. The planner, in the end, sees and understands only those things for which they can provide expression.
Autonomous constraints
Autonomous or independent constraints do not derive from the problem as given and understood…there was nothing in the problem statement, or brief, that required any reference be made to it. This constraint, introduced by the designers, usefully transcended the givens of the problem situation.
Unless the entire problem at hand can be solved using strictly problem-oriented constraints, we have to step outside the known problem context in order to continue problem solving activity.
The strange familiar and the familiar strange
The problem solver, when confronted with a new and yet unsolved problem, overlays the structure of the unsolved problem with an apparently similar problem with which he or she is experienced.
Making the strange familiar and the familiar strange are also principally based on the use of analogy.