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.
The Interpretation of Microstructures of Metallic Artifacts
The meaning of objects
The meaning of objects is harder to grasp than that of words.
Something akin to style
My own attitude toward the microstructure of metals is not unlike that of an art historian regarding a painting or sculpture. There is something akin to style even in a photomicrograph. There are aspects of structure that are not immediately apparent to the untrained eye, and quite minor features may be clues to a deep meaning.
From the head of Jove
A complex structure is a result of, and to a large extent a record of, its past. Though a proton and an electron may, as a pair, be able to spring full-panoplied from the head of Jove, more complex things cannot, or at least do not.
Everything complicated must have had a history, and its internal structural features arise from its history and provide a specific record of it. One might call these structural details of memory “funeous,” after the unfortunate character in Borge’s story “Funes the Memorious” who remembered everything.
The objects themselves speak loudly
The written record is very little help in determining the techniques used by ancient metalworkers, though the objects themselves speak loudly to an educated ear.
The way it has been made
The internal structure of a work of art in metal can often throw as much, or more, light on its origin as can be derived from stylistic analysis. Moreover, the techniques employed can provide clues to the habits of mind of the people who originated them.
…Perhaps the most important reason for structural studies of museum objects is that the intimate knowledge so derived as to the way in which an object has been made adds so greatly to the aesthetic enjoyment of it. Very often some detail and sometimes the whole of an effective design arises directly in the exploitation of the merits and the overcoming of the difficulties of a specific technique, in the reaction between the artist’s fingers and his material.
Aesthetically motivated curiosity
It seems that the first and most imaginative use of practically every material was, before quite modern times, in making something decorative. People are experimentally minded when looking for decorative effects, but they can’t experiment with the established techniques on which their livelihood depends.
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It is of basic significance for human history that, from the cave paintings on, almost all inorganic materials and treatments of them to modify their structure appear first in decorative objects rather than in tools or weapons necessary for survival. Aesthetically motivated curiosity, or perhaps just play, seems to have been the most important stimulus to discovery.