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.
Gods of the Word
Imagine that we had no voice and no tongue
Socrates: Imagine that we have no voice and no tongue, but want to communicate with one another. Wouldn’t we like the deaf and the dumb make signs with the hands and the head and the rest of the body?
Hermogenes: There would be no choice, Socrates.
Socrates: We would imitate the nature of the thing: lifting the hands to heaven would mean lightness and upwardness. Heaviness and downwardness would be expressed by letting them drop toward the ground...
Hermogenes: I don’t see that we could do anything else.
Socrates: And when we want to express ourselves with the voice or tongue or mouth, the expression is simply their imitation of what we want to express?
Hermogenes: I think, it must be so.My name
“I am the utterance of my name.”
— Thunder, Perfect Mind, The Nag Hammadi Library
Reference and Is-ness
There are at least two aspects to what we have traditionally called the meaning of a word. One aspect is reference, and the other is something I will call ‘inherent meaning’ following Ullman (1963). Inherent meaning is ‘Is-ness’ meaning. Inherent meaning is a word’s identity, and reference merely its resumé, where it has gone and what it has done, an itemization of its contexts. ‘Is-ness’ is unifying. Each word has a single pronunciation, a single inherent meaning. But reference is divisive. It makes what was one thing – the word – appear to be many things – its senses. It is inherent meaning which gives all those multifarious senses the power of being a single word.
It flows out and fills
This deeper meaning of a word isn’t confined to what we think of as a dictionary definition. Rather it flows out and fills all the space available to it. Although a basic sense does affect the dynamics of a word, it has no power over its essence. Like the captain of a ship, it can control the crew’s actions, but not their minds. Each word has an aspect of meaning which lies deeper than any of its senses, and it is fundamentally on this meaning that all the senses depend.
No less than a Zeus
I too am a true believer in the autonomy of the archetype. A
/t/
or an/h/
is no less than a Zeus. The consonants are not essentially physical, but they live, evolve and influence human affairs. We overlook something essential if we deny that they can get up and walk around. This is not to say that their existence is independent of the human psyche. But then everything depends on everything.Like a prism
When you look at phonemes, you look through the perspective of morphemes, which are one linguistic level higher. The higher level is like a prism that splits the light in two. What was one thing, like ‘length’ at the phoneme level, looks like two opposite things ‘long’ and ‘short’ from the perspective of the morphemes. In practice, when you find both a word and its opposite, then the phoneme is not about either of these two things, but about what is common to them.
Fracturing
If we step back and view from afar this process of One-ness and Is-ness to fracturing and interpretation – of inherent meaning to reference, it follows that what lies at the foundation of language is simply what it is – sound – free of reference and interpretation. What makes what we know as language from its sound is fracturing and interpretation or using a word for a function other than what it simply is.
To evolve the language itself
So in the process of talking, we might say we are putting words in slightly new contexts, and then testing them against our peers to see if our experiment in juxtaposition had ‘meaning. If we succeed, we have introduced new contexts for the words we use. These contexts will be taken up by our listeners, and will gradually become clearly enough defined to be thought of as referents. Once our words gain new referents, they start affecting the underlying phonosemantic structure of the language, the clustering patterns, the network of semantic relations. That is, the purpose of talking in the long run is to evolve the language itself.
Scooting over
There is at this point no evidence that acquired characteristics can be inherited. It is held that all changes to a genome are random, and cannot be subject to any higher principle. However, when a word is used in a new context, as it is whenever we say something new, a new sense is permitted. This does affect the phonosemantic structure, the linguistic DNA. Words in the vicinity of this word ‘scoot over’ to make room and allow themselves to be influenced by its philosophy. The language itself is now different.
The element becomes a sign
Each unit can be seen purely as form, as what it is. Or it can be viewed as having a function. Its function is only understandable within the next higher level of organization. And in every case, function must succumb to the constraints of form. Once this worldly function is assigned, the element becomes a ‘sign’. It falls into the realm of concept. There is a mapping from one thought system to another.
The demand of a new word
Why are these phonosemantic classes enough, and we need neither more nor less? Why are these consonants enough, and we need neither more nor less? What determines the need for a new word? How is this demand ‘felt’ by a language? How did the metabolic pathways of American English recognize that ‘jerk’ and ‘twerp’ and ‘punk’ and ‘nitwit’ and ‘dork’ and ‘ass’ and ‘goon’ and ‘twit’ and ‘dodo’ and ‘bum’ and ‘nerd’ and ‘dunce’ and ‘turd’ and ‘boob’ and ‘chump’ and ‘bitch’ and ‘bastard’ and ‘prude’ and so on and so forth simply were not equal to the task? We had to add ‘turkey’ and ‘squirrel’ as well?