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 Blue Cliff Record
Mountains are mountains
Before attaining enlightenment,
mountains are mountains, rivers are rivers.At the moment of enlightenment,
mountains are no longer mountains,
nor are rivers rivers.After attaining enlightenment,
mountains are mountains, rivers are rivers.No door at which to knock
Penetrating the Blue Cliff, you will open
the eye within eyes
and realizing life through the Blue Cliff,
you yourself will become a torch,
the light beyond light.Being so, you will find no door at which to knock, nor any door to be opened.
The miraculous bones of the ancients
A Ch'an master once wrote that the wise enshrine the miraculous bones of the ancients within themselves; that is, they do not regard teachings of ways to enlightenment as an external body of knowledge or information to be possessed as an acquisition or believed or revered as inflexible dogma, but rather apply it as far as possible to themselves and their situations, vivifying the way of enlightenment with their own bodies and lives, not just in their thoughts. It is therefore a matter of course that new Buddhist literature has been produced; for the Buddhist canon is not closed, as long as people continue the search for enlightenment.
Ch'an was referred to by its followers as the "school of the patriarchs" because it was transmitted by a living succession of human exemplars; not a school of doctrine, or philosophical or scholastic interpretation, it was not based on any particular scripture, but on the direct experience of the enlightened mind, by whatever means currently necessary.
What did you see when you were there?
Ch'an masters tell us to look right where we stand and step: they ask what is there, is there anything, who are you and where do you come from, what did you see and hear when you were there?—in a hundred ways they try to make us wake up, clear our minds, and face reality directly.
I have pacified your mind
Bodhidharma, the first Patriarch of Ch'an in China, was asked by Hui K'e, the future Second Patriarch, so pacify his mind for him; Bodhidharma said, "Bring me your mind and I will pacify it for you." Hui K'e said, "When I search for my mind, I cannot find it." Bodhidharma said, "I have pacified your mind for you." At this Hui K'e was enlightened.
In one there are many; in two, one
The ultimate path is without difficulty;
Speech is to the point, words are to the point.
In one there are many kinds;
In two there is no duality.It's dark outside
As soon as Te Shan crossed the threshold he said, "Long have I heard of Lung T'an (Dragon Pond), but now that I've arrived here, there's no pond to see and no dragon appears." Master Lung T'an came out from behind a screen and said, "You have really arrived at Lung T'an." Te Shan bowed and withdrew.
During the night, Te Shan entered Lung T'an's room and stood in attendance till late at night. Lung T'an said, "Why don't you go?" Te Shan bade farewell, lifted up the curtain, and went out; he saw that it was dark outside, so he turned around and said, "It's dark outside." Lung T'an lit a paper lantern and handed it to Te Shan; as soon as Te Shan took it, Lung T'an blew it out. Te Shan was vastly and greatly enlightened.