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 Sense of Style
Classic style
The guiding metaphor of classic style is seeing the world. The writer can see something that the reader has not yet noticed, and he orients the reader’s gaze so that she can see it for herself. The purpose of writing is presentation, and its motive is disinterested truth. It succeeds when it aligns language with the truth, the proof of success being clarity and simplicity.
The truth can be known, and is not the same as the language that reveals it; prose is a window onto the world.
The assumption of equality
Classic writing, with its assumption of equality between writer and reader, makes the reader feel like a genius. Bad writing makes the reader feel like a dunce.
Nominalization
The English language provides bad writers with a dangerous weapon called nominalization: making something into a noun.
Instead of affirming an idea, you effect its affirmation; rather than postponing something, you implement a postponement.
"Comprehension checks were used as exclusion criteria” would be better said as “we excluded people who failed to understand the instructions.”
“There is not any anticipation there will be a cancellation” would be better as “I don’t anticipate that I will have to cancel.”
Zombie sounds, unlike the verbs whose bodies they snatched, can shamble around without subjects. That is what they have in common with the passive constructions that also bog down these examples.
The curse of knowledge
The better you know something, the less you remember about how hard it was to learn.
The curse of knowledge is the single best explanation I know of why good people write bad prose. It simply doesn’t occur to the writer that her readers don’t know what she knows - that they haven’t mastered the patois of her guild, can’t divine the missing steps that seem too obvious to mention, have no way to visualize a scene that to her is as clear as day. And so she doesn’t bother to explain the jargon, or spell out the logic, or supply the necessary detail.
Structural parallelism
If the new phrase has the same structure as the preceding one, its words can be slotted into the waiting tree, and the reader will absorb it effortlessly. The pattern is called structural parallelism, and it is one of the oldest tricks in the book for elegant (and often stirring) prose.
“He maketh me to lie down in green pastures; he leadeth me beside the still waters.”
You live only once
The logician would argue, You only live once should be rewritten as You live only once, with only next to the thing it qualifies, once.
The logician would be unbearably pedantic, but there is a grain of good taste in the pedantry. Writing is often clearer and more elegant when a writer pushes an only or a not next to the thing that it quantifies. In 1962 John F. Kennedy declared, “We choose to go to the moon not because it is easy but because it is hard.” That sounds a lot classier than “We don’t choose to go to the moon because it is easy but because it is hard."
Such tortuous syntax
How does a writer manage to turn out such tortuous syntax? It happens when he shovels phrase after phrase onto the page in the order which each one occurs to him.
The problem is that the order in which thoughts occur to the writer is different from the order in which they are easily discovered by a reader. It’s a syntactic version of the curse of knowledge. The writer can see the links among the concepts in his internal web of knowledge, and has forgotten that a reader needs to build an orderly tree to decipher them from his string of words.