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.
Several Short Sentences About Writing
Here, in short, is what I want to tell you.
Know what each sentence says,
What it doesn't say,
And what it implies.
Of these, the hardest is knowing what each sentence actually says.
Sonorisms V
Leave space between them for the things that words can't really say.
To suggest more than the words seem to allow.
Perhaps it renames the world.
The Anxiety of Sequence.
It was all change until the very last second.
The debris of someone else's thinking.
You'll never run out of noticings.
Names that announce the whatness of the world.
What were you trying to protect?
You were protecting the memory.
The tyranny of what exists.
Do any of them sound first?
It sets an echo in motion.
Try writing for the reader in yourself.
So call it "perfection enough".
Toward the name of the world—yours to discover.Both models are completely useless
In your head, you'll probably find two models for writing.
One is the familiar model taught in high school and college—a matter of outlines and drafts and transitions and topic sentences and argument.The other model is its antithesis—the way poets and novelists are often thought to write.
Words used to describe this second model include "genius", "inspiration", "flow", and "natural", sometimes even "organic".Both models are useless.
I should qualify that sentence.
Both models are completely useless.The shape of the sentence
You've been taught to overlook the character of the prose in front of you in order to get at its meaning.
You overlook the shape of the sentence itself for the meaning it contains,
Which means that while you were reading,
All those millions of words passed by
Without teaching you how to make sentences.The Anxiety of Sequence
Much of what's taught under the name of expository writing could be called "The Anxiety of Sequence."
Its premise is this:
To get where you're going, you have to begin in just the right place
And take the proper path,
Which depends on knowing where you plan to conclude.You can get anywhere from anywhere
And if you can get anywhere from anywhere,
You can start anywhere
And end anywhere.
There is no single necessary order.Significant everywhere
Writing isn't a conveyor belt bearing the reader to "the point" at the end of the piece, where the meaning will be revealed.
Good writing is significant everywhere,
Delightful everywhere.It was all change until the very last second
Every work of literature is the result of thousands and thousands of decisions.
Intricate, minute decisions—this word or that, here or where, now or later, again and again.
It's the living tissue of a writer's choices,
Not the fossil record of an ancient, inspired race.Attention requires a cunning passivity.
A renaming of the already named
A true metaphor is a swift and violent twisting of language,
A renaming of the already named.
It's meant to expire in a sudden flash of light
And to reveal—in that burst of illumination—
A correspondence that must be literally accurate.The debris of someone else's thinking
A cliché isn't just a familiar, overused saying.
It's the debris of someone else's thinking.How each sentence got that way
When the work is really complete, the writer knows how each sentence got that way.
This small internal quaver
Pay attention now:
No matter how much you know or learn about syntax, grammar, or rhetoric,
This small internal quaver, this inner disturbance,
Is the most useful evidence you'll ever get.
Someday, you'll be able to articulate what causes it.
But for now, what's important is to notice it.
Noticing is always the goal....the faint vertigo caused by an ambiguity you can't quite detect.
What matter is what it points to.
Find out what's causing it and fix it
Even if you're not sure how.The urge to be done
"Flow" is often a synonym for ignorance and laziness.
It's also a sign of haste, the urge to be done.Talking and writing
Talking is natural.
Writing is not.It may seem strange that the manual dexterity needed to hold a pencil—or use a keyboard—comes later than the lingual and mental dexterity needed to speak.
But it does.What were you trying to protect?
As the piece evolves, you try to protect those original, effusive sentences.
Only to realize, at last, that what you're writing won't come together until they've been removed or revised.What were you trying to protect?
The memory of the excitement you felt when those words "came to you."
(Where did they "come" from?)
You were protecting the memory
of the excitement of really concentrating,
of paying close attention to your thoughts and, perhaps, your sentences,
the excitement of feeling the galvanic link between language and thought.The discoveries you make in the making
Style is an expression of the interest you take in the making of every sentence.
It emerges, almost without intent, from your engagement with each sentence.
It's the discoveries you make in the making of the prose itself.Where ambiguity rules, there is no "style"—or anything else worth having.
Pursue clarity instead.
In the pursuit of clarity, style reveals itself.The virtue of already existing
It can be overwhelming—the inertia of the paragraphs and pages you've already composed, the sentences you've already written,
No matter how rough they are.Whether you love what you've written or not,
Those sentences have the virtue of already existing,
Which makes them better than sentences that don't exist.
Or so it seems.Composition and revision
Revise at the point of composition.
Compose at the point of revision.
Think of composition and revision as the same thing.Squander your material
Squander your material.
Don't ration it, saving the best for last.
You don't know what the best is.
Or the last.Do any of them sound first?
Just try out some sentences.
Lots of them.
See how they sound.
Do any of them sound first?You're holding an audition.
Many sentences will try out.
One gets the part.We have testimony
Proof is for mathematicians.
Logic is for philosophers.
We have testimony.When you're interested in what you're working on
It's never hard to work when you're interested in what you're working on.
But what if you hate what you're working on?
It helps to examine the content of your loathing.
What is it you hate?The work selects its audience
Imagine a cellist playing one of Bach's solo suites.
Does he consider his audience?
(Did Bach, for that matter?)
Does he play the suit differently to audiences
Of different incomes and educations and social backgrounds?
No. The work selects its audience.