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.
Rationality: From AI to Zombies
The Tao of rationality
If you would learn to think like reality, then here is the Tao:
Since the beginning
not one unusual thing
has ever happened.Everyone sees themselves as behaving normally
To understand why people act the way they do, we must first realize that everyone sees themselves as behaving normally.
Argue against the best
To argue against an idea honestly, you should argue against the best arguments of the strongest advocates.
It’s all too easy to argue that someone is exhibiting Bias #182 in your repertoire of fully generic accusations, but you can’t settle a factual issue without closer evidence. If there are biased reasons to say the sun is shining, that doesn’t make it dark out.
Let the meaning choose the word
What is above all needed is to let the meaning choose the word, and not the other way around.
Probably it is better to put off using words as long as possible and get one’s meaning as clear as one can through pictures and sensations.
People can stand for what is true, for they are already enduring it
Knowing the design
Knowing the design can tell you much about the designer; and knowing the designer can tell you much about the design.
Cutting through to the truth
The essential thing in the art of epistemic rationality is to understand how every rule is cutting through to the truth in the same movement.
Your sword has no blade. It has only your intention. When that goes astray you have no weapon.
Joy in the merely attainable
But the really fundamental problem with desiring the unattainable is that as soon as you actually get it, it stops being unattainable. If we cannot take joy in the merely available, our lives will always be frustrated.
I have failed my art
The novice goes astray and says, “The art has failed me.”
The master goes astray and says, “I have failed my art."Taboo your words
Albert says that people have “free will.” Barry says that people don’t have “free will.” Well, that will certainly generate an apparent conflict. Most philosophers would advise Albert and Barry to try to define exactly what they mean by “free will,” on which topic they will certainly be able to discourse at great length. I would advise Albert and Barry to describe what it is that they think people do, or do not have, without using the phrase “free will” at all.
Your map of reality
Reality is very large—just the part we can see is billions of lightyears across. But your map of reality is written on a few pounds of neurons, folded up to fit inside your skull. I don’t mean to be insulting, but your skull is tiny. Comparatively speaking. Inevitably, then, certain things that are distinct in reality, will be compressed into the same point on your map. But what this feels like from inside is not that you say, “Oh, look, I’m compressing two things into one point on my map.” What it feels like from inside is that there is just one thing, and you are seeing it.
What you're trying to swim
There is an art to using words; even when definitions are not literally true or false, they are often wiser or more foolish. Dictionaries are mere histories of past usage; if you treat them as supreme arbiters of meaning, it binds you to the wisdom of the past, forbidding you to do better. Though do take care to ensure (if you must depart from the wisdom of the past) that people can figure out what you’re trying to swim.
Applause lights
Most applause lights are much more blatant, and can be detected by a simple reversal test.
For example, suppose someone says: We need to balance the risks and opportunities of AI.
If you reverse this statement, you get: We shouldn’t balance the risks and opportunities of AI.Since the reversal sounds abnormal, the unreversed statement is probably normal, implying it does not convey new information. There are plenty of legitimate reasons for uttering a sentence that would be uninformative in isolation. “We need to balance the risks and opportunities of AI” can introduce a discussion topic; it can emphasize the importance of a specific proposal for balancing; it can criticize an unbalanced proposal. Linking to a normal assertion can convey new information to a bounded rationalist—the link itself may not be obvious. But if no specifics follow, the sentence is probably an applause light.
Reality just seems to go on crunching
I once met a fellow who thought that if you used General Relativity to compute a low-velocity problem, like an artillery shell, General Relativity would give you the wrong answer—not just a slow answer, but an experimentally wrong answer—because at low velocities, artillery shells are governed by Newtonian mechanics, not General Relativity. This is exactly how physics does not work. Reality just seems to go on crunching through General Relativity, even when it only makes a difference at the fourteenth decimal place, which a human would regard as a huge waste of computing power. Physics does it with brute force. No one has ever caught physics simplifying its calculations—or if someone did catch it, the Matrix Lords erased the memory afterward.
Mystery exists in the mind
Mystery exists in the mind, not in reality. If I am ignorant about a phenomenon, that is a fact about my state of mind, not a fact about the phenomenon itself. All the more so if it seems like no possible answer can exist: Confusion exists in the map, not in the territory. Unanswerable questions do not mark places where magic enters the universe. They mark places where your mind runs skew to reality.
Lost purposes
There’s chocolate at the supermarket, and you can get to the supermarket by driving, and driving requires that you be in the car, which means opening your car door, which needs keys. If you find there’s no chocolate at the supermarket, you won’t stand around opening and slamming your car door because the car door still needs opening. I rarely notice people losing track of plans they devised themselves.
It’s another matter when incentives must flow through large organizations—or worse, many different organizations and interest groups, some of them governmental. Then you see behaviors that would mark literal insanity, if they were born from a single mind. Someone gets paid every time they open a car door, because that’s what’s measurable; and this person doesn’t care whether the driver ever gets paid for arriving at the supermarket, let alone whether the buyer purchases the chocolate, or whether the eater is happy or starving.