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.
On Love
Not knowing quite what they mean
"Do you understand all the symbolism?"
"Not really, besides its being Venus and Cupid."
"I didn't even know that, so you're one up on me. I wish I'd read more about ancient mythology," she continued. "But actually, I like looking at things and not knowing quite what they mean."
We're not children, you know
We're not children, you know.
And with these words, she placed her lips on mine and we embarked on one of the longer and more beautiful kisses mankind has ever known.
To think
Few things are as antithetical to sex as thought. Sex is instinctive, unreflective, and spontaneous, while thought is careful, uninvolved, and judgmental. To think during sex is to violate a fundamental law of intercourse.
A cruel paradox
When we look at someone (an angel) from a position of unrequited love and imagine the pleasures that being in heaven with them might bring us, we are prone to overlook a significant danger: how soon their attractions might pale if they began to love us back. We fall in love because we long to escape from ourselves with someone as ideal as we are corrupt. But what if such a being were one day to turn around and love us back? We can only be shocked. How could they be as divine as we had hoped when they have the bad taste to approve of someone like us? If in order to love we must believe that the beloved surpasses us in some way, does not a cruel paradox emerge when we witness this love returned? "If s/he really is so wonderful, how could s/he love someone like me?"
Shoes
It was perhaps a pedantic matter over which to come to such a decision, but shoes are supreme symbols of aesthetic, and hence by extension psychological, compatibility. Certain areas and coverings of the body say more about a person than others: shoes suggest more than pullovers, thumbs more than elbows, underwear more than overcoats, ankles more than shoulders.
I marshmallow you
Then I noticed a small plate of complimentary marshmallows near Chloe's elbow and it suddenly seemed clear that I didn't love Chloe so much as marshmallow her. What it was about a marshmallow that should suddenly have accorded so perfectly with my feelings toward her, I will never know, but the word seemed to capture the essence of my amorous state with an accuracy that the word 'love', weary with overuse, simply could not aspire to.
From then on, love was, for Chloe and me at least, no longer simply love, it was a sugary, puffy object a few millimeters in diameter that melts deliciously in the mouth.
A social animal
What does it mean that man is a "social animal"? Only that humans need one another in order to define themselves and achieve self-consciousness, in a way that mollusks and earthworms do not. We cannot come to a proper sense of ourselves if there aren't others around to show us what we're like.
"A man can acquire anything in solitude except a character," wrote Stendhal.
Possible lives
Watching Alice talk, light a candle that had blown out, rush into the kitchen with the plates, or brush a strand of blonde hair from her face, I found myself falling victim to romantic nostalgia, which descends whenever we are faced with those who might have been our lovers, but whom chance has decreed we will never properly know. The possibility of an alternative love story is a reminder that the life we are leading is only one of a myriad of possible lives, and it is the impossibility of leading them all that plunges us into sadness.
The threat of happiness
Dr. Saavedra had diagnosed a case of anhedonia, a disease defined by the British Medical Association as a reaction remarkably close to mountain sickness resulting from the sudden terror brought on by the threat of happiness. It was a common disease among tourists in this region of Spain, faced in these idyllic surroundings with the sudden realization that earthly happiness might be within their grasp, and prey therefore to a violent physiological reaction designed to counteract such a daunting possibility.
I don't see a wall
"I don't know, really. It's just a sense I have that ever since about the middle of September, we haven't really been communicating. It's like there's a wall between us and you're refusing to acknowledge it's there."
"I don't see a wall."
"That's what I mean. You're refusing to admit there was ever anything other than this."
"Than what?"
Romantic terrorism
Once a partner has begun to lose interest, there is apparently little the other can do to arrest the process. Like seduction, withdrawal suffers under a blanket of reticence. The very breakdown of communication is hard to discuss, unless both parties have a desire to see it restored.
This leaves the lover in a desperate situation. Honest dialogue seems to produce only irritation and smothers love in the attempt to revive it. Desperate to woo the partner back at any cost, the lover might at this point be tempted to turn to romantic terrorism, the product of irredeemable situations, a gamut of tricks (sulking, jealousy, guilt) that attempt to force the partner to return love, by blowing up (in fits of tears, rage, or otherwise) in front of the loved one. The terroristic partner knows he cannot realistically hope to see his love reciprocated, but the futility of something is not always (in love or in politics) a sufficient argument against it.
Certain things are said not because they will be heard, but because it is important to speak.
Walls
As the plane pierced through the clouds, I tried to imagine a future: a period of life was coming brutally to an end, and I had nothing to replace it with, only a terrifying absence.
What would life mean from now on? Though we continued holding hands, I knew how Chloe and I would watch our bodies grow foreign to each other. Walls would be build up, the separation would be institutionalized, I would meet her in a few months or years, we would be light, jovial, masked, dressed for business, ordering a salad in a restaurant—unable to touch what we were now revealing: the sheer human drama, the nakedness, the dependency, the unalterable loss. We would be like an audience emerging from a heart-wrenching play but unable to communicate any of the emotions they had felt inside, able only to head for a drink at the bar, knowing there was more, but unable to touch it.
In order that
No philosophy is further from the thought that what happens to us is random than psychoanalysis. I did not simply love Chloe and then she left me. I loved Chloe in order that she would leave me. It was not for the shape of her smile or the liveliness of her mind that I had chosen Chloe. It was because the unconscious, perverse casting director of my life had recognized in her a suitable character to leave the stage after inflicting the requisite amount of suffering.
The significance of love's burden
There is an Arabic saying that the soul travels at the pace of a camel. While most of our self is led by the strict demands of timetables and diaries, our soul, the seat of the heart, trails nostalgically behind, burdened by the weight of memory. If every love affair adds a certain weight to the camel's load, then we can expect the soul to slow according to the significance of love's burden.
What does wisdom counsel?
We start trying to be wise when we realize that we are not born knowing how to live, that living one's life is a skill that has to be acquired, like learning to ride a bicycle or play the piano. But what does wisdom counsel us to do? It tells us to aim for tranquility and inner peace, a life free from anxiety, fear, idolatry, and harmful passions. Wisdom teaches us that our first impulses may not always be trustworthy, and that our appetites will lead us astray if we do not train reason to separate vain from genuine needs.
Once again begun to fall
Such lessons appeared all the more relevant when Rachel accepted my invitation for dinner the following week, and the very thought of her began sending tremors through the region the poets have called the heart, tremors that I knew could have meant one thing only—that I had once again begun to fall.