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 Architecture of Happiness
A few millimeters apart
Aesthetes force us to consider whether happiness may not sometimes turn on the presence or absence of a fingerprint, whether in certain situations beauty and ugliness may not lie only a few millimeters apart, whether a single mark might not wreck a wall or an errant brush stroke undo a landscape painting.
Tragic colors
Life may have to show itself to us in some of its authentically tragic colors before we can begin to grow properly visually responsive to its subtler offerings.
Classical absurdity
Corbusier observed that the requirements of flight of necessity rid airplanes of all superfluous decoration and so unwittingly transformed them into successful pieces of architecture. To place a Classical statue atop a house was as absurd as to add one to a plane, he noted, but at least by crashing in response to this addition, the plane had the advantage of rendering its absurdity starkly manifest.
Ideas of a good life
In essence, what works of design and architecture talk to us about is the kind of life that would most appropriately unfold within and around them. They tell us of certain moods that they seek to encourage and sustain in their inhabitants. While keeping us warm and helping us in mechanical ways, they simultaneously hold out an invitation for us to be specific sorts of people. They speak of visions of happiness.
To describe a building as beautiful therefore suggests more than a mere aesthetic fondness; it implies an attraction to the particular way of life this structure is promoting through its roof, door handles, window frames, staircase, and furnishings. A feeling of beauty is a sign that we have come upon a material articulation of certain of our ideas of a good life.
The people we love
The objects we describe as beautiful are versions of the people we love.
The search for happiness
If the search for happiness is the underlying quest of our lives, it seems only natural that it should simultaneously be the essential theme to which beauty alludes.
An equivalence
In both early Christianity and Islam, theologians made a claim about architecture likely to sound so peculiar to modern ears as to be worth of sustained examination: they proposed that beautiful buildings had the power to improve us morally and spiritually. They believed that, rather than corrupting us, rather than being an idle indulgence for the decadent, exquisite surroundings could edge us towards perfection. A beautiful building could reinforce our resolve to be good.
Behind this distinctive claim lay another astonishing belief: that of an equivalence between the visual and ethical realms.
Inwardly to resemble
What we want, at the deepest level, is inwardly to resemble, rather than physically to possess, the objects and places that touch us through their beauty.
We can conclude from this that we are drawn to call something beautiful whenever we detect that it contains in a concentrated form those qualities which we personally, or our societies more generally, are deficient.
The gathering darkness of a Sunday evening
Our jobs make relentless calls on a narrow band of our faculties, reducing our chances of achieving rounded personalities and leaving us to suspect (often in the gathering darkness of a Sunday evening) that much of who we are, or could be, has gone unexplored.
What we don't like
A grasp of the psychological mechanism behind taste may not change our sense of what we find beautiful, but it can prevent us from reacting to what we don’t like with simple disbelief.
Our understanding of the psychology of taste can in turn help us to escape from the two great dogmas of aesthetics: the view that there is only one acceptable visual style or (even more implausibly) that all styles are equally valid.
Over-imagination
An architect intent on being different may in the end prove as troubling as an over-imaginative pilot or doctor. However important originality may be in some fields, restraint and adherence to procedure emerge as the more significant virtues in a great many others.
We rarely wish to be surprised by novelty as we round street corners. We require consistency in our buildings, for we are ourselves frequently close to disorientation and frenzy.
The extremes of order and complexity
Such works emphasize the truth of the ancient maxim that beauty lies between the extremes of order and complexity.
It follows that the balance we approve of in architecture, and which we anoint with the word ‘beautiful’, alludes to a state that, on a psychological level, we can describe as mental health or happiness. Like buildings, we, too, contain opposites which can be more or less successfully handled.
Beauty and strength
Yet the bridge testifies to how closely a certain kind of beauty is bound up with our admiration for strength, for man-made objects which can withstand the life-destroying forces of heat, cold, gravity or wind.
We respond with emotion to creations which transport us across distances we could never walk, which shelter us during storms we could not weather, which pick up signals we could never hear with our own ears and which hang daintily off cliffs from which we would fall instantly to our deaths.
With grace and economy
Both bridges accomplish daring feats, but Maillart’s possesses the added virtue of making its achievement look effortless - and because we sense it isn’t, we wonder at it and admire it all the more. The bridge is endowed with a subcategory of beauty we can refer to as elegance, a quality present whenever a work of architecture succeeds in carrying out an act of resistance - holding, spanning, sheltering - with grace and economy as well as strength; when it has the modesty not to draw attention to the difficulties it has surmounted.
The dignity of age
Wood and stone, and now concrete and wood, age slowly and with dignity. They do not shatter hysterically like glass, or tear like paper, but discolor with a melancholy, noble air.
Apportioning value
Contrary to the Romantic belief that we each settle naturally on a fitting idea of beauty, it seems that our visual and emotional faculties in fact need constant external guidance to help them decide what they should take note of and appreciate. ‘Culture’ is the word we have assigned to the force that assists us in identifying which of our many sensations we should focus on and apportion value to.
There was no fog in London
It is books, poems and paintings which often give us the confidence to take seriously feelings in ourselves that we might otherwise never have thought to acknowledge. Oscar Wilde referred to this phenomenon when he quipped that there was no fog in London before Whistler started painting the Thames. Likewise, there must have been little beauty in old stones before Japanese priests and poets began writing about them.
To the worms and the trees
We owe it to the fields that our houses will not be the inferiors of the virgin land they have replaced. We owe it to the worms and the trees that the buildings we cover them with will stand as promises of the highest and most intelligent kinds of happiness.