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 Nature and Aesthetics of Design
Any imaginable shape
The thing which sharply distinguishes useful design from such arts as painting and sculpture is that the practitioner of design has limits set upon his freedom of choice. A painter can choose any imaginable shape. A designer cannot.
Useless work on useful things
Anyone can verify by simple observation two important facts.
The first is, that whenever humans design and make a useful thing they invariably expend a good deal of unnecessary and easily avoidable work on it which contributes nothing to its usefulness.
The second fact is that all useful devices have got to do useless things which no one wants them to do. Who wants car to get hot? Or to wear out its tires? Or to make a noise and a smell?
Presentable
I have sometimes wondered whether our unconscious motive for doing so much useless work is to show that if we cannot make things work properly we can at least make them presentable.
The principle of arrangement
It is really rather remarkable that, while anyone can tell whether a thing is a pocket-knife because, presumably, anyone can recognize the principle of arrangement which constitutes the similarity between all pocket knives, no one can visually abstract that arrangement. We recognize it when we 'see' it embodies, we can describe it disembodies, but we cannot visualize it disembodied.
The minimum condition
When a device is so designed that its component parts are only just strong enough to get the intended result without danger of failure, we may say it is in its minimum condition.
I suspect that the functionalists sometimes meant by functional design simply design aimed at the minimum condition for a device. In that case 'form should follow function' would mean that every system should be in its minimum condition, thus having certain limitations imposed on its form.
The requirements of economy
Economy is the mother of most inventions, not necessity, unless in the sense of poverty and hardship. A requirement for convenience is simply a diluted requirement for ease and economy.
It seems to be invariably true that those characteristics which lead people to call a design functional are derived from the requirements of economy and not of use.
When design gets too easy
Design has invariably exhibited styles because some clear limitations on freedom of choice are psychologically necessary to nearly all designers. When design gets too easy it becomes difficult.
The one best way
It is a most diverting spectacle to see the experts in work study exercising their considerably ingenuity to find the one cheapest way of doing operations which could perfectly well be dispensed with; for example, getting shiny surfaces on furniture. The 'one best way' of doing things like that is not to do them.
The versatility of flat surfaces
The versatility of flat surfaces is not commonly seen in nature.
The works of God
Via reddit
Thus the first and most important stratagem adopted to cheapen construction was the squaring and turning of components in order to eliminate offering up and individual fitting. The flatness, straightness, and squareness which more than any other characteristic distinguish man's construction from the works of God, derive from economy. We see the mark of economy in every building of squared masonry however magnificent it may be. Only the few remnants of ancient polygonal masonry remind us that the pattern of stone work where each stone is individually fitted can be very different from the cheap squared pattern to which we are accustomed."
Skill vs. knowledge
We should say that anybody has skill enough to build a good dry-stone wall but that few know how to design one, for the placing of the stones is a matter of knowledge and judgment, not of dexterity.
6 methods for economical design
- Use readily available materials.
- Use easily worked ('wasted') materials.
- Avoid dexterous labor.
- Use standardized materials or components.
- Avoid intermediate states (get straight to the final product).
- Use standardized language and geometry. Design only what can be easily communicated.
Old solutions
Where the problem is old, the old solutions will nearly always be best.
The bloodless ghosts of memory
The bloodless ghosts of memory.
A strangely negative character
Utility has a strangely negative character. We speak of the secret of happiness, for its causes are elusive; but there is no secret about the causes of unhappiness: thirst, hunger, want of sleep, exhaustion, pain, constraint of movement and too great heat and cold, are evils which can effectively prevent happiness. Utility has a negative character, because useful devices are adopted in the main for the sake ultimately of avoiding such evils.
From the fact that deadly injury, pain, and exhaustion prevent the fulfillment of the universal wish for happiness, we have always tended to infer that if only life were safe, comfortable, and effortless, we would be happy. It does not follow.
Sine qua non
What we see of a device is rarely the essential part, the sine qua non, but nearly always the superstructure which economy has imposed on it.
It seems that the work we call purely utilitarian is not more useful than its more ornamental counterpart. It is merely more economical.
The contribution that something in them yet compelled them to make
Makers and designers must gradually have come inwardly to believe that half their work had been mere frivolity because it had been avoidable, and because some of it had contributed nothing to the satisfaction of people's material wants. This must have affected them like a conviction of original sin.
The idea that utility was the purpose of work overpowered them and seemed unanswerable. From that time on perhaps the artist and workman have been weakened by an inward suspicion that they are supporting a lost cause. They have perhaps half-believed that the world could get on very well without the contribution that something in them yet compelled them to make.
No more than a sketch
The quality of a musical performance depends on the performers as much as on the score. The performers are said to be interpreting the score, but in fact they are adding intention of their own to those of the composer, recognizing that no score can in practice ever fully express the intentions of a composer, that it can never be more than an indication, a sketch; and no designer can in practice ever produce more than a sketch.
Purpureus
Our way of talking about surface quality as 'texture' is rather like the ancient Roman way of calling anything bright colored 'purpureus' on the principle perhaps that any bright color was much the same as any other.
The weather in the space
The architect's special preoccupation is first to decide what kinds of spaces shall be enclosed.
All manner of different considerations will influence an architect's decisions about the shape of the spaces they are to enclose, but the chief of them will always be the probable activities of the people who will enjoy the weather in the space.
Holding together a civilization
It is only in the present age that it has been asserted that 'architecture is not an art' or 'should not be an art': and that strenuous efforts are made to made a distinction between design and art. And nowadays we build cities of such a quality that no one likes living in them, everyone who can do so gets a motor car to escape from them. Because of the multitude of motor cars, escape is now denied us, the country is destroyed, and the cities become still less tolerable to live in.
All that is the consequence of contempt for art. Art is not a matter of giving people a little pleasure in their time off. It is in the long run a matter of holding together a civilization.
A cumulative effect
It is a cumulative effect, this character. It results from the combined impact of the design of a great many separate things, none of which is so very atrocious but too many of which are flatly negative, wanting. The design of each single thing in the environment, however small it may be, is really important.
Scenery
What is designed and made outlasts the people for whose profit and for whose use it was made.
We may think we are designing furniture of motor cars, but we are not. If we are designing a motor car for one man, we are designing scenery for fifty thousand others.
Something more is required
Efficiency, the capability of performing effectively, never made anything beautiful yet and it justifies no design in itself. To say of a design 'it works, it does its job', or 'it gets the intended result' no more commends or excuses it than to say of a man 'he has never actually defrauded anybody'. That is not what virtue means! Something more is required.
Beauty is like a joke
If some story makes you laugh aloud, then something in it causes the experience which issues in laughter. But can you describe that something to a person who does not think it funny in such a way as to make him see the joke and experience just what you have experienced?
A being-without
Not having a toothache is no goal for a lifetime. Happiness, however one defined it, is not something negative, a being-without.
Some emptiness in us
Whenever we encounter beauty we become aware, each time with a sense of shock and pleasure, faint though it may be, that some emptiness in us, not consciously felt but continually present, has been assuaged and fulfilled. We have a sudden high sense of completeness and harmony.
The matrix of all we know
Man's species has existed for an immensely longer period, unimaginably longer, in an unmodified natural environment. That unmodified environment was the matrix of all man knows of beauty. All the means of his experience of beauty evolved in it. Now, in the artificial environment, art creates an equivalent for that beauty, for it is a need of man's spirit.
The evolution of devices
All the first antecedents of man's devices were given him by Nature. Every one of his devices is traceable back to something in nature which suggested the first remote and primitive beginnings of its evolution. And every feature in art that man's mind conceived is conceived by a mind that has evolved as a part of nature: that grew out of nature.
The evolution of devices is as much a natural process as the evolution of organisms.
Deliberate acts
I do not know what one should call the landscape of a long cultivated countryside, or the enchanting pattern of lights which shows at night time in a modern city seen from overhead. Are these not works of art? It is scarcely justifiable to say that these things have taken shape by chance. Each part of them has been made as it is by what seemed a deliberate act, and it need not necessarily be assumed to be a matter of chance that the results of many acts of many men over a considerably period of time should harmonize together aesthetically.
The skill of perception
The newborn baby and the [blind man suddenly gifted with sight] do not have to learn to see. Sight is given to them. But they do have to learn to perceive. Perception is learnt and learnt slowly. Skill is required for perception as for speech. We are largely unaware of the skill we exercise. None of the things we have to learn to perceive are self-evident, or, apparently, instinctively evident. No doubt, however, we have an instinctive aptitude for this learning, and once we have learnt we cannot easily see as though we had not.
As Ruskin says, one has to strive, if one is to see with the 'Innocent Eye'.
Who did the teaching, then?
It has been contended sometimes that our response to works of art is entirely learnt and in no way innate; but the questions 'Who did the teaching, then? and how?' have not, I fancy, been much investigated. This contention is very true of our responses to styles and fashions, but it is not true of our response to beauty.
No kind
No kind of shape, no kind of design or kind of picture or other work of art can be beautiful. No kind of color is beautiful. Beauty comes always from the singularity of things. Two things which happen to be closely similar in size, color, insurance value, smell, weight, or shape, may both seem equally beautiful. It is not therefore to be deduced that, say, a smell of turpentine is a necessary prerequisite of beauty; and nor is the fact that the two things' shapes are measurably within a millimeter of each other. They might still be as different as chalk and cheese: they might differ hugely in surface quality so that one lived and the other was dead. One judges a man by what he is, by his individuality, his idiosyncrasy; not by his measurable properties or measurable behavior or by the shape of his nose or the description in his passport. So with a work of art.
Tradition
Change is of the essence of tradition. Our declining civilization has largely lost the conception of tradition as continuous change by small variations – as evolution, in other words – and can produce only fashions which, one after another, appear, live for a little while, and die without issue.
What a greenhouse was for
The new-found ability to make a wall all of glass had advantages, undoubtedly, in certain particular cases, but not in nearly so many as the Bauhaus stylists pretended. It is not forgotten by those who have to work in buildings with these glass walls that their propagators must have known quite well what a greenhouse was for and what it did. That knowledge counted for nothing beside the imperative necessity of showing how new the 'new architecture' was, by doing something obvious different from the fenestrated walls of the styles which had preceded it.
The act of creation
What I suggest has usually happened [during the act of creation] is this: the artist has glimpsed something: he has seen, perhaps fleetingly and indistinctly, some particular relation or quality of visible features which had previously been disregarded, and which impressed itself on him by its beauty. By means of making a work of art he then seeks as it were to fix isolate and concentrate what he has seen.
No one has ever succeeded in demonstrating in principle how this is done, but done it is; and when we see it done we find it hard to understand why it should have been so intensely difficult to do.
The imprint of a man
Art is the imprint of a man: a creature whose nature is idiosyncrasy sparring with conformity.
Déjà vu
The artists expression may make us aware for the first time of something we had too little regarded or had not been fully conscious of, presenting us with something which is quite new to us and yet at the same time disturbingly familiar – déjà vu.
The signature
It has long been understood that striving for originality as an end in itself is the mark of an inferior artist. The personal style of a good artist is never something that has been deliberately cultivated and forced but something that has appeared unsought as inevitably as the personal style of a man's handwriting.
But since artists of note are seen to have a distinct personal style, no artist can hope to make a reputation in a competitive society unless he too can show a distinctive style which easily differentiates his work from that of other artists and draws attention to it. Therefore artists of little capability or uncertain vocation will take great care to make their work look 'different', whereas those with any certainty in them will know that their work cannot help but look different from that of other people any more than signatures can.
It is worth reflecting that the fact of the unmistakable individuality of each man's signature is one foundation of modern commerce everywhere. To establish the individuality of it one need not write it vertically up the page in letters two inches high. And yet there are only twenty six letters, and everyone else uses them too.
It will not stand still to be pointed at
The cause of the experience of beauty is a series of events, not a state of affairs existing continuously. That perhaps is why the cause of the experience is something we find impossible to point out. It will not stand still to be pointed at. We can point out only what we perceive. We can never point out or describe what we see.