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 Evolution of Useful Things
Here, then, is the central idea: the form of made things is always subject to change in response to their real or perceived shortcomings, their failures to function properly. This principle governs all invention, innovation, ingenuity.
Spike and spon
Using an older, pointed knife and spoon, a "spike and spon," to keep the fingers from touching food may have given us the phrase "spic and span" to connote a high standard of cleanliness.
Shaped and reshaped
[Inventions] do not spring fully formed from the mind of some maker, but, rather, become shaped and reshaped through the (principally negative) experiences of their users within the social, cultural, and technological contexts in which they are embedded.
Form follows failure
Imagining how the form of things as seemingly simple as eating utensils might have evolved demonstrates the inadequacy of a "form follows function" argument to serve as a guiding principle for understanding how artifacts have come to look the way they do. Reflecting on how the form of the knife and fork has developed, let alone how vastly divergent are the ways in which Eastern and Western cultures have solved the identical design problem of conveying food to mouth, really demolishes any overly deterministic argument, for clearly there is no unique solution to the elementary problem of eating.
What form does follow is the real and perceived failure of things as they are used to do what they are supposed to do. Clever people in the past, whom today we might call inventors, designers, or engineers, observed the failure of existing things to function as well as might be imagined. By focusing on the shortcomings of things, innovators altered those items to remove the imperfections, thus producing new, improved objects. Different innovators in different places, starting with rudimentary solutions to the same basic problem, focused on different faults at different times, and so we have inherited culture-specific artifacts that are daily reminders that even so primitive a function as eating imposes no single form on the implements used to effect it.
Their wrongness is somehow more immediate
In general, a successful design, which Alexander terms a good fit between form and context, can be declared only when we can detect no more [points that conform to the standard against which we judge]. It is "departures from the norm which stand out in our minds, rather than the norm itself. Their wrongness is somehow more immediate than the rightness."
A small corner of the world of things
It need not be only the likes of engineers, politicians, and entrepreneurs who have a hand in shaping the world and its things, for we are all specialists in at least a small corner of the world of things.
My job is simply to design gadgets that I like
Inventors are people who not only curse, but who also start to think of what can be done to eliminate the bother...When I see something that I don't like, I try to invent a way around it. My job is simply to design gadgets that I like.
More easily asked than definitively answered
Some design questions are more easily asked than definitively answered. Inventors are seldom at a loss for problems, and so they must choose which ones they will work on.
Only a commercial and utilitarian view
By World War II, we seem to have come to take new gadgets for granted or relied upon advertising to inform us of what was new. Whereas our great-grandparents apparently found the latest improvement on the fountain pen or the bicycle of intellectual interest, most people in our generation take only a commercial and utilitarian view of such things.
Stick like hell
When the Wizard of Menlo Park called invention 10 percent inspiration and 90 percent perspiration, he was speaking not only about the creative act of inventing but also about the whole inventive process needed to bring more than intellectual success. Edison warned against discouragement during the perspiration phase in the following way, reminding us that we get things to work by the successive removal of bugs:
Genius? Nothing! Sticking to it is the genius! Any other bright-minded fellow can accomplish just as much of he will stick like hell and remember nothing that's any good works by itself. You've got to make the damn thing work!...I failed my way to success.
This tactile form of doodling
Paper clips have also served as objects of more inwardly directed aggression by providing something for the fingers to twist grotesquely out of shape during phone calls, interviews, and meetings. This tactile form of doodling may consume only a fraction of the twenty billion paper clips produced each year, but it underscores the almost limitless functions to which a single form can lead.
Materials and how to employ them
Forming a paper clip presents a common dilemma encountered by engineers and inventors: the very properties of the material that make it possible to be shaped into a useful object also limit its use. If one were to try and make a paper clip out of wire that stayed bent too easily, it would have little spring and not hold papers very tightly. On the other hand, if one were to use wire that did not stay bent, then the clip could not even be formed. Thus, understanding the fundamental behavior of materials and how to employ them to advantage is often a principle reason that something as seemingly simple as a paper clip cannot be developed sooner than it is.
Sonorisms VI
A small corner of the world of things
This tactile form of doodling
The crowded past of reality
Infundibular cores
Whose form our hands have often grown to gloveThe need to dispense a product properly
Difficulties in getting Scotch tape off the roll, for example, prompted the development of a dispenser with a built-in serrated edge to cut off a piece squarely and leave a neat edge handy for the next use. (This provides an excellent example of how the need to dispense a product properly and conveniently can give rise to a highly specialized infrastructure.)
Intrapreneurship
The characteristic of 3M that enabled it to attain such diversity in its product line is a policy of what has generally come to be called "intrapreneurship". The basic idea is to allow employees of large corporations to behave within the company as they would as individual entrepreneurs in the outside world.
...It is 3M's policy (and that of other enlightened companies) to allow its engineers to spend a certain percentage of their work time on projects of their own choosing, a practice known as "bootlegging".
Customers will confer a favor on us
A leaflet printed in March, 1906, tacitly confessed to many difficulties. The instructions for applying the fastener were wordy and complicated. The sponsors of C-curity betrayed their own lack of security by stating: "Customers will confer a favor on us by reporting any difficulty in applying fastener, in which case we will send more detailed instructions." The "instructions for using" were not merely wordy but worried.
The inventive process was often a nonverbal one
Through the ages, the professional users of tools by and large have not needed to, been able to, or wanted to talk to outsiders about their implements. They did not need to because tools themselves are used to make other tools, and thus users could very often fashion a new tool with their old ones. If they did need to communicate the design for a new tool to someone outside their trade, they could do so without having to reveal the tool's intended use...Besides, the inventive process of conceiving a new tool was often a nonverbal one. Finally, craftsmen were unwilling to share information about their specialized tools because to do so would have been to give up their competitive edge and their value to those outside the craft.
The reflective craftsman
Specialized tools like bench shears have proliferated throughout history in part because craftsmen necessarily do the same task with the same tool over and over. After a while, the task becomes routine, and the craftsman is able to perform it with predictable skill. The most creative of artisans is frequently one who, in the midst of routine, pays attention to the details of the work and the tools that effect that work, and so it is that the reflective craftsman develops ideas for new and improved tools in the course of working with those that he perceives to limit his achievement or efficiency.
Form eschews function
Many of the most contemporary silverware patterns appear to be designed more for how the pieces look than for how they work...There is a kind of design that can ignore function entirely. We might say that this is a "form eschews function" school of design, and one that places considerations of aesthetics, novelty, and style above everything else.
But to design from the handle is to shoot from the hip when it comes to silverware, for the business end of the individual pieces is where the action is going to be. Though Emily Post may not have perceived that tradition emerges out of the minimization of failure, there is no excuse for a designer to overlook the fact. Yet this is exactly what modern product designers seem to do when they strive so hard for a striking new look that they throw out function with tradition.
Style consists in distinction of form
Writing about style in architecture, the nineteenth-century theorist Viollet-le-Duc asserted that "style consists in distinction of form," and complained that animals expressed this better than the human species. He felt that his contemporaries had "become strangers to those elemental and simple ideas of truth which lead architects to give style to their designs," and he found it "necessary to define the constituent elements of style, and, in doing so, to carefully avoid those equivocations, those high-sounding but senseless phrases, which have been repeated with all that profound respect which most people profess for that which they do not understand."
Such an unholy alliance
Something was wrong, according to Raymond Loewy, who admitted that, "with few exceptions, the [competitors'] products were good." He was "disappointed and amazed at their poor physical appearance, their clumsiness, and...their design vulgarity." He found "quality and ugliness combined," and wondered about "such an unholy alliance."
...Loewy was also "shocked by the fact that most preeminent engineers, executive geniuses, and financial titans seemed to live in an aesthetic vacuum," and he believed that he could "add something to the field." But, not surprisingly, the people he approached were "rough, antagonistic, often resentful."
Most advanced yet acceptable
There is an apparent reluctance among consumers to accept designs that are too radically different from what they claim to supersede, for when, for when familiar things are redesigned too dramatically the function they perform can be less than obvious and thus possibly suspect. Loewy summarized the phenomenon by using the acronym MAYA, standing for "most advanced yet acceptable."
Combinations and arrangements
Everything designed has an element of arbitrariness in its form. Loewy described how groups of his designers used to go about designing a new model automobile. Different groups were given different tasks, such as the front and rear of the car, and the conceptual work began, to be cut off at some predetermined time by deadlines that were imposed at the outset. After a time, there were "piles of rough sketches," and Loewy saw the design proceed as follows:
Now the important process of elimination begins. From the roughs, I select the designs that indicate germinal direction. Those that show the greatest promise are studied in detail, and these in turn are used in combination or arrangements with one another. A promising front treatment can be tried in combination with a likely side elevation sketch, etc. From this a new set of designs emerges. These are then sketched in detail. After careful analysis, they boil down to four or five.
Economy of material and labor
Whatever the comparative merits of [various bed framing methods], what is clear from Aristotle's Mechanica is that economy of material, and labor, was as much an issue in ancient times as it is now.
More profitable and a better buy
The bottom line is certainly of concern, both to those seeking profit and to those seeking value, but neither of these can be measured solely by the amount of dollars spent on production or product. The nonquantitative word "quality" conveys countless ways in which a more expensive thing might be more profitable and yet a better buy as well. The advantages of thicker metal in an automobile body can clearly be argued from various points of view, including resistance to denting and even simple snob appeal. Whereas the manufacturer can use these as selling points and also as justification for a higher price tag, the buyer can easily justify spending more for a car that will keep its appearance longer and provide a status symbol.
Substance over style
By the 1930s, the teardrop shape, known since the turn of the century to be the form of least resistance, was incorporated into Boeing and Douglas aircraft, and, being the contemporary artifact that best symbolized the future, the airplane set the style for things generally. The most static of mundane objects were streamlined for no functional purpose, and chromed and rounded staplers, pencil sharpeners, and toasters were hailed as the epitome of design.
...Though all design is necessarily forward-looking, all design or design changes are not necessarily motivated by fickle style trends. The best in design always prefers substance over style, and the lasting concept over the ephemeral gimmick.
When engineers refuse to leave well enough alone
In a column entitled "March of the Engineers," the humorist and social critic Russell Baker lamented the complexity and sophistication of his office's new telephone system...Baker closed his column by defining the new telephone system as "another bleak example of the horrors created when engineers refuse to leave well enough alone."
In The Design of Everyday Things, Donald Norman wrote that "new telephone systems have proven to be another excellent example of incomprehensible design."
To anticipate all the uses and abuses
Success depends wholly on the anticipation and obviation of failure, and it is virtually impossible to anticipate all the uses and abuses to which a product will be subjected until it is in fact used and abused not in the laboratory but in real life. Hence, new products are seldom even near perfect, but we buy them and adapt to their form because they do fulfill, however imperfectly, a function that we find useful.
Lighter, thinner, cheaper
The evolution of form begins with the perception of failure, but it is propagated through the language of comparatives. "Lighter", "thinner," and "cheaper" are comparative assertions of improvement, and the possibility of attaching such claims to a new product directly influences the evolution of its form. Competition is by its very nature a struggle for superiority, and thus superlative claims of "lightest," "thinnest," "cheapest" often become the ultimate goals. But the goals more often than not are incompatible. Thus, the lightest and thinnest crystal can be expected also to be the most expensive. But limits on the form of artifacts are also defined by failure, for too light and too thin a piece of crystal might hardly be usable.