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 Beauty of Everyday Things
The Beauty of Kasuri
An EssayWashi
An EssayHandmade washi (traditional Japanese paper) is replete with appeal. Looking at it, touching it, fills me with an indescribable sense of satisfaction. The more beautiful it is, however, the more difficult it is to put to use. Only a master of calligraphy could possibly add to its beauty; it is exquisite just as it is. This is wonderfully strange, for it is merely a simple material. Yet plain and undecorated as it is, it is alive with nuanced beauty. Good washi makes possible our most ambitious creative dreams.
The Characteristics of Kogin
An EssayHandicrafts and Sesshu
I have almost never judged a work of art by first looking at its signature. This way of assessment holds no interest for me. If what I see is good, it is good with or without a seal.
Whether it is a painting or a pot, you must first look at the thing itself.
What is Folk Craft?
An EssayThe Beauty of Miscellaneous Things
An EssayA Painted Karatsu as Food for Thought
Recently there is a tendency to pursue distortion in art, but in the case of this jar, natural deformation has raised distortion to the level of spontaneous beauty.
Okinawa's Bashofu
The users of bashofu cloth are ordinary people, not the wealthy. It is used for the kimonos they wear every day. It is not something they buy with a highly appreciative aesthetic eye, comparing one piece with another as objects of art. It is bought as a mundane item and worn as a part of mundane life. Still, bashofu is beautiful just as it is. Here the idea that you get what you pay for does not apply. The cheap is the good and beautiful.
Seeing and Knowing
An EssayThe results of intuition can be studied by the intellect, but the intellect cannot give birth to intuition.
What is Pattern?
Since a pattern is the depiction of the fundamental nature of an object, it is what remains of an object’s form after all that is unnecessary has been removed.
Since a pattern is a crystallization, it is also an exaggeration. But it is not merely that; it is an accentuation of the truth.
Woodblock Prints
An EssayIt seems to me that many printmakers are suffering under a delusion. Looking at current trends, it appears that recent prints are simply copying fine art and painting. Some printmakers are working in the nanga style of painting. Others are attempting to reproduce the effects of oil. Some cleverly contrived prints are often difficult to distinguish from paintings done with a brush. The question arises: Why are these printmakers working in the medium of woodblock printing at all?
For prints to follow in the footsteps of painting has very little meaning. The art of the brush and palette should be left to the brush and palette.
The Japanese Perspective
An EssayGenerally speaking, the Western perception of art has its roots in Greece. For a long time its goal was perfection, which is particularly noticeable in Greek sculpture. This was in keeping with Western scientific thinking; there are no painters like Andrea Mantegna in the East. I am tempted to call such art ‘the art of even numbers’.
In contrast to this, what the Japanese eye sought was the beauty of imperfection, which I would call ‘the art of odd numbers’. No other country has pursued the art of imperfection as eagerly as Japan.