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.
Stealth Architecture: The Rooms of Light and Space
To absorb it or build your own
Robert Smithson and other so-called land artists simply disengaged from architecture, placing their works in America's open landscape, leaving behind the museums and galleries Smithson referred to as "tombs". A new "expanded field" allowed artists to contextualize their work beyond the institutional frame of the museum or the commercial structure of a gallery. Richard Serra, who also began to move outdoors, at times chose to "attack" architecture, creating structures that disrupted or overwhelmed the buildings around them.
The artists of the Light and Space movement took another tack. Rather than fight or flee the architecture, they explored and manipulated it, approaching architecture as a kind of found object, creating a series of rooms that incorporated architecture and architectural structures directly into their art. Bruce Nauman summarized it well: "When you work in a gallery or museum, the architecture is a given. If you wanted to have a show, you didn't have a choice, except to deal with it. You had to find a way to either absorb architecture into the piece of build your own."
A stealth architect
By the 1970s, Irwin was in effect a stealth architect. We often talk about the ephemeral qualities of light and space in Irwin's installations, but what make those qualities palpable to our perception are practical structures—windows, walls, corridors, doorways, and skylights—in other words, architecture. And Irwin was keenly aware of how best to use all of those structures. One of his greatest talents has been to engage bad or benign architectural situations, disappearing into their details, changing them, and creating and entirely new quality of space.
The measuring unit of all space
The piece was titled The Portal, referring to a large opening in the center of the wall. Whether you want to call it art or architecture, it was a testament to the amazing presence that can be shown by a simple wall, which [Tadao Ando] has referred to as "the measuring unit of all space."
The walls are reserved for the sun
Maria Nordman always insisted, "Nothing should hang on a wall. The walls are reserved for the sun." It was like being inside a large cardboard box that had been gently slit open with an X-Acto knife, allowing thin planes of light to emerge. It is well known that Nordman avoided using the camera to document her installations, feeling that it abstracted and framed various aspects of the experience, which is best absorbed more holistically. It is ironic that Nordman's rooms often took the form of a kind of architectural camera in which slits in walls and corners created mysterious apertures that allowed light to leak into a room at a glacial pace. Being inside one of Nordman's spaces is like being inside a camera operating in exceedingly slow motion.
A little too something
As Irwin had chosen a stairwell for his UCLA installation because it was curious in its banality and innocuousness, Bruce Nauman became interested in corridors and shafts as overlooked and slightly eccentric spaces. He was particularly interested in those that had "a kind of constriction that wasn't natural or was a little too long or a little too something—like the architect just hadn't really thought it out."
Various titles of Bruce Nauman artworks
- Sound Breaking Wall
- Get Out of My Mind, Get Out of This Room
- False Silence
- Flayed Earth Flayed Self (Skin/Sink)
- Room with My Soul Left Out, Room That Does Not Care