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.
Matter versus Materials: A Historical View
Atoms and aggregates
I see science reversing the trend toward atomistic explanation that has been so triumphant in the last 400 years, and I predict a more human future based on the symbiosis of exact knowledge (which is by its very nature limited) and experience.
...Matter cannot be understood without a knowledge of atoms; yet it is now becoming evident that the properties of materials that we enjoy in a work of art or exploit in an interplanetary rocket are really not those of atoms but those of aggregates...It is not stretching the analogy much to suggest that the chemical explanation of matter is analogous to using an identification of individual brick types as an explanation of Hagia Sophia.
Whose eyes had seen and whose fingers had felt
Aristotle’s 18 qualities of homoeomerous bodies that he chose to explain in detail in his Meteorologica, are just those fine points of behavior that would be noticed in a workshop. They are:
solidifiable
meltable
softenable by heat
softenable by water
flexible
breakable
fragmentable
capable of taking an impression
plastic
squeezable
ductile
malleable
fissile
curable
viscous
compressible
combustible
capable of giving off fumesThis redundant list of properties is not the neat classification of a philosopher. It reads more as if it were based on a conversation with a workman whose eyes had seen and whose fingers had felt the intricacies of the behavior of materials.
The alchemists in their mixings
Many wonderful things must have been seen by the alchemists in their mixings.
A holograph of itself
All [physical properties of matter] derive from the different patterns of the interaction of electrons and photons within the fields of the positively charged atomic nuclei, stabilized in a particular morphology by the interaction of the levels themselves. Matter is a holograph of itself in its own internal radiation.
To worship at the shrine of mathematics
The new [physics-based] viewpoint is so potent that it has perhaps, caused too many metallurgists to forsake their partially intuitive knowledge of the nature of materials to worship at the shrine of mathematics, a trend reinforced by the curious human tendency to laud the more abstract.
What the advancing interface leaves behind
I see in the complex structure of any material—biological or geological, natural or artificial—a record of its history, a history of many individual events each of which did predictably follow physical principles. Nothing containing more than a few parts appears full panoplied, but it grows. And as it grows, the advancing interface leaves behind a pattern of structure perfection or imperfection which is both a record of historical events and a framework within which future ones must occur.
A realization that this leaves out something essential
Nothing so fundamental lies in the realm of concern to us aggregate humans, where the need is, now, for the study of real complexity, not idealized simplicity. In every field except high-energy physics on one hand, and cosmology on the other, one hears the same. The immense understanding that has come from digging deeper to atomic explanations has been followed by a realization that this leaves out something essential. In its rapid advance, science has had to ignore the fact that a whole is more than the sum of its parts.