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.
Spatial Interfaces
Software applications can utilize spatial interfaces to afford users powerful ways of thinking and interacting. Though often associated with gaming, spatial interfaces can be useful in any kind of software, even in less obvious domains like productivity tools or work applications. We will see spatial interfaces move into all verticals, starting with game-like interfaces for all kinds of social use-cases.
There is no app that replicates a deck of cards
It seems that the only way to build an app that replicates the full functionality a deck of cards is to build a 3D simulation or game. Model the cards in 3D and put them on a 3D table. As long as you have controls for reaching out and picking up the cards, and moving them in space, you can do anything you can do in real life.
Preserving the higher dimensionality makes it simpler and more intuitive, not less.
Humans are spatial creatures
Humans are spatial creatures. We experience most of life in relation to space. We sit in a circle with our family. We drive down the left side of the road in Ireland and try to stay in our lane. We ride scooters down a path along the water. We sculpt a human body out of stone. We follow signs to our train. We walk, through arches, or on crosswalks beside bike lanes, or to follow our brother across a bridge. We position our camera to frame the shot while our friend leans out over a balcony. We sense ourselves in space in relation to all of the other objects in our environment.
And this is powerful knowledge that we've left out of lots of software. In fact, while most obvious in 3D, this thinking does work outside of just three-dimensional software. Almost any software can use spatial concepts to become easier to understand.
Web trails
There's more room for spatial concepts to become part of our web browsing experience.
One example is an idea I call "trails." It's based on the story of Hansel and Gretel walking through the forest and leaving a trail of breadcrumbs behind them, so that they could find their way back later. What if you could do this on the web?
A breadcrumb in this case is a single pixel that you can place in a precise location on a webpage. Placing a breadcrumb could be as simple as Option + click. While navigating the web, you could leave breadcrumbs on different pages you find interesting over the course of a browsing session. When you're done, that sequential "trail of breadcrumbs" would be saved. You could then jump back into the trail and navigate "forward" and "backward" through the things you found interesting in that browsing session. Or share the trail with a friend, and they could step through your spatial path of navigating the web.