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.
Winning by Design: The Methods of Gordon Murray
A case study of the working methods of one particularly successful designer in a highly competitive design domain - Formula One racing car design. Gordon Murray was chief designer for the very successful Brabham and McLaren racing car teams in the 1970s and 1980s. His record of success is characterised by innovative breakthroughs, often arising as sudden illuminations, based on considering the task from first principles and from a systemic viewpoint. His working methods are highly personal, and include intensive use of drawings. Personality factors and team management abilities also appear to be relevant. There are some evident similarities with some other successful, innovative designers
You need to make the step forward
Throughout a racing season there is constant, relentless pressure on the designer to keep making design improvements. But there is a limit to what can be achieved with any car design, before a jump has to be made to basically a new design, an innovation. As Gordon Murray says, ‘Given the situation and the pressure at any one time, you do get to the brick wall...I mean you're doing all these normal modifications, you know you can't go any quicker, you need to make the step forward.’
In the midst of the pressure, the fervour, the panic, he ‘used to get breakthroughs, I mean I used to get like suddenly a mental block's lifted.’
Drawing the bits
That's what is great about race car design, because even though you've had the big idea - the “light bulb” thing, which is fun - the real fun is actually taking these individual things, that nobody's every done before, and in no time at all try and think of a way of designing them. And not only think of a way of doing them, but drawing the bits, having them made and testing them.
Like designing things for the first time
Gordon Murray insists on keeping experience 'at the back of your mind, not the front' and to work from first principles when designing. For instance, in designing a component such as a suspension wishbone, 'it's all too easy - and the longer you're in design the easier it is - to say, I know all about wishbones, this is how it's going to look because that's what wishbones look like.' But if you want to make a step forward, if you're looking for ways of making it much better and much lighter, than you have to go right back to load path analysis. It is like designing things for the first time, rather than the nth time.
Wonder Plots
Working from first principles, and working in a highly organized way seem to come naturally to him, but his personal design process is much less structured than the results might suggest. Although he can tightly organize his team and run a complex racing organisation, his personal ways of designing are relatively unstructured, based on annotated, thumb-nail sketches. ‘I don't sit down and say, OK, now I've had the idea, let's see, this is a solution, these are the different ways to go, if I do this, and do that; I do lots of scribbles just to save it, before I forget.’
Gordon’s design process is based on starting with a quick sketch of a whole idea, which is then developed through many different refinements. ‘I do a quick sketch of the whole idea, and then if there's one bit that looks good, instead of rubbing other bits out, I'd put that bit to one side; I'd do it again and expand on the good bit, and drop out the bad bit, and keep doing it, doing it; and end up with all these sketches, and eventually you end up throwing ninety percent of these away.’ He also talks to himself - or rather, writes notes to himself on the sketches; notes such as ‘rubbish’, ‘too heavy’ or ‘move it this way 30mm.’ Eventually he gets to the stage of more formal, orthographic drawings, but still drawing annotated plans, elevations and sections all together, ‘Until at the end of the day the guys at Brabham used to call them “Wonder Plots”, because they used to say “It's a wonder anybody could see what was on them”!’
I never have engineers that aren't designers
Although Gordon Murray carried immense personal responsibility for the design work of his racing cars, inevitably it involved a lot of teamwork. Clearly he has been successful in inspiring others to work with him. He likes to involve team members in the design problems, and for that reason prefers to recruit all-rounders to his team; ‘I never have engineers that aren't designers.’
The problem with CAD
He also likes to work collectively, standing around a drawing board discussing problems and trying ideas.
For this kind of teamwork, and especially for conceptual design work, he finds computer aided design systems too restrictive. For the McLaren F1 super-car, he installed a five-metre long drawing board in the design office, so that the car could be drawn full size. ‘The problem with CAD for this sort of stuff is that you can never have a full-size drawing, unless you do a print, and by the time you do a print it's out of date in the concept stage.’ He also does not like the one-person emphasis of CAD screens; ‘You can only ever talk to one person at once - you stand behind and look over somebody's shoulder, which is not very good for a boss-designer relationship anyway, to have somebody standing behind you is never a good thing. To look over somebody's shoulder at a tiny little screen, it's just wrong, it's totally wrong.’
(On the other hand, he fully acknowledges that tasks like a complex suspension plot to determine the wheel envelope are ideal for CAD.)
Drawing as a means of thinking
Two-dimensional plans or sections can be seen with sketches and more diagrammatic marks all on the same piece of paper in what appears a confusing jumble.’ These sound like Gordon’s ‘wonder plots’. The architects also use their drawings as a means of thinking ‘aloud’, or ‘talking to themselves’, as Gordon put it. For example, Lawson reports the architect Richard MacCormac as saying, ‘I use drawing as a process of criticism and discovery’; and the engineer-architect Santiago Calatrava as saying, ‘To start with you see the thing in your mind and it doesn’t exist on paper and then you start making simple sketches and organizing things and then you start doing layer after layer.... it is very much a dialogue.’
The common elements in these similar descriptions are the use of drawing not only as a means of externalising cognitive images but also of actively ‘thinking by drawing’, and of responding, layer after layer and view after view, to the design as it emerges in the drawings. These observations also confirm Schön’s observation of designing as a ‘reflective conversation’ between the designer and the emerging design. It is the reliance on drawing, and the preference for the immediacy of the interaction and feedback that manual drawing gives, that makes the architects, like Gordon Murray, unenthusiastic about CAD as a conceptual design tool.
A new gestalt
The innovator has a systems mind, one that sees things in terms of how they relate to each other in producing a result, a new gestalt that to some degree changes the world.
Intense activity, then relaxation
The working style is based on periods of intense activity, coupled with other periods of more relaxed, reflective contemplation. This working style may not be a reflection of a particular personality trait, but a necessary aspect of creative work, which requires alternating intense effort with relaxation.
Strategic, not tactical
The working methods of the innovative designer are, for the most part, not systematic; there is little or no evidence of the use of systematic methods of creative thinking, for example. The innovative designer seems to be too involved with the urgent necessity of problem solving to want, or to need, to stand back and consider their working methods. Their design approach is strategic, not tactical.
Drawing for parallel design thinking
An important feature of their strategy is parallel working - keeping design activity going at many levels simultaneously. The best cognitive aid for supporting and maintaining parallel design thinking is drawing. Drawing with the conventional tools of paper and pencil gives the flexibility to shift levels of detail instantaneously; allows partial, different views at different levels of detail to be developed side by side, or above and below and overlapping; keeps records of previous views, ideas and notes that can be accessed relatively quickly and inserted into the current frame of reference; permits and encourages the simultaneous, non-hierarchical participation of co-workers, using a common representation.
The drawing of partial solutions or representations also aids the designer’s thinking processes, and provides some ‘talk-back’. As well as drawing, innovative designers frequently like to undertake practical work related to the design solution, such as building models or mock-ups, or participating in construction.
A small team of committed coworkers
The innovative designer also likes, perhaps needs, to work with a small team of committed co-workers who share the same passions and dedication.