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.
Seeing With Fresh Eyes
Meaning
Space
Data
Truth
The problem with trees
Many systems are organized hierarchically. The CERNDOC documentation system is an example, as is the Unix file system, and the VMS/HELP system. A tree has the practical advantage of giving every node a unique name. However, it does not allow the system to model the real world. For example, in a hierarchical HELP system such as VMS/HELP, one often gets to a lead on a tree such as:
HELP COMPILER SOURCE_FORMAT PRAGMAS DEFAULTS
only to find a reference to another leaf: Please see
HELP COMPILER COMMAND OPTIONS DEFAULTS PRAGMAS
and it is necessary to leave the system and re-enter it. What was needed was a link from one node to another, because in this case the information was not naturally organized into a tree.
Content-responsive space
Content-responsive spaces in text can be as meaningful as spaces and line breaks in computer code, poetry, math, dialogues, cartoons.
For 1500 years, printed text has used grids indifferent/hostile to meaning. Content-responsive grids are better than imperious grid-possessed layouts. To clarify and intensity meaning, authors and editors can remodel relations between spaces and words...insisting on control of line breaks by authors (who, after all, know the content).
Personal annotations
Yehudi Menuhin, a great violinist, marked up a score for Bach's Sonata No. 2 for Solo Violin. Penciled annotations show real-time performance strategies. To outsiders, insider markups appear chaotic and cryptic, but these personal annotations are for Menuhin's eyes, the only eyes that matter. All can learn from this useful workaday grid strategy: a relevant and intense data layer can become a coherent substrate scaffold upon which to overlay additional information. Maps do this all day long.
What excellence is
Learn what excellence is, how to identify it...This is not a big reading assignment – excellence is scarce, lognormal, long-tailed. Acting on this knowledge is liberating, freeing oneself from vast piles of triviality, knock-offs, petty connoisseurship, over-publishing, and the short-sighted, trendy, greedy. Excellence is long-term knowledge, even forever knowledge.
Excellence, like good taste, is perhaps a universal quality. Analytical thinking is about the relationship between evidence and conclusions, and is fundamental to all empirical work, regardless of field, discipline, specialty. Thus it is possible at times to assess credibility of nonfiction work without being a content expert. Thinking eyes may well have an eye for excellence, regardless of field or discipline.
Sentences and words do not exist by themselves
Sentences and words do not exist by themselves, but have natural, inevitable, unavoidable interactions with their surrounding spaces, words, and other sentences. Sentences are not independent of their spatial context, and interactions can create meanings and harms. Sentences survive content-indifferent and content-hostile spacings, but surviving is not thriving. Text space should not be owned and governed by generic productions grids, which make for convenient production but inconvenient meaning. Space can and should be content-responsive, actively contributing to meaning – forever practices in poetry, maps, math, computer code, comics, theater/movie scrips, posters. Subtle visual spacing differentiates and clarifies sentences, and meaning becomes more consequential, memorable, retrievable.
Central-axis text
Central-axis provides a clear signal of the next line, so that readers and speakers don't have to search on the left margin, sometimes accidentally skipping down a line. Ragged-left typography is used for dialogue in novels and scripts. In central-axis, each line is activated at both left and right margins – unlike squared-off conventional text. Readers/speakers are aware of the length of the next line at both its beginning and end. That knowledge may also help readers detect the pace and rhythm of the words, as in reading poetry aloud.
Idiosyncratic paragraphs
Text-only paragraphs differ from one another only in their words. All the words are typographically the same – typeface, spacings, line-lengths piled up into long deep columns. Systematic regularity of text paragraphs is universally inconvenient for readers, who are unable to find and read once against a specific string of words in previously-read paragraphs. All readers have encountered this problem in essays, articles, novels, news reports. Idiosyncratic paragraphs assist memory and retrieval by readers, by uniquely activating the relevant neural substrates for retaining visual memories. Nearly every paragraph in this book is deliberately unique.
No more LittleDataGraphics
Small data sets should be shown directly...LittleDataGraphics (pie charts, bar charts) translate and encode data into areas and colors. Viewers must then mentally translate codes back into numbers. These codes are unique to the local sets of data graphics, and do not repay learning. Instead, just directly show numbers as numbers. No more LittleDataGraphics. Data visualizations are at their best when there is so much data that the only way to see it...is to see it.
An immense wordy diagram
In ~1560 Ettore Ausonia, a polymath with interests from mathematics to mirror-making, constructed an immense wordy diagram depicting reflections from concave spherical mirrors. Then, between 1592 and 1601, while teaching at the University of Padua, Galileo made this handwritten copy of the diagram, which was fortunate since Ausonio's original has since gone missing. Three helpful architectures for the off-the-grid sentences are deployed – word trees, stacklists, annotated linking lines.
Stacklists
Stacklists organize and clarify complex material in 2-space. Readers read more slowly, and that's good: to think, look again, and connect words vertically within each stack and horizontally between stacks. Instead of polyphony, conventional inline lists are a freight train of words along a one-way narrow track, making it difficult to identify which words belong to which list and to link and compare elements within and between lists.
Observe data collection at the moment of measurement
See, observe, learn how data are collected at moment and place of measurement. "You never learn more about a process than when you directly observe how data are actually measured," said Cuthbert Daniel, a superb applied statistician. See with fresh eyes. Walk around what you want to learn about. Talk to those who do measurements. See how numbers came to be.
Do those measuring know the desired answer? Are those measuring skilled, alert, honest, biased, incompetent, sloppy, tired and emotional?...Artifacts and errors in measurements measured? How are outliers adjudicated?
46 data quality issues in spreadsheets
Outcomes decide
High levels of U.S. patient satisfaction are mainly associated with hospitality (greeters at the door, empathetic staff, comfortable rooms) – but also with more treatments, high costs, and substantially higher mortality even after adjusting for baseline health and comorbidities. Several plausible stories explain these big n and replicated observational findings. Whatever the case, post-treatment patient satisfaction/gratitude does not measure whether a treatment works or not. Patient outcomes decide.
Which half?
One day when I was a junior medical student, a very important Boston surgeon visited the school and delivered a great treatise on a large number of patients who had undergone successful operations for vascular reconstruction. At the end of the lecture, a young student at the back of the room timidly asked, "Do you have any controls?" Well the great surgeon drew himself up to his full height, hit the desk, and said, "Do you mean did I not operate on half of the patients?" The hall grew very quiet then. The voice at the back of the room hesitantly replied, "Yes, that's what I had in mind." Then the visitor's fist really came down as he thundered, "Of course not. That would have doomed half of them to their death!" God, it was quiet then, and one could scarcely hear the small voice ask, "Which half?"
Good annotation
Information displays should be annotated, combining words, images, graphics, whatever it takes to describe and explain something. Annotation calls out and explains information and, at the same time, explains to viewers how to read data displays. Good annotation is like a knowledgeable expert/teacher at the viewer's side pointing and saying, "Now see how this works with that, how this might explain that..."
Signs seen and unseen
Direct instructions at point of need may encourage writers and programmers to divert diversions. Or not, because signs are seen only a few times before becoming unseen.
What is the strongest visual element?
Preparing to write the novel Catch 22, Joseph Heller composed a storyboard, a 2-dimensional list with 3,650 words arrayed in 34 × 21 = 714 interacting cells. Rows are ordered in time, and each row records when each character does what. Some cell entries are erased. It took 7 years to complete the novel's 758-page typescript.
The Catch 22 plotchart works better upon replacing optically noisy grids with ghost grids. Lightness of framing lines creates soft boundaries to maintain order and also allows words to spill across cells naturally...More generally, ask of information displays and interfaces, "What is the strongest visual element?" The correct answer is not "grid lines".
Your only language is vision
To see with fresh, uninstructed eyes and an open mind requires a deliberate, self-aware act by the observer. Abstract artworks represent themselves and should be first viewed for themselves. When looking at outdoor abstract pieces, concentrate initially on the unique optical experience produced by the artworks. See as the artist saw when making the piece.
A focus on optical experience does not deny stories, it postpones them. Viewing an artwork may evoke interesting narratives – or just tedious artchat recalling similar art or artists, concocting playful tales, realizing how scrap metal was repurposed into art, making judgments about the artist's intentions or character, or contemplating an artwork's provenance, price, politics. Let the artwork stand on its own. Walk around fast and slow, be still, look and see from
up down sideways close afar above below
, enjoy the multiplicity ofsilhouettes shadows dapples clouds airspaces sun earth glowing
. Your only language is vision.Lists consist of whatever it takes
Lists consist of whatever it takes – nouns, proper nouns, verbs, graphics, images, numbers.
...In lists, spaces have meaning, locating elements in relation to other elements. Lists are often free and independent from conventional rules of stylesheets / grammar / typography / punctuation. Lists also help us escape from the personal internalized mash-up stylesheets of every writer and reader – a continuous low-level background buzz checking to see if word usage, spelling, punctuation, grammar are 'correct'. Lists are all content – about the substance contained, not the container. An empirical theory here for reasoning about lists includes
selection of list items
list quality and completeness
comparing list models
comparing list architecturesNo wonder you think it's complicated
We were very proud of our user interface and the fact that we had a way to browse 16,000 (!!) pages of documentation on a CD-ROM. But browsing the hierarchy felt a little complicated to us.
So we asked Tufte to come in and have a look, and were hoping perhaps for a pat on the head or some free advice. He played with our AnswerBook for 90 seconds, turned around, pronounced his review:
"Dr. Spock's Baby and Child Care is a best-selling owner's manual for the most complicated 'product' imaginable – and it has only 2 levels of headings. You have 8 levels of hierarchy and I haven't stopped counting yet. No wonder you think it's complicated."
Verb List
Documents vs. decks
Decks are easier to prepare than documents, however. Documents require coherence, thinking, sentences. But convenience in preparing decks harms the content and the audience. Optimizing presenter convenience is selfish, lazy, and worst of all, replaces thinking.
Books are meant to be used
During most of the course students have their books open, either to read or to follow along. Students are encouraged to annotate the books: "Books are meant to be used. Dog-ear the pages, mark them up, put notes in the spacious margins."
Learning via teaching
The course material changes 15% each year, as the book currently in progress becomes part of the course years before it is finally published. I detect incoherencies and mistakes in the new material while teaching. This leads to refinements or even throwing stuff out from the forthcoming book. A good way to learn about something is to teach it.
Self-publishing, self-exemplifying
I sought to design [my first book] so as to make it self-exemplifying – that is, the physical object itself would reflect the intellectual principles advanced in the book. Publishers seemed appalled at the prospect that an author might govern design. Consequently I decided to self-publish the book.
...[Howard Gralla and I] spent the summer in his studio laying out the book, page by page. We integrated graphics into the text, sometimes in the middle of sentences, eliminating the usual segregation of text and image – one of the ideas Visual Display advocated.
My view on self-publishing was to go all out, to make the best and most elegant and wonderful book possible, without compromise. Otherwise, why do it? The next 4 books were financed by the previous books. I have never written a grant application.
A history of content and sources
Not all that many readers go to the back matter and look up the source for a single sentence. But the back matter can also be read as ordinary text, revealing a history of content and sources. And images and illustrations from the book in the back matter create a lovely visual/verbal summary quilt of the entire book, enjoyed by all.