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.
Interaction of Color
The deception of color
In order to use color effectively it is necessary to recognize
that color deceives continually.What counts here – first and last – is not so-called knowledge
of so-called facts, but vision – seeing.Practice before theory
Instead of mechanically applying or merely implying laws and rules
of color harmony, distinct color effects are produced
– through recognition of the interaction of color –
by making, for instance,
2 very different colors look alike, or nearly alike.The aim of such study is to develop – through experience
– by trial and error – an eye for color.
This means, specifically, seeing color action
as well as feeling color relatedness.As a general training it means development of observation and articulation.
This book, therefore, does not follow an academic conception
of “theory and practice.”
It reverses this order and places practice before theory,
which, after all, is the conclusion of practice.50 reds
If one says “Red” (the name of a color)
and there are 50 people listening,
it can be expected that there will be 50 reds in their minds.
And one can be sure that all these reds will be very different.Not the what but the how
Our concern is the interaction of color; that is, seeing
what happens between colors.We are able to hear a single tone.
But we almost never (that is, without special devices) see a single color
unconnected and unrelated to other colors.
Colors present themselves in continuous flux, constantly related to
changing neighbors and changing conditions.As a consequence, this proves for the reading of color
what Kandinsky often demanded for the reading of art:
what counts is not the what but the how.Scotopic seeing
The sensitivity
and consequently the registration of the retina of an eye is different
from the sensitivity and registration of a photographic film.Normally, black-and-white photography registers all lights lighter
and all darks darker than the more adjustable eye perceives them.
The eye also distinguishes better the so-called middle grays,
which in photography are often flattened if not lost.This shows what a higher key in light can lose in photography.
The greatest advantage the eye has over photography
is its scotopic seeing in addition to its photopic seeing.
The former means, briefly, the retinal adjustment to lower light conditions.Disliked colors
We try to recognize our preferences and our aversions –
what colors dominate in our work; what colors, on the other hand,
are rejected, disliked, or of no appeal. Usually a special effort
in using disliked colors ends with our falling in love with them.Color intervals
The tune of “Good morning to you” consists of 4 tones. It can be sung
in a high soprano, a low basso, and in all in-between voices, as well as
on many levels and in many keys. It can be played on innumerable instruments.In all possible ways of performance, this melody will keep its character
and it will be recognized instantly.Why? The intervals of the 4 tones, that is, their acoustical
constellation (again comparable with a topographical relationship),
remains the same.Although it is not common practice, one can also speak of intervals
between colors.
Colors and hues are defined, as are tones in music, by wavelength.Any color (shade or tint) always has 2 decisive characteristics:
color intensity (brightness) and light intensity (lightness).
Therefore, color intervals also have this double-sidedness, this duality.Time and space
Tones appear placed and directed predominantly in time from before to now to later.
Their juxtaposition in a musical composition is perceived
within a prescribed sequence only.
Horizontally, the tones follow each other,
perhaps not in a straight line, but of necessity in a prescribed order
and only in 1 direction – forward.
Tones heard earlier fade, and those farther back disappear, vanish.
We do not hear them backward.Colors appear connected predominantly in space. Therefore,
as constellations they can be seen in any direction and
at any speed. And as they remain, we can return to them repeatedly
and in many ways.
This remaining and not remaining, or vanishing and not vanishing,
shows only 1 essential difference between the fields of tone
and color.The accuracy of perception in one field is matched
by the durability of retention in the other, demonstrating
a curious reversal in visual and auditory memory.A cook with taste
Observe the interior and exterior, the furniture and textile decoration
following such color schemes, as well as commercialized color “suggestions”
for innumerable do-it-yourselves.Our conclusion: we may forget for a while those rules of thumb
of complementaries, whether complete or “split”, and of triads and
tetrads as well.
They are worn out.Second, no mechanical color system is flexible enough
to precalculate the manifold changing factors, as named before,
in a single prescribed recipe.Good painting, good coloring, is comparable to good cooking.
Even a good cooking recipe demands tasting and repeated tasting
while it is being followed.
And the best tasting still depends on a cook with taste.Flexible imagination
By giving up preference for harmony,
we accept dissonance to be as desirable as consonance.Besides a balance through color harmony, which is comparable
to symmetry, there is equilibrium possible between
color tensions, related to a more dynamic asymmetry.Again: knowledge and its application is not our aim;
instead, it is flexible imagination, discovery, invention – taste.Does it have color?
Whether something “has color” or not is as hard to define verbally as are
such questions as “what is music” or “what is musical."The Weber-Fechner law
Exponential increases in physical stimuli produce linear perceptual increases.
Thinking in situations
Naturally, practice is not preceded but followed by theory.
Such study promotes a more lasting teaching and learning
through experience. Its aim is development of creativeness
realized in discovery and invention – the criteria of creativity,
or flexibility, being imagination and fantasy. Altogether
it promotes “thinking in situations,” a new educational concept
unfortunately little known and less cultivated, so far.Not of method but of heart
In the end, teaching is a matter not of method but of heart.
The teacher actually is right and always will gain confidence
when he admits that he does not know, that he cannot decide, and,
as it often is with color, that he is unable to make a choice
or to give advice.Besides, good teaching is more a giving of right questions
than a giving of right answers.Results of a search
This book presents results of a search, not of what is academically called research.
In addition to the dedication of this book, I should like to state that my students in color have taught me more color than have books about color.
Simple forms
The concept that “the simpler the form of a letter the simpler its reading” was an obsession of beginning constructivism. It became something like a dogma, and is still followed by “modernistic” typographers.
This notion has proved to be wrong, because in reading we do not read letters but words, words as a whole, as a “word picture.” Ophthalmology has disclosed that the more the letters are differentiated from each other, the easier is the reading.
Without going into comparisons and the details, it should be realized that words consisting of only capital letters present the most difficult reading—because of their equal height, equal volume, and, with most, their equal width. When comparing serif letters with sans-serif, the latter provide an uneasy reading. The fashionable preference for sans-serif in text shows neither historical nor practical competence.