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.
My Life as an Architect in Tokyo
World renowned architect Kengo Kuma presents an enlightening tour of Tokyo, expressing his personal thoughts and reflections on the city's most influential buildings and its rich architectural heritage.
A collection of villages
I became a 'border person', as defined by the sociologist and philosopher Max Weber, viewing Tokyo from an outsider's perspective. Observing the city while walking around its streets enabled me to discover a wide variety of location, cultures and people, and that Tokyo is a collection of small villages, rather than one big city.
...When I design a building in any city, I believe that the world is a collection of villages, instead of a group of nations.
Low wooden silhouettes
While [Kenzo] Tange aspired to verticality, we looked to horizontality, believing that pre-1964 Tokyo, with its low wooden silhouettes, was a better model for the city of the future.
Occupied by a void
Roland Barthes wrote that the centre of Tokyo is occupied by a void...it is a quiet forest that lies at Tokyo's heart.
...The centre of Tokyo is certainly a void, but one that is protected by a circular train line, the Yamanote, which forms a 40-km (25-mile) loop around it. It seems to me that this ring of steel emphasizes the importance of the void, and the depth of its significance.
Such an enormous machine
In cities across the world, industrial zones beside rivers and canals have become the focus of attention, with their unique vivacity associated with places where things are made.
...Because the area is designated as a semi-industrial zone, we were able to get away with such an enormous machine inside [the Starbucks Reserve Roastery].
A more spiritual place
In the centre of the forest is the sandō, leading up to the shrine. It follows an L-shaped curve, and is very different to the straight processional pathways found in religious buildings in the West or in China. Curves ensure that the view changes constantly, helping visitors make the transition to a deeper, more spiritual place.
The building as less important than the path
In the design of Japanese tea houses, the building is seen as less important than the path (roji) leading up to it, and tea masters of the past believed that the journey along the roji allowed participants to better immerse themselves in the slow time of the tea ceremony.
The gentle light of shoji screens
Le Corbusier, the greatest architect of the last century, noted that 'architecture is the learned game, correct and magnificent, of forms assembled in light', demonstrating to what extent light has been prioritized in the Western tradition. Tanizaki, on the other hand, spoke of the important of shadows, of extended eaves. Rather than the light that shines directly into a room, he praised the soft light that penetrates a space after being reflected off the floor, and again from the ceiling.
...In Japanese architecture, the gentle light that passes through shoji screens serves a key purpose. It reaches right to the back of the room, so that the space feels bright, even without the aid of artificial light. The soft light filtering through the white film at Takanawa Gateway Station represents a form of light that was forgotten about by Japanese Modernism.
The thin lip of a teacup
To give the building a sense of the delicacy associated with such crafts, as well as a feeling of warmth, I designed louvres from white porcelain panels, and used them to cover the outer walls. The louvres are tapered, to make their tips as fine as possible. (In fact, making tips as thin as possible is one of my key design principles: the thin lip of a teacup allows a better experience of the subtleties of tea - this is always at the forefront of my mind when I pay such close attention to edges.)
Skyscrapers are frowned upon
During the twentieth century, much importance was attached to things that were big and tall, but, as we moved into the twenty-first century, I felt that being big and tall had become embarrassing.
...Today, skyscrapers are frowned upon in Japan, and are seen as the product of the mistaken mindset that prevailed during the country is period of enhanced growth.
The Metabolist philosophy
Tange put Tsukiji as the centre of his plan, which now seems grandiose and delusional. His design for the Dentsu building had much in common with the Metabolist philosophy of the 1960s, which maintained that buildings needed to continually evolve in a flexible way.
...With [the Nakagin Capture Tower], Kurokawa's Metabolist philosophy was fully realized. After it was completed, however, it became almost impossible to switch over the capsules - indeed, since its completion, not one of the capsules has been moved. As a result, the Metabolist movement has been forgotten. Yet its core principles, which sought to draw architectural lessons from living organisms, has much inspiration to offer society today.
A city of hills
Many of the stations [on the Yamanote Line] have one entrance on the uphill side and another lower down, and the neighbourhoods around them have a totally different feel, depending on which exit you use to leave the station.
The hilly areas in Tokyo are mostly made up of quiet, well-to-do residential districts, while the lower sections often have more of a populist feel, with shopping arcades and small urban factories. As a result, the atmosphere outside the entrances are dramatically different in character. Take the wrong exit, and you might find yourself lost in a completely different kind of neighbourhood than you were expecting. In Tokyo, elite and working-class cultures exist alongside one another and mix together. I think the fundamental cause of this is the complexity of the city's topography.
...Tokyo is a city of hills, with most of it lying on an alluvial plain between the Tama and Kanda rivers. It is via these hills that the upland, elite neighbourhoods are connected with the more working-class areas down below. The slopes are thus a key part of the co-existence of these two worlds, used by people to come and go between them. Kagurazaka is particularly notable in this respect.
They can smell the wood
All of the wooden shelves used for storing books were on the warehouse's first floor. We decided to keep these shelves as they were to form a library, and we also created a small lecture hall for holding talks by writers and makers. Although contemporary society is moving away from books and towards computers and information technology, people nevertheless have a strong feeling of connection to – and nostalgia for – trees and things that are made from wood. La kagu is a space where visitors can really get a sense of the culture of books. When they step inside, some even say that they can smell wood.
As a kind of gateway
Historically, Japan's shrines have been built in order to worship the gods who live in the sacred mountains or seas; They don't reside in the shrine itself, but in the space beyond it. This belief that the spirits and deities exist beyond the confines of the shrine, and that the shrine itself acts not as a centre, but as a kind of gateway, is very different to the grand, imposing churches and cathedrals of Christianity.
The majority of shrines are not found in the mountains or in the middle of the fields, therefore, but at the borders of mountain villages – which is to say, at what is seen as the edge of the mountains. The tori gate, marking the entrance to a shrine, indicates that there are gods on the other side of it.
The golden poo
On the opposite bank of the Sumida River lies the Asahi Beer headquarters (1989), a strange building with a golden sculpture mounted on top of a granite-plated black box. It was designed by Philippe Starck, and completed in 198g when the Japanese economy was still going strong. The sculpture, with no clearly defined use, is a clear representation of its time. Today, the building is known as the 'golden poo', a reference to the shape of its crowning object.
Like crossing the sea
The Sumida is a symbol of Tokyo, but is not like the Thames in London or the Seine in Paris, or other rivers that are woven into the geography of the city. Its banks were pushed back, so that the river became extremely wide and travelling across it feels liberating, like crossing the sea.
These thrown-away items
I decided to furnish the restaurant [Tetchan] with the kinds of discarded items one wouldn't normally use in interior design, from recycled LAN cables to acrylic by-products.
When using discarded objects in interior design, it gives even brand-new places the feeling that they have always been there. I think this is due to the inherent history of these thrown-away items, which lives on inside of them.
Kengo Kuma's sketches