Some sketches

Despite good intentions, I find that there is no sketch or watercolor attached to a lot of my posts. So here are some. Most of these are from small notebooks.

In 1959 after I had graduated I visited my parents in Oregon and caught this lovely oast house (where they dried hops). I was impatient so my drawings were quite loose in those days.

Oast House in Oregon, 1959

Soon after that I was drafted, and spent 6 months at Fort Ord, near Monterey California. On weekend passes I and a friend would take the steam-powered train to Monterey and have espresso at the Sancho Panza coffee house, where we could read the NY Times. We would walk around the beautiful laid-back town after a martini or two, and wander along Cannery Row. After the canneries closed in the 1940’s, the buildings decayed and many burned in the 1960’s – I caught them in between. The web tells me that the area has be “revitalized” into a popular tourist destination (like so many others) that attracts 4 million visitors a year. I have no interest in revisiting. These are charcoal sketches about 9 x 12. The harbor view is pen and ink.

Cannery Row, 1959
Cannery Row 1959
Harbor at Monterey CA, 1959

Just after Ellen and I were married, we traveled to Scotland, England and Italy. The sketch of the English village is my favorite.

Outbuilding at Cardney-Dunkeld, 1968
Village of Fletching near Piltdown, East Sussex, 1985
Lago de Garda, 1968

As we waited at a cafe for St. Martin in the Fields to open at Trafalgar Square, I made this sketch of a building at the intersection of Duncannon Street and the Strand with a pen on a napkin, about 4 x 5. It is all made of inked dots, which is the only practical way to draw on a paper napkin. (Thanks to Google Maps for reminding me where the building is).

Building in London, 1968, pen on napkin

In 1985 we went to England and Germany with the kids, aged 14 and 11. My drawing got much more precise. I think these were all pencil sketches, but am not sure. The originals are packed somewhere.

Tower Bridge, London, 1985
Canterbury Cathedral, 1985
Lanthorn Tower, Tower of London, 1985
Ruins of 11th Century Limburg Abbey, converted from 9th Century castle, Bad Durkheim, Germany
Limburg Abbey

In 2001, we made a trip to France, where we rented a “penichette” houseboat and motored up the Mayenne River from Angers. I sketched along the way. It was a glorious trip. I recommend your own boat versus a big river boat; you rent bikes and can get off and explore the countryside. These are about 5 x 7, pen.

Our penichette in a lock, 2001
Penichette docked at a mill on the Mayenne River, 2001
Hospital and Bridge on the Mayenne, 2001
Lock on the Mayenne, 2001

We had wine at a cafe on the plaza in Chateau Gontier; it was light well past 10, being mid-June, and as I recall there was a parade.

Cafe at Chateau Gontier, 2001

Naturally, there were old mills all along the river, at the locks.


Mill on the Mayenne, 2001


Mill at la Benatre, 2001. I outlined the corners for some reason, which is a no-no.

After the river trip, we went to Paris. The highlight for me was the Eiffel Tower. I had seen thousands of images and little models of the structure, but as we approached it along the Champs de Mars it loomed ever larger until at its base we were in its awesome presence, under its spreading legs. Nothing prepared me for its incredible scale.  No way to capture its essence graphically or even cinematically. You simply have to be there.

Tuilleries Garden, 2001
Cafe at the foot of the Rue de Maurice Utrillo stairway east from Sacre Coeur, 2001. These are the steps in the famous Brassai photo, I believe.

Hope you enjoyed sharing these sketches!



The Foundation of Civilization: Trust

Rowayton harbor on a misty morning

A structural engineer friend once told me that you can’t safely use welded steel framing in India because so many of the weld inspectors take bribes. Welding is subject to failure, and trustworthy inspection is the only way to find the failed welds and repair them. Likewise, I am told that LEED certification in India is a farce, as consultants who must verify performance simply invent results for a small fee. Most LEED-certified project in India receive the highest rating, Platinum: if you are cheating, cheat big.

Civilization has always been based to some extent on technology, but we have become like parasites in a technological host, utterly helpless without it. Even ISIS, determined as it is to send us all back into the 14th Century, is completely dependent upon electronic media, along with modern weapons, explosives and vehicles. Shut off the electricity and everything else shuts down.

As the examples from India suggest, corruption that undermines trust also undermines the technology on which modern civilization is based. Quality control standards are at the heart of establishing trust. It is unfortunately one of the least sexy aspects of technology. The procedures prescribed by ISO (International Organization for Standards) are mind-numbingly thorough, requiring repeated testing, recording and re-checking. But ISO is the gold standard of quality control, and manufacturers who meet ISO standards can profitably use this fact in their marketing.

The Wikipedia entry on ISO notes that Microsoft pushed through a fast-track procedure now approved by ISO. A criticism of ISO’s decision to approve this procedure elicited this comment from Computer security entrepreneur (and investor in a Linux based operating system that completes with Windows), Mark Shuttleworth:

When you have a process built on trust and when that trust is abused, ISO should halt the process… ISO is an engineering old boys club and these things are boring so you have to have a lot of passion … then suddenly you have an investment of a lot of money and lobbying [by Microsoft] and you get artificial results. The process is not set up to deal with intensive corporate lobbying and so you end up with something being a standard that is not clear.

Volunteer standards like ISO are necessary but not sufficient in a marketplace based on economic competition. A current example is the Millennium Tower in San Francisco, which as of 2016 had sunk 16 inches and tilted 6 inches at the top. The causes likely were the use of 60- to 90-foot long friction pilings sunk in the unstable mud and sand, rather than 200-foot pilings down to bedrock; combined with the crazy decision (in an earthquake-prone area) to use a heavy concrete frame rather than lighter steel framing to construct the tallest building in the city. I speculate that these decisions reduced first cost, greed trumping quality and common sense, as it so often does. No one knows what will happen to the building in a major earthquake, when the mud and sand supporting the building may turn into jelly.

It is this perverse incentive to ignore reality for short-term gain that requires regulation by disinterested third parties, typically established by governments. The Millennium Tower was built before the city had set up the current structural review procedure, relying instead on the word of the structural engineer (hired by the developer).

It is highly disturbing that our political right-wing is bent on dismantling such regulations. Yes, they slow down development, burden companies with paperwork, and add cost. That’s the whole point, to insist on quality in order to establish trust. The alternative is, in the long run, to undermine the foundations not only of tall buildings but of the technology that holds up modern civilization.

It is ironic that the companies that fund the radical right are themselves completely dependent upon the trust they are undermining – yet another example of the very phenomenon (setting reality aside in favor of short-term gain) that the regulations are designed to counteract!



Bucky Fuller’s Dymaxion Car: What Was He Thinking?

Bucky Fuller was what we would now call a “futurist,” someone wedded to the myth of progress, with unlimited faith in the power of technology. He was also a highly gifted amateur, never afraid to apply his intuition to a problem, but always taking a short-cut to avoid the difficult road to truly useful innovation, one paved with hard won evidence.

He leapt into any endeavor from the top, determined that his genius could solve problems that plagued ordinary smart people who were working in the trenches building from a solid foundation. Now it does happen, very rarely, that a gifted amateur stumbles on a solution to an important problem. Amateurs indeed contribute much, yet seldom without devoting the time to become thoroughly knowledgeable about the science and technology behind the problem they are trying to solve.

Two examples come to mind. The great physicist Richard Feynman, always determined to derive insight from first principles, delved into evolutionary theory and correctly deduced a number of insights, which he proudly put before the famous evolutionist Steven Jay Gould for comment. Gould noted that he could have learned everything he deduced from a textbook on evolution. So in regard to this subject, Feynman was an amateur, and had over-valued his insight in his attempt to make an end-run around the hard work of learning the trade.

In contrast I would point to the highly respected amateur ethologist, Ellen Dissanayake. She never achieved a bachelor’s degree, but by dint of hard work over many years, she self-educated to become a leading investigator on the subject of the evolution of art, and was admitted to the company of scholars in the field despite her lack of credentials.

As far as I know, Bucky Fuller never invented anything that was particularly useful. His famous geodesic domes were a solution looking for a problem (and he didn’t invent them). They were lightweight and could cover large spans, as long as the foundation was circular. They did find limited use covering a few sports stadiums and exhibit spaces, and for radar installations, temporary shelter, and even a few homes. I was involved in the construction of a geodesic dome birdcage in Oakland, California, an ideal use because it did not involve the difficulties in attaching a weatherproof skin to the framework, and took maximum advantage of the volume enclosed within a spherical shape.

Bucky did however popularize a number of ideas that have implanted themselves in the cultural milieu (what Richard Dawkins would call “memes”). He got deeply involved in the geometry of geodesic solids, which he applied to his domes; his name became closely enough associated with geodesic geometry that when scientists fabricated strong carbon materials exhibiting geodesic geometry, they named the materials “fullerenes”. I think his primary claim to fame is popularizing this branch of mathematics.

In the 1930’s, the technology of both cars and airplanes was advancing rapidly and garnering in inordinate amount of attention. Bucky, in his thirties, got a bee in his bonnet that the world needed a land-sea-air vehicle, which would in his view provide people with unlimited freedom of movement.

Putting aside its utter impracticality if realized and the anti-social Libertarian philosophy underlying it, the idea had no basis in physics. Flight then and now requires rigid wings, but Bucky envisioned inflatable wings, along with “slots” for future jet-power of some sort. It was a Flash Gordon fantasy, an adolescent dream.

Acknowledging the difficulty of resolving the flying and floating aspects of the vehicle, he concentrated on its terrestrial mode. With the help of a kindred spirit, the inventor, aviation pioneer and yacht designer William Starling Burgess, he put his ideas for a car together and over time built three prototypes.

As the photos show, it was innovative in shape, adopting the “streamlining” that was being popularized in the U.S. by the industrial designers Henry Dreyfuss, Otto Kuehler and Raymond Loewy. Its superstructure (including the thin members supporting the extensive glass) was lovingly crafted from wood, no doubt courtesy of yacht-designer Burgess. It had an engine in the rear that drove the front wheels, and was a whopping 19 feet long. It had no rear windows and no rear-view mirrors.

For reasons only known to Bucky’s unconscious, he decided that rear-wheel steering was a good idea. Anyone who has handled a cart or vehicle with rear steering, or tried to back a car, knows its fatal defect: any steering error is magnified in a runaway feedback cycle. Rear steering cannot be made self-correcting, as can front steering (with appropriate camber and toe-in). The only vehicles made with rear steering are fork-lifts and street-sweepers, which are operated at low speeds and need to make sharp turns.

Somehow or other, a few highly skilled people learned how to drive the Dymaxion car at moderate speeds. On an infamous occasion in 1933, Francis Turner, a famous race driver who had learned to operate the Dymaxion car, was driving aviation pioneer William Sempill and the French Air Minister Charles Dollfuss to a rendezvous with the Graf Zeppelin when a Chicago official crowded the car trying to get a close look. Turner sped up to 70 mph to evade the rubbernecker.

The official’s car accidentally bumped into the rear of the Dymaxion, causing Turner to lose control of the steering. The car entered its fatal feedback cycle, turning sideways and rolling over, killing Turner and seriously injuring Sempill. Astonishingly, the court did not find the design of the car to be a factor in the crash. If something like this happened today, Fuller would be in prison.

Not only did Fuller promote a fundamentally deadly design, he hyped its capabilities, claiming that its weight was low, that it got very high gas mileage, and that it could go over 100 mph, all of which were untrue.

To get a more in-depth account of what it is like to drive a Dymaxion car, go to or

Here are a few quotes from the latter source:

“With the steering’s self-centering action non-existent and the epic amounts of tiller-spinning still required, crowned roads, bumps and potholes can present life-threatening challenges.”

“…no one in his or her right mind would ever venture above 45 miles per hour because of the lousy handling…”

“So this is the stuff that some automotive legends are made of – a wacky idea, a shameless promoter’s dream and a credulous press, excited to herald the coming of a wonderful new future.”

“Shameless promoter” and “living in an alternative reality” are good ways to describe Fuller (and Donald Trump), yet he remains a fascinating character, and his lectures were utterly riveting to this impressionable college student. Building the geodesic birdcage in Oakland, California was a highlight of my life.

Bucky acted out the same dreams I dreamed, and perhaps I am chastising myself for being unrealistic. I admit that I am being unfair to Bucky: he was a man of his time –  and of mine. And compared to the devastation left in Mr. Trump’s wake, contributing to one death is a small crime indeed. The time was out of joint, dislocated by the myth of progress that still enthralls us.

Symbolism – Part One

Lago de Garda, 1968
Lago de Garda, 1968

The subject of symbolism has intrigued me for 60 years. I am not talking about symbols like the Christian cross or a national flag, but about the symbolism of language and mathematics, and how this applies to art.

I was strongly influenced by two books by the aesthetician Suzanne K. Langer, “Feeling and Form” and “Philosophy in a New Key,” both of which were published around the time I was in college. She had an unusually practical mind, and at the same time had a deep understanding of all the (fine) arts, not just easel painting, so often the sole focus of aestheticians.

Langer developed her aesthetic theory out of earlier work on symbolic logic, a field developed by the great logicians of the late 19th and early 20th Centuries and one very fresh and vital at the time she wrote. She was strongly influenced by Alfred North Whitehead, who along with Bertrand Russell prepared much of the foundation for symbolic logic. Both symbolic logic and mathematics are precise formalizations of discursive language (the language of statements), and language is universally acknowledged to be the quintessential example of symbolism (Langer’s lucid writing made her little book ”Symbolic Logic” close to a page-turner).

Symbols point to things: naming is of their essence. When Langer extended to art the language-based concept of symbolic representation, she naturally looked for something concrete that the art symbol could represent. Her solution was to postulate that the art symbol pointed to the form of the life of feeling, something she saw as complex patterns of rising and falling tensions, so complex as to be forever impossible to depict using ordinary language. Only the art symbol could present the life of feeling in a way that made it understandable.

I believe that Langer is correct that a work of art symbolizes patterns of feelings that occur in the lives of people, but I believe she made a mistake when she characterized these patterns as forms. I am an architect, and the word immediately conjures up an image of the forms into which concrete is poured. Forms are rigid, while the patterns of felt experience are fluid and difficult to pin down.

By her insistence that a work of art symbolized some kind of form, Langer created a problem for herself: how do you decipher the work of art to reveal the symbolized form? Langer invoked intuition, a famously vague term, and argued that the work of art must be intuitively seen as a whole, all at once (of course in music, dance, drama and literature the work unfolds over time, but we can still imagine the work as a whole after it is completely presented or read).

The form to which the symbol refers is the meaning of the work of art; in her later work, Langer substituted “import” and especially “significant form” for “meaning,” but the sense was the same: the art symbol referred to some kind of pattern representing the ebb and flow of feelings. (To confuse matters, the expression “significant form” was used by art critic Clive Bell in a famous 1913 article “Art and Significant Form” at Langer strongly disagreed with Bell’s ideas, as do I, but the confusion of terms did not help her argument.)

The problem of how to decipher the underlying form to which the symbol refers was clarified for me through an article I found on the web by chance. It was written by Berel Lang, who I find is a philosopher at Wesleyan University, and is entitled “Langer’s Arabesque and the Collapse of the Symbol” (see In this essay, Lang argues that it is not possible to distinguish the art symbol from what Langer argued that it represents, the form of the life of feeling. She notes that this problem was raised in a review of Langer’s seminal book “Philosophy in a New Key” by philosopher Ernest Nagel, and that Langer took his criticism seriously and tried to resolve the problem in her later writings, in Lang’s view without success.

Summarizing Lang’s critique, a work of art is not a symbol because instead of referring to an objective content, a meaning, the work is itself the content. There is no symbolism because it is impossible to abstract the symbolic form from the details of the work. Here Langer herself makes this point:

But a work of art does not point us to a meaning beyond its own presence.  What is expressed cannot be grasped apart from the sensuous or poetic form that expresses it. In a work of art we have the direct presentation of a feeling, not a sign that points to it.” (Principles of Art, pp.133-34)

Put another way, the patterns that one detects in a work of art are far too complex to be perceived as a single form, no matter how elaborate. Think about how you take in a work of art: at each viewing or hearing, it always takes time, and all art lovers comment that you can come back to a familiar painting or piece of music hundreds of times and never cease to discover something new. How, if we perceive the whole work as a single form (a gestalt, as Langer would put it), can we continue to reconstruct it as we experience it over time? Would we not each time find a new overall form?

If all the intricate details of a work are cast into the form, it must be a very flexible form indeed to accommodate the reinterpretations by each perceiver, not to mention the dramatic reinterpretations that take place as cultural norms change over the decades – so flexible as to lose its character as a form. And Nagel’s and Lang’s telling criticism also applies, that the form is so intricately trapped in the actuality of the work of art that you cannot separate the work from its import. The work, as the quote from Langer above suggests, is the import.

The solution to this dilemma, I maintain, is that the pattern and flow of human feeling cannot be symbolized as a form, a conceptual object, something that is remembered in the same way that we remember objective things. Instead, the art symbol needs to be thought of as a process rather than an objective thing, as a verb instead of a noun. What this means is not obvious, but I will try to make my idea clear in subsequent essays in this series.

Essay I on Function: Does Reality Have a Purpose?


Leaving aside ceremonial dinners and mathematical expressions, “function” is a synonym for purpose, or is the role something plays, whether by itself or as part of a larger assembly. So if we substitute “role or purpose” for “function” we will not be far off. My ultimate goal is to explore the concept of function in the context of architecture, but in this first essay, I want to explore function in the context of nature.

Does the sun or a rock or the atmosphere play a role or have a purpose? Well, if you believe that this land was made for you and me, then the sun functions to light and heat us, rock provides us with building material, and the atmosphere delivers oxygen to breathe and protects us from ultraviolet radiation. If you adopt the Enlightenment notion of a clockwork universe, then each has a function as part of God’s grand clock that he wound up and let run according to the Newton’s laws. Einstein had such a view of God; he was certain God made orderly, logical laws, and didn’t at all like the uncertainty that is fundamental to the theory of quantum mechanics he helped formulate.

A modern scientific point of view rejects such ideas. Stars and rocks and gases are things that emerge naturally, given the laws of physics. You can trace cause and effect step by step back to the big bang, beyond which for the moment we are blind. If you want to interject a deity, you can be a Deist and have God set off the Big Bang or design the laws of physics, but I don’t think this kind of deity would satisfy many people’s spiritual needs, although it seemed to satisfy Einstein. For today’s science, there is no one behind the scenes deciding what comes next. Arguments on the subject continue unabated; I stake my claim on the side of materialism.

A role or purpose needs context. To define the role or purpose of a rock, you need to have some kind of goal, for example, making a rock wall. When you set a goal, everything in the universe suddenly has a role or purpose relative to the goal. Rocks, yourself, tools, a plan, and a location all have essential functions, while the Andromeda galaxy takes no part, and the sun functions to keep you warm while you work and to provide light. Invent a goal, and function follows; if there is no goal, nothing has a function.

But does this line of thinking apply to life? It seems natural to think of one’s heart as having an essential function, doing its part to keep you alive so you can reproduce. Richard Dawkins argued in “The Selfish Gene” that it is your genes that have the goal, and you are just the vehicle they use to make more of themselves (as a chicken is an egg’s way to make another egg, or a scholar is a library’s way to make another library).

I maintain that we have invented the goal. Genes don’t have goals, they are just doing their thing, even if it seems to us that they “want” to persist into the next generation. It is a compelling metaphor, but it is just that: a gene is the pathway through time taken by the atoms that make up the gene, a part of the unfolding evolution of our universe.

The injection of purpose, intention, role, agency, goals and the like into our thinking about things is probably essential for understanding. We seem compelled to personify things, to think of them as if they were people, applying our hyper-developed social skills as a tool for understanding. In this way, we endow objects and events with an essence, a thingness that helps us makes sense of the world.

This is a highly useful tendency (which is no doubt why it evolved), since on our planet, nature clumps into identifiable objects and events. But our planet is highly unusual: very little of nature clumps into identifiable objects and events, important as these are to us. Most of nature consists of undifferentiated aggregations of dust, gas, plasmas, particles and fields. Only a few percent of the mass of the universe is in the form of what we call matter. But we didn’t know that when our mental equipment evolved. It was tuned to a pre-scientific world.

Nature doesn’t have a purpose, and being a part of nature, neither do you. This idea creates cognitive dissonance, so we seek meaning and purpose in our lives: it’s the only way we can avoid existential despair. But when we try to find out how nature actually works, we need temporarily to abandon our search for meaning and purpose and accept reality as it presents itself.

Life in a Bubble

Norwalk boatyard

I see four fundamental characteristics of all living things: they process resources from the environment and return waste products (i.e. they metabolize); they reproduce; they evolve; and (the focus of this essay) they are separated from their environment by some kind of membrane.

The origin of the chemical processes that make life possible, and the place or places of their origin, are the subject of active research. But all agree that these processes need to be concentrated so they don’t dissipate into the outer environment. Even if life processes can arise in an open environment that is rich with the necessary energy and chemical resources, evolution is not possible until there is some kind of “it” to evolve.

The key component required to contain life is a “lipid bilayer” that forms a membrane. This remarkable structure surrounds each living cell. It is formed of a double layer of molecules packed side by side, with other molecules embedded in the membrane that act as gates for passing chemicals through the membrane. You can learn about them in the Wikipedia article on lipid bilayers at

Because of this need to be separated from its outer environment, a cell is its own mini-universe. Whatever behavior it exhibits results from processes occurring within that universe, informed by information and chemicals from outside. This in turn requires the cell to form a model of what is both inside and outside its boundary.

Information and chemicals pass through the membrane in both directions, informing the cell of what is going on outside its membrane; internal machinery tracks what is going on within the cell. Each cell is specialized to process only certain kinds of chemicals and information.

Assemblies of specialized cells can process a great deal more information and chemicals than a single cell. A single-cell bacterium can configure itself as a lens to focus light on the inside of its membrane, allowing it to move toward the light. But a human’s eye and brain can form incredibly more elaborate models of what is visible.

Yet the principle is that same: only certain kinds of information and chemicals can get through the barriers surrounding individual cells and the vast collections of cells forming a complex organism, and only certain processes within the cell are monitored. It is impossible for a living organism to form a perfect model of itself or its outer environment.

Each organism’s model of its self in its environment is created out of a rich and constantly changing stew of inheritance and experience – “nature versus nurture” is entirely void of meaning. The model is volatile; what astonishes is not its malleability, but its apparent stability. If a human brain decides every tenth of a second what its brain-body is to do next, which seems to be approximately the case, the model created by a person my age (80) must have been updated 25 billion times through exceedingly complex neural transformations. This is either a miracle, or we are kidding ourselves regarding the stability of the model.

I leave you with two take-aways. One is that indeed our sense of self and our memories are transient and inaccurate, something science is probing with unsettling consequences. The other is that the organism must have some sort of criteria to decide on the appropriate next behavior. Without such criteria, no consistent model could be formed, and therefore no consistent behavior.

I feel that not enough attention is being paid to the subtleties of these criteria, and that art is a way of laying bare at least some of them. This leads me to believe that art can be a window into the workings of human brains, a subject I will pursue in other essays.

Oakland Bird Cage

In 1956, my good friend and excellent sculptor Bill Underhill organized a team of architecture students to design and build an all-aluminum geodesic dome birdcage, which is still extant in Merritt Park in Oakland, CA. It was funded by the Kaiser [Aluminum] Foundation, and Don Richter of Kaiser was our engineer. The following writeup can be found on the localWiki

Oakland Dome Photo

The Geodesic Bird Dome is a part of the Lake Merritt Wildlife Refuge in Oakland, California. Built in [1956] with Kaiser Foundation supplied materials, it was used for years as an exhibit cage for a variety of wild birds, but in later years has been utilized as a cage for sick and injured birds. 1

The birds in here have long been considered unwell and poorly kept. It is not clear if the birds are actually being warehoused or if this is a sanctuary.

A plaque on it says “designed by Buckminster Fuller”, which is incorrect. Fuller did the math behind geodesic domes and did much to popularize them, but was not the inventor of them. This particular dome was designed by William Underhill, Gordon F. Tully, Dick Schubert, Dan Peterson, and Marshall K. Malik, who were architecture students at UC Berkeley.

As I worked out the geometry and saw the project through to the end, I thought it worthwhile to set down the details of how it was designed. I have drawn over the photograph of the dome to help explain its geometry. The meanings of the lines, dots and diamond shapes are explained in the text.

Oakland Dome

Geodesic domes are based on the icosahedron, a 20-sided figure that is one of the five “Platonic” solids, the only convex solids with equilateral faces. The other Platonic solids are the tetrahedron, with four triangular faces, the cube with six square faces, the octahedron with eight triangular faces and the dodecahedron with 12 pentagonal faces. Bucky Fuller’s riveting lectures on the relationships among the Platonic solids inspired the dome, and the geometry continues to fascinate both Bill and myself (see the Wikipedia entry on Platonic solids at )

Five triangles meet at each of the 12 vertices of the icosahedron. By spotting these five-spoked intersections, you can figure out the geometry of any geodesic dome. In the diagram, these intersections are shown with large white dots. In a complete icosahedron, there are 30 edges connecting the 12 vertices. The diagram highlights ten of these edges with yellow lines; our dome has a total of 20 such edges.

The vertices of all Platonic solids lie on the surface of an imaginary circumscribing sphere. The icosahedron is chosen as the basis for geodesic domes because it has the most faces, and they are all structurally rigid triangles. However, an icosahedron creates a crude, pointy structure with long edges. Since the edges of the underlying icosahedron become the straight members that form the dome’s structure, the larger the dome, the longer the edges. So for both practical and aesthetic reasons, it becomes necessary to subdivide the edges of the icosahedron to create shorter members and additional vertices which, when brought out to the surface of the imaginary circumscribing sphere, make the dome more nearly spherical.

The number of times each edge of the underlying icosahedron is subdivided is the “frequency” of the dome. To create reasonably sized members and screen panels for the dome, we chose to make it a “third frequency” structure, with each icosahedral edge subdivided into three segments. You can see in the diagram that each yellow edge is broken into three segments that bend outward. A complete sphere subdivided in this way has 180 triangular faces, 90 edges and 80 vertices.

Even if you subdivide the edges of the icosahedron into three equal lengths, the triangles will not be equal in size (if they were, this would be a Platonic solid with 180 sides, which does not exist). The best you can do in a third frequency design is to have two edge lengths and two sizes of triangles.

As we conceived of the dome to be a flight cage, it made sense to maximize its volume. We did this by designing it as a bubble-shaped three-quarter sphere, incorporating 15 of the 20 icosahedral faces. Each face of the icosahedron is subdivided into nine triangles. If you do the math, you will find that our three-quarter sphere, third-frequency structure has 135 triangular faces, 75 edges and 65 vertices.

Tables were available later on to aid designers in calculating the lengths of the struts. Not having such tables, I did the the complex spherical geometry calculations on a Marchant electro-mechanical calculator owned by the Oakland Park Department, in whose office we did our design work. Marchant calculators were much faster and more sophisticated than any others on the market at the time. (Wikipedia records that the firm was bought by Smith Corona in 1958 and the new firm, unsuccessful in switching to electronics, was gone by 1980).

The ingenious aspect of the design was suggested by our engineer from Kaiser, Don Richter. His idea was to join pairs of triangles to form 65 diamond-shaped panels (plus five infill triangles at the base). The two triangles forming each diamond bend around the dome like the covers of an opened book, and so are not in the same plane. When the screen is stretched across the diamond, it naturally forms a doubly curved surface (a  “hyperbolic paraboloid” or “hypar”). The screen panels are cut so that one set of screen wires runs from end to end of the diamond, curving outward, while the perpendicular strands run across the diamond, curving inward. As opposed to a flat screen, which has to deform before it can take pressure, our screens are already bent and can resist pressure immediately from either side. The diagram below shows how it works – naturally the screening is closer woven than is shown in the diagram.

Screen Panel

The doubly-curved screening also contributes  to the strength of the dome. Richter had constructed such a structure for Kaiser Aluminum out of a 1/16th inch thick corrugated aluminum panel, reinforced at the edges by beams. It was about 20 feet corner to corner. They tested the structure, which supported 18″ of sand before one of the beams buckled (it buckled upward, showing that the failure was due to compression and not bending). You can make a crude model by folding a piece of aluminum foil into corrugations, then flattening and doubling over the corrugations on opposite edges to create the edge beams. The foil pops into a hypar shape – a neat party trick.

There are two sizes of diamond-shaped panels. 20 symmetrical panels cut across the middle of each icosahedral edge, shown in light yellow in the diagram. The other 45 panels form the corners of each icosahedral face, shown in light red. These panels are asymmetrical. The diamond panels meet in the center of each icosahedral face, forming six-pointed vertices.

Across the center of each diamond, we designed a tubular strut to complete the triangular structural grid. If the strut were straight (like the spine of the folded book), it would lie outside the screening. Amadee Sourdry, our supervisor at the Oakland Park Department, vetoed this configuration because the exposed struts would form a perfect jungle gym, which would be an attractive nuisance and therefore a liability to the city. We were forced to put the struts on the inside of the screening, bending them inward to stay inside the curved screening. Each end of the tubular strut was flattened and bent, then bolted to the inside flanges of the C-shaped frame members that form the edges of each diamond.

The screen is clamped between the outside flange of the C-shaped frame member and a neoprene gasket held in place by (as I recall) a 3/8″ x 1″ aluminum plate, all secured by closely-spaced bolts. The outside flanges of the diamond panel frame members is bent inward so that the flanges of two adjoining frame members lie in the same plane, allowing them to be bolted snugly together. At the points of the diamonds, the bottom flanges are likewise bolted together. The complex construction of the screen panels and struts made it necessary to hand-craft all the members and hand-assemble each screen  panel. If you built hundreds of these, you could use specialized machinery to do the job. This being a one-off project, we instead relied on a wonderful Polish metalworker. The cross section through a typical screen frame member shows all these parts.


After the foundation was poured, assembly of the dome took one long day, after which we celebrated with a spaghetti dinner. Unfortunately, Bill had been drafted into the army and could not be there. The dome was assembled from the top down, suspended from a crane that raised it up as diamond panels were added and bolted to adjacent panels. The weight was supported only at the top (I don’t recall whether they used a single cable or five separate ones, as the pictures of the dome under construction have disappeared).

As a result of the limited number of supports while the dome was being assembled, bolts began to pop as sections were added near the equator , causing considerable alarm. Luckily, enough survived until the dome was set on its foundation, where it was held up at a sufficient number of points that it became (more) structurally sound. I speculate that any dome which is more than a hemisphere is structurally suspect because it wants to bulge at its equator.

The bulbous configuration of the dome was a result of it being designed as a flight cage, with a small footprint and a large volume. Only well into design and construction, did we learn from Sourdry that it would house waterfowl instead of perching birds. It would have made more sense to widen the base somewhat by eliminating the bottom layer of triangles.

Five triangular infill panels were required at the bottom to complete the enclosure, one of which made a natural entry. The blue triangle in the diagram shows the nearest such panel – the actual entry is on the opposite side. To keep the birds from escaping, an entry lock was needed with a door at each end. Because the panel leans toward the ground, if the panel was used as a door, it would have to open inward and upward.

[The following paragraph was revised in February 2017] Instead, we built an awkward-looking but serviceable tunnel between two rectangular doors, transitioning to fit into the triangular sector as it passes through the dome. My memory here is somewhat fuzzy, but you can check out the actual solution by visiting the dome (which I haven’t done since it was built).

All this is still vividly imprinted on my memory 60 years after its construction in 1956. It was a most rewarding experience, one of the highlights of my career.

PS: I can’t resist inserting a reference to an extraordinary item I found on the web at , a verbatim transcript of one of Bucky Fuller’s long lectures. He gave several of these lectures while I was at Berkeley, and held his audience from 2 PM to 11 PM, with a break for dinner. In it there is a reference to Don Richter, which I insert below. I also found a patent filed by Richter in 1955 and granted in 1959 for the corrugated hypar roof that he had tested. at

One of my boys at the Institute of Design in Chicago was Don Richter. Don was an extraordinary man and he stayed with me during all the early years of the developing of the geodesic dome, after he graduated from the Institute of Design. He had been a sailor in the Merchant Marine during the war. Please hold the pictures for a minute. Don’t do anymore with them for a second. And Don wanted to really go on. Many architectural students asked me what they ought to do, and I would say, what I think you ought to do is to get production engineering. And the only way you can do that, to really get it first class, would be in the aircraft industry. Don did work for a while with Kaiser Aluminum and he then got a job in Texas with the Republic Aircraft. They were building an enormous bomber and he began he did so well in general engineering that he did get into production engineering, and he lived with the Head of the Production Engineering and developed extraordinary capability.

Don, then, Kaiser Aluminum Company were looking for somebody with design capability and I recommended Don and he went to them, and Don had made his small geodesic dome of aluminum and had it on his desk. He made it at home, and brought it in one day and put it on his desk, and Henry Kaiser, old Henry Kaiser walked by the desk and he thought this was a Kaiser product and he simply said, “I’d like to have one of those built for Hawaii,” and he had just been building a big hotel out there, and so everybody just takes Henry’s orders and so they had to make deals with Don, and there was a great deal of negotiating from there on`. The Kaiser patent attorneys came in to get license from my patent attorney.

The Art of the Body

The oldest art medium may be the human body. The most familiar body art is clothing, including armor, jewelry, accessories and headdresses. Here I want to discuss using the body itself as a medium instead of an armature on which to hang dress and adornment, although there is really no sharp boundary between direct manipulation of the body and adornment hung on the body.

Modern forms in this genre are make-up, hair dyeing and styling, and perfume, but the practice likely predates the evolution of modern humans. Evidence that cannot endure the millennia has vanished, but the mineral red ochre has been found in prehistoric sites dating back 200,000 to 250,000 years ago ( Although no one knows what it was used for, it was not a local material at the sites, and since nearly every pre-modern society we know of uses it for body painting, it is not unreasonable to assume it was so used by ancient hominins (including Neanderthals).

Body painting was widely used by pre-modern peoples as beautification and as a part of rituals (the past tense reflects the inevitable modernization of dress and decoration after a people are assimilated into modern global civilization). This practice persists, sometimes in surprising ways not associated with the theater, as we will see.

Mutilation is another branch of body art. Common contemporary examples include pierced ears for earrings and tattooing. Pre-modern examples include the insertion of large disks into the skin of lips and ears, the deformation of a girls’ clavicle bones by a stack of decorative hoops exaggerating the length of the neck, the thankfully abolished practice of Chinese “lily feet”, scarification, insertion of bones through the skin and the like. Contemporary self-mutilation includes weights hanging from hooks through penises or buried in the skin, evidently to act out masochistic urges. Many contemporaries (few from my generation) sport rings through the nose, belly, or other body parts.

Child's face painting

Children love to have their faces painted, but other forms of body painting thrive in contemporary society. In my search for information on pre-modern examples, I innocently Googled “body painting” and was startled to find endless images of elaborately painted, typically attractive and entirely nude women. This has apparently been an organized practice at least since the 1960’s, an outgrowth of our newly (and incompletely) liberated attitudes toward nakedness and sex. Men are also painted, but it is hard to integrate their exposed genitals into a convincing image, while a shaved pubis can be readily disguised, at least from a respectful distance.

The countess-model Veruschka turned body painting into an art form, as can be seen on her website There she is “painted into” elaborate backgrounds until she is virtually invisible, enticing the viewers to play a sort of “Where’s Waldo” game. Many other artists have picked up this theme. In my post “A New Way to Think About Art” I show a photo of Veruschka’s head disguised as a rock.

Another iconic example is the follow-up cover of Vanity Fair magazine on the anniversary of Demi Moore’s famous 1991 nude photo while 7 months pregnant. In the follow-up, the newly trim Demi is painted as if wearing a three-piece suit. And Sports Illustrated’s swimsuit issue now features models with painted-on bathing suits. Perhaps the fad will spread, but paint doesn’t last as long as bathing suits, however abbreviated. I have questions about using the toilet while painted, a subject we shall politely pass over.

Today’s body painting covers a broad range: Verushka’s serious art; other attempts at high art including happenings where painted nudes are used as tools to make images; soft-core porn where painted-on scanty clothing enhances the sexiness of the models; borrowings from fantasy genres; animal imagery; sports team wear; and decorative patterns. There are regular body painting festivals (held in warm weather one assumes). Chasing down links in the Wikipedia entry on body painting, I found Guido Daniele, who has made some extraordinary trompe-l’oeil hand-paintings photographed for use in advertising campaigns, found at

I insist that painting, clothing, adorning, and mutilating the body, and using the body as a tool, be accounted for in any comprehensive theory about the nature and evolutionary origins of art.

Authenticity – Part Two

Village in Pays de la Loire region, France
Photo of village in Pays de la Loire region, France

To probe more deeply into the concept of authenticity, I want to present three examples.

Example one: You are walking through a beautiful garden. Flowers are blooming, the trees are in leaf, the grass is green. You lean over to smell a flower and discover to your surprise that it is a highly convincing imitation made of plastic. Suspicious, you check the grass and the trees, and behold, they also are plastic. At a certain level of detail, consistency and continuity abruptly break down. Were the garden real, you could continue to drill down, verifying the authenticity of the environment at every level of detail.

Example two: You live in a Modernist house, with 1950’s modern furniture and decorations. It is a real gem, on everyone’s list of historic houses. You are well-off, and are driving your guests in your BMW to a fine restaurant, where again everything is coordinated and consistent – both your house and the restaurant are works of environmental art. The interior of your BMW is likewise well-designed, and you all step out of the vehicle into the restaurant, leaving the car with the valet. The experience is consistent and continuous. On the way to and from the restaurant, you pass through Typical Suburbia. Most of the buildings you see are pathetic imitations of something else. Behind the rusted and bent guardrails along the road lie abandoned paper cups, plastic bags and weeds. The roads are vast expanses of asphalt, and the vehicles on the roads are each things unto themselves, as a group without coordination or consistency.

The experience is consistent and continuous only because you have shut out most of the environment. Seen from the outside, the experience is profoundly discontinuous, uncoordinated and inconsistent except for the three environments you were able to cherry-pick out of the continuum. Where in the first example there was an abrupt end to the apparent authenticity as you drilled down in scale, here each of the designed elements – the house, the car and the restaurant – are  islands of order in a sea of chaos. You have to tune out everything except the private cocoons of good design that your high income allow you to enjoy. You must exercise selective attention because there are gaps in the environmental continuum

Example three: You are comparing a Model T with a modern SUV. The form of the Model T reflects separate parts each with a clear function: the passenger compartment, wide at the top and narrower at the bottom, just like the human form; the engine in a separate little house with fold-up doors on each side; the wheels, lights, fenders, running boards and spare tire all clearly articulated. Each function is either shrouded with its own skin, or is a separate element attached to the skin or supporting the whole ensemble. If you look at the car from underneath, you see more functional elements. The vehicle is all of a piece. Also, a dent here and there doesn’t spoil the effect, as the car is not trying to be perfect.

1919 Ford Model T Highboy Coupe
1919 Ford Model T Highboy Coupe

The SUV is a far more complex piece of machinery, but the entire vehicle except the wheels is shrouded with a single, perfect shell. While the shell does in some way reflect function its most striking characteristic is how smoothly the surfaces are blended into a perfect shell, one where the slightest scratch is noticeable and spoils the intended perfection. If you look at the car from underneath, you see something entirely different, a bewildering array of functional elements, entirely discontinuous with the perfect carapace above. As a Model T is to a steam locomotive, the SUV is to a diesel locomotive. Some kind of authenticity is lost between the first pair and the second, a third kind of discontinuity.

We routinely cherry-pick the elements of an object or event that we consider relevant to its significance. In the three examples, it seems that something is wrong with the criteria we use to separate the relevant from the accidental, the signal from the noise. So what constitutes a “right” set of criteria? You can’t consider all aspects of a situation to be relevant. To do so you would have to include the moon, social justice, biological evolution, animal rights, baseball stats – there really are no limits to the potentially relevant contexts. Clearly we have to make choices about which contexts are relevant and which aren’t. To do this, we need criteria.

This conundrum exposes a deeper level to the concept of authenticity. The authenticity of an object or event seems to be determined by the criteria used to establish relevance. They must in some way be “right”; the puzzle is figuring out what “right” means. I believe that if we can identify these criteria, we can learn a great deal about art in particular, and human experience in general.

Authenticity – Part One

Monterey, California 1959
Monterey, California 1959

There is strong evidence from cognitive psychology that we are by nature highly attuned to detecting cheaters. This makes evolutionary sense, because one of our distinctions as a species is our cooperativeness, our ability to trust and share with others. Cheaters naturally arise in an environment of trust, and our obsession with cheater detection limits the number of cheaters in a group. We punish cheaters, as well as those who fail to punish cheaters.

We are also obsessed about genealogy. One of the reasons languages tend to develop arbitrary rules that have little to do with transmitting meaning is to sharpen the ability of experienced speakers to detect outsiders. In “My Fair Lady,” ’Enry ‘Iggins claimed to be able to tell what London block someone came from by their accent.  Early humans evolved in small groups, and it was important to know who was “in” and who was “out,” not only to know who was on your side, but to avoid incest. Likewise, we are exquisitely sensitive to nuances of the appearance and behavior of others. We are deeply interested in whether the person across from us is genuinely smiling, or just putting on a smiley face while wishing you would go away. The battle between our drive to reproduce and our need to establish the authenticity of our partner’s professions of love figures in the plot of many a tale (sadly, this obsession with whether we are insiders or outsiders underlies our xenophobia).

Both art and science rely heavily on these two traits working together. Doing science is intensely cooperative and based on trust, so science cheaters make headlines and lose their credibility, and often their jobs. Also, it must be possible in principle to trace the genealogy of evidence back through all the actions taken to arrive at it. Scientists are highly skeptical and openly critical of each other, and as a result, there is a culture of cooperation and trust that makes it unnecessary to check every step in a discovery unless someone smells a rat, In the end, the findings of science are the consensus of experts (I borrow this from a lecture at the Marine Biological Laboratory in Woods Hole by Harvard historian of science Naomi Oreskes). I have more to say on this subject elsewhere.

Genealogy is important in art for similar reasons. Much of the value of a work of art depends on how sure you are that it is genuine. Musicologists research and argue about whether a note in a score was intended to be an A or an A#, how and when improvised decorative figures were originally used in the Baroque and Classical eras, or whether a newly discovered chorale was written by Bach, a contemporary of Bach, or a present-day charlatan. Never mind that the average listener can’t tell the difference – the experts can. Art that can be faked needs a pedigree to maintain its monetary value, and as these are not always easy to document, cheating is more widespread in art than in science.

One much-discussed subject is why an art work with a pedigree is better in some way. Can’t a work be judged simply through your senses? Imagine three Federal style silver cups. They are all exquisite and physically identical, except that the first is by an unknown silversmith, the second is by Paul Revere, and the third, also by Paul Revere, was owned by George Washington. No doubt their monetary value is in the same order, and escalates dramatically from one to the other. Yet without their pedigrees (no fair turning them over to see the silversmith’s marks on the bottom) they are indistinguishable.

You can see that provenance is much more important in science than in art, because the whole enterprise of science is founded on authenticity, whereas a work of art can in principle stand on its own, although its monetary value and part or most of its public appeal lies in its provenance.

When talking about authenticity, I think most of us would equate it with whether something is genuinely what it is advertised to be. But there are other aspects to authenticity that are more subtle and more interesting, and I want to focus on those in subsequent installments.