Most blogs become defunct, and sooner of later so will mine. But I intend to keep posting. We continue to be in the throes of moving, and I don’t have the mental energy or the time to attend to the website.
I’ll be back soon!
Most blogs become defunct, and sooner of later so will mine. But I intend to keep posting. We continue to be in the throes of moving, and I don’t have the mental energy or the time to attend to the website.
I’ll be back soon!
I love messing with language. I subscribe to “A Word a Day” found at http://Wordsmith.org that posts a new word every weekday and has a weekend compilation of letters, puns, anagrams and limericks based on the week’s words (there are other links worth checking out – the Instant Anagram Server at http://Anagram Server is especially useful). The author of the website, Anu Garg, is an outspoken critic of Trump, as are most of his readers.
When I was a kid, I looked forward to every issue of Collier’s magazine, which had a kid’s page that included aphorisms and stories by “Colonel Stoopnagle.” He was a master of Spoonerisms, expressions in which phonemes are exchanged, named after a professor Spooner who purportedly made such mistakes. Here is an example:
The Orned Howl: He inks his bleyes, and weems very size, but is astoop as boutid as a beed can burr.
The Horned Owl: he blinks his eyes, and seems very wise, but is about as stupid as a bird can be.
Once you get the hang of it, the trick is to make the Spoonerism interesting and natural sounding, especially when you can rearrange the phonemes to come up with new words. Playing cards are useful, as in: who of tarts, hoar of farts, hive of farts, Spain of queeds, spack of jades, clicks of subs, space of aids, and so forth. Some Spoonerisms seem so natural that they stick in the mind: roak porst, sweet-streeper, flied crams.
In a typical Spoonerism (Spoonical typerism), you simply switch the initial phonemes of two words as in the above examples. If you exchange phonemes among more than two words with more than two syllables you can come up with elaborate concoctions that can baffle and irritate your audience, which rather defeats the purpose (for example “econcorate lab octions” for “elaborate concoctions”). Such concoctions require careful bookkeeping.
This reminds me of words with multiple consecutive double letters, with sub-bookkeeper being a prize-winner. A similar game is finding words with the most consecutive consonants. I know of two winners with six consonants each: Knightsbridge and catchphrase.
Another form of wordplay is substituting words that sound similar. In this hilarious version of Little Red Riding Hood, every word is a substitute: https://www.exploratorium.edu/exhibits/ladle/ Be sure to click on the spoken version. For those who don’t want to go to the site, here is the ending:
“Mural: yonder nor sorghum stenches shut ladle gulls stopper torque wet strainers.” Which translates: “Moral: under no circumstances should little girls stop or talk with strangers.”
A true masterpiece is “Mot d’heure, gousse, rames,” (Mother Goose Rhymes) which is best explained in the Wikipedia entry https://en.wikipedia.org/wiki/Mots_d’Heures , which has links and references to other similar works. Per the Wikipedia entry, the book “is purportedly a collection of poems written in archaic French with learned glosses. In fact, they are English-language nursery rhymes written homophonically as a nonsensical French text (with pseudo-scholarly explanatory footnotes).”
When our kids were young, we played a game during dinner in which we would start with a word, then one of us would come up with a definition of a similar sounding word. The next person would break in with “no, that’s (word),” then a new definition, new word, and so on. We would end up weeping with laughter.
Edward Gorey was an artist and writer known for “His characteristic pen-and-ink drawings [that] often depict vaguely unsettling narrative scenes in Victorian and Edwardian settings” to quote his Wikipedia entry. We own a wonderful little book he wrote called “15 Two”, now a collectors item. It imitates didactic Victorian friezes that one would apply around the walls of the nursery, in a format that would allow you to copy it as a frieze.
The book shows strange animals that look like a cross between a pig and a hippo following each other, each with a speech balloon containing a word. Large block capital letters standing in the bleak landscape spell out “The Nursery Frieze Edward Gorey,” perversely starting with the “E” in “nursery.”
The words in the speech balloons form rhymes, and are mix of obscure and common words:
Archipelago, cardamon, obloquy, tacks
Ignavia, samisen, bandages, wax
Gavelkind, turmeric, imbat, cedilla
Cassation, hendiadys, quincunx, vanilla
Corposant, madrepore, ophicleide, paste
Jequirity, tombola, sphagnum, distaste
Aceldema, lunistice, yarborough, cranium
Febrifuge, ampersand, hubris, geranium
Opopanax, thunder, dismemberment, baize
Hellebore, obelus, cartilage, maze
Antigropelos, piacle, occamy, whistle
Maremma, accismus, badigeon, epistle
Quodlibet, catafalque, hiccup, remorse
Idioticon, gibus, botargo, divorce
Phylactery, gegenschein, clavicle, sago
Ballonion, thurible, aphthong, plumbago
Amaranth, rhoncus, pantechnicon, hymn
Diaeresis, purlicue, sparadrap, whim
Cicatrix, salsify, palindrome, Bosphorus
Narthex, betrayal, chalcedony, phosphorus
Ligament, exequies, spandrel, chandoo
Gehenna, etui, anamorphosis, glue
Wapentake, orrery, aspic, mistrust
Ichor, ganosis, velleity, dust.
The ominous ending is typical Gorey. I’ve just scratched the surface of wordplay, and hope you will explore its pleasures.
It is common to ask yourself what music you would choose to take with you to a desert island. Instead, I will use this criterion: if I were listening to a broadcast while working on a project, what music would cause me to stop working and listen intently? This by no means is a list of music I revere, it is just those particular pieces that I have to stop and listen to no matter what. I am partial to fugues, luscious orchestral sound, and piano music I used to (attempt to) play.
I can’t imagine why anyone would be interested in this list.
Vivaldi: six violin concertos “La Stavaganza” Opus 4 (almost never played for some reason – I had a recording of it in college that I played incessantly)
Bach: both books of the “Well-Tempered Clavier”; “The Art of Fugue”; the Goldberg Variations; the choral fantasias from the “St. Matthew Passion” that open and close the first part, and the closing fantasia; the two great fugues on Kyrie Eleison and the Crucifixus from the Mass in b minor; and the choral fantasia that opens Cantata 8, “Liebster Gott, wenn werd ich sterben?.” Plus….
Mozart: Barbarina’s lament for a lost pin that begins Act IV of “Le Nozze di Figaro”; the opening Kyrie of the Great Mass in c minor; most of his piano sonatas. Plus….
Beethoven: The first movement of Opus 130 string quartet (top of my list along with Bach’s Well-Tempered Clavier); the first (fugue) and middle (theme and variations) movements of Opus 131 string quartet; piano sonata opus 106 “Hammerklavier”, especially the closing fugue; Diabelli Variations opus 120; many other string quartets and piano sonatas
Mendelssohn: complete incidental music to “Midsummer’s Night’s Dream” with singing and recitation
Chopin: Preludes; 4th movement, Sonata #2
Brahms: Requiem; “Variations and Fugue on a Theme by Handel” opus 24; “Liebeslieder Waltzes” Opus 52
Borodin: Polovtsian Dances; “In the Steppes of Central Asia”
Dvorak: “Serenade for Wind Instruments” opus 44; opening movement of “Stabat Mater”.
Tchaikovsky: “Romeo and Juliet”
Kodaly: “Hary Janos” (recording out of print, narrated by Peter Ustinov)
Bartok: Opening fugue of “Music for Strings, Percussion and Celesta”; String Quartet #4.
Ravel: “Gaspard de la Nuit”
Stravinsky: “Le Sacre du Printemps”; “Le Chant du Rossignol”; “L’Histoire du Soldat”; “Symphony of Psalms”
Hindemith: “Symphonic Metamorphosis of Themes by Carl Maria von Weber”
Prokofiev: Symphony #5, second movement; “Cinderella” ballet, clock scene; Toccata; “Alexander Nevsky” cantata
Khachaturian: Piano Concerto (second-rate piece I happen to have loved as a youth) played by William Kapell
Britten: “Rejoice in the Lamb”; Serenade for Horn, Tenor and Strings; Four Sea Interludes from “Peter Grimes”
Weill: “September Song”
Beatles: “Fool on a Hill”
Our son David inherited his grandmother’s 1969 white Pontiac convertible with red vinyl upholstery. It was sexy, but it was a beast, a gas-guzzler and perfect example of the dangerous designs Ralph Nader successfully fought against during the same era. Just to list a few of its more egregious features, it had a rigid frame, lap belts, bench seats, tiny rear view mirrors, manual windows and door locks, carburetor, spongy suspension and a big V-8 motor that drank a half-pint of fuel every mile. Its heating, air conditioning and ventilation system hardly functioned and the radio was a joke. The bumpers could not withstand a 5 mph collision. It did have power steering and power brakes – drum brakes. It was huge.
In 2015, when we sold the Pontiac, I still owned a 1992 Camry station wagon I had bought used many years before. In the 23 years between 1969 and 1992, all major mechanical, safety and comfort issues had been addressed. It had a transverse V-6 engine with fuel injection driving the front wheels that delivered 25 mpg on the highway (its in-city mileage is not that great because of the big engine, but still was more than twice that of the Pontiac). It had a driver’s side air bag, over the shoulder seat belts, individual adjustable front seats, electric windows and door locks, rear window wipers, big motor-adjusted mirrors, cruise control, an impact-absorbing frame, highly effective rust protection, ABS disk brakes, impact-resistant bumpers, a modern suspension system, and an excellent radio with tape and CD players. It was quiet, comfortable, capacious, compact and reliable. In short, it was a fully modern car.
In the twenty-three years between 1992 and 2015, what important features have been added? More air bags, endless electronics and other small refinements, some very nice but of minimal importance and at a substantial increase in maintenance. All aspects have been refined, but there have been no innovations that increase safety, durability or efficiency remotely comparable to the dramatic changes that took place between 1969 and 1992. In addition, the progress made in compacting the car between 1969 and 1992 has been reversed. We are now used to tall huge vehicles instead of the low and wide huge vehicles we loved in the 60’s.
This is an example of one way in which “progress” is a misnomer: adding expensive, high-maintenance incremental changes to mature products. An important factor in the escalation of medical care in advanced societies is expending large sums to add a small increment of time at the end of life. I am very glad I have a titanium knee and a piece of cow in my heart, because otherwise I would not be nearly as active as I am, and might well have expired. But these surgeries cost tens of thousands of dollars, and it is simply impossible to extend such benefits to billions of people; we can’t even do so for a large fraction of our own citizens.
And can anyone argue that the difference between iPhones six and seven is remotely comparable to the difference between a flip phone and an iPhone? Or between no cell phone and a cell phone?
Another form of spurious progress is the loss of functionality in the service of reducing costs. Not only is ours a throw-away economy, but the products often break down after only a few hours or even minutes of use. In countless cases, new products are less reliable, less durable and less functional than those they replace.
We purchased a cheap set of wooden lawn furniture made with some kind of tropical hardwood (Americans import 95% of the tropical hardwoods, a large fraction of which are harvested unsustainably). To reduce the assembly cost of the chairs, the slats that formed the seats were designed with joints that were guaranteed to fail if a robust adult sat on them. Edward O. Wilson once commented that cutting tropical rain forest for profit was like burning a Renaissance painting in order to cook dinner. Our tropical chairs made a nice fire one chilly evening.
To keep the economic growth engine running, governments, corporations and consumers (previously known as citizens) need to buy more stuff. Some of the new stuff is indeed useful, notably electronic devices. Even in that case, functional innovations are becoming marginal, and sometimes (as in the case of Microsoft operating systems) run in reverse.
There are exceptions. One is scientific instruments, where innovation and refinement allow us to uncover entirely new layers of natural phenomena, genome sequencing and deep space imaging being only two examples. Another is technology that has enhanced the arts.
It’s too bad all this extra stuff isn’t making us happier: even though at the moment Americans are the safest people who have ever lived on earth, we don’t feel safe. And the religion of growth is making us much less safe in the long run.
As I argue in other essays, the survival of human civilization requires moving from the economy we have to one where growth is parceled out in ways that are productive, equitable and sustainable.
This will require a fundamental shift from an emphasis on competition to one of cooperation, sharing, redistribution and making hard choices to shed amenities we can’t support. No one has yet figured out how to make these hard choices in a way that will be acceptable to a free people in a liberal economy.
My wife and I, my daughter and her family, and almost everyone I know continue to consume like mad, just like everyone else who can afford it and way too many who can’t (our son is admirably abstemious). We don’t know what else to do, and I suspect you are in the same boat.
It is time to put the sapiens back in Homo sapiens, but it will require us to work together, give up many luxuries, and base our actions on science. This is not exactly a recipe for getting elected to public office.
Born in 1935, I began to be vaguely aware of the wider world in 1941, at age 6, when we declared war on Japan. So 75 years of sentience have passed for me as I write in 2016.
It is hard for me to realize that had I been born just one lifetime earlier, in 1855, the Civil War would have replaced WWII, and I now would be living in the depths of the Great Depression. Further, my life expectancy at birth would have been 43 years instead of 65 years. Another lifetime earlier, and from then all the way back, it would have been in the 30’s.
So it seemed useful to look back in time in 75-year steps – my long lifetime – beginning in 2015 to keep the numbers simple. As I go back in time, there is less detail. Partly this is a matter of perspective, but mainly it is that events with crucial implications for the future are occurring much more rapidly. A decade today is equivalent to a century a few hundred years ago, millennia before that, and hundreds of millennia as humans evolved from earlier species.
The timeline begins at the present, and steps back for 10 lifetimes. After that, the steps become progressively longer. Prior to the point at which the human lineage splits off from that leading to chimps, the steps become snapshots of key evolutionary events.
YBP means “years before the present”. I will omit “CE” for common era dates (formerly AD), and stick with “BC” instead of “BCE” (before the common era).
My Lifetime: 1940-2015
Human population more than triples
Humans and their livestock and pets now account for 98 percent of the world’s vertebrate biomass (mammals, marsupials, reptiles, amphibians, fish, birds). 10,000 years ago, they accounted for one-tenth of one percent.
Global warming is confirmed and accelerates
Sixth Extinction is confirmed and accelerates
Communism, Nazism, WWII, Chinese Revolution, Cold War, numerous regional wars
Invention, use and proliferation of nuclear weapons – humans gain the capacity to exterminate all complex life on earth through “nuclear winter”
Plastics revolutionize material culture, create extensive ocean pollution
Tens of thousands of man-made chemicals created and disseminated
Electronics revolutionizes communication and becomes the primary information storage medium
All technology becomes dependent upon electricity
Consumerism and the necessity of growth becomes a dominant global force
Free trade and air travel link all people economically
Muslim radicalism erupts
Growing migration of populations displaced by war
Lifetime Two: 1865-1940
Height of Nationalism and Colonialism
All science is integrated around seminal discoveries in physics
Western countries are electrified
Coal-based railroads and steamships revolutionize transportation
Petroleum-based technology revolutionizes transportation again (automobiles and aircraft)
Modern medicine matures, extending life expectancy
Russian Revolution, WWI
Lifetime Three: 1790-1865
Industrial revolution becomes a dominant force
Coal-based technologies arise
Middle classes expand in Europe and America
Darwin undermines the foundation of religion
Nationalism and Colonialism growing
Most of eastern U.S. deforested
Lifetime Four: 1715-1790
American and French revolutions
Early industrial revolution
Lifetime Five: 1640-1715
Newton revolutionizes physics and invents the calculus
Louis XIV guides France to its apex of power and sets the stage for its decline
Wars in Europe continue
Lifetime Six: 1565-1640
Galileo and the rediscovery of experimental science
Luther triggers the Reformation, which leads to the wars of religion
Age of Elizabeth and Shakespeare
Descartes revolutionizes philosophy and mathematics
Spanish are the dominant power in Europe
Lifetime Seven: 1490-1565
Voyages of discovery
Destruction of Pre-Columbian civilizations and their written history by the Spanish
Lifetime Eight: 1415-1490
Ottomans conquer Byzantium
High Renaissance in Europe
Lifetime Nine: 1340-1415
Black Death kills one in three Europeans
Schism fractures the Catholic Church
Tamerlane invades middle Asia
Rise of Ottoman Empire
Lifetime Ten: 1265-1340
Islam in retreat, under attack in east and west
Late Middle Ages in Europe
Nation states arise in Europe
Increased contact between East and West
Lifetimes 20-10: 515 – 1265
Feudalism dominates for 500 years in Europe after Germanic invasions
Catholic church becomes the predominant power in Europe
Medieval revival of European culture
Origin and explosive expansion of Islam
Golden age of Islamic science
Lifetimes 30-20: 235 BC to 515
The Roman Empire replaces the Roman Republic and disintegrates after 500 years of dominance
The invasion of the Huns sets off the Great Migration of Germanic tribes into Europe
Christianity arises and becomes the dominant religion in Europe.
Roman culture survives in the Byzantine Empire
Lifetimes 50-30: 1735 BC to 235 BC
Written history arises in China
Urban civilizations arise in India and Meso-America
Classical Greek culture
Alexander the Great reaches India
Roman republic arises and conquers Italy
Deforestation for agriculture, ship-building and fuel begins, continuing to the present
Lifetimes 75-50: 3610 BC to 1735 BC
Invention of writing and recorded history
First civilization – Sumer, c. 3,300 BC
First Egyptian dynasty c. 3,100 BC
Harrapan civilization arises in the Indus Valley c. 2,600 BC
Pre-literate civilizations arise in Meso-America
First Chinese dynasty c. 2,000 BC
Lifetime 100-75: 5,485 BC to 3,610 BC
Pre-literate civilizations arise in Iraq, Egypt and Pakistan
Lifetimes 150-100: 9,235 BC to 5,485 BC
End of the last glacial period
Invention of agriculture and animal husbandry
The first towns appear
Lifetimes 200-100: 12,985 BC to 9,235 BC
Humans colonize North America via the land bridge across the Bering Strait.
As occurred elsewhere soon after colonization by humans, nearly all the megafauna in the Americas become extinct.
Lifetimes 500-200: 35,500 YBP to 12,985 BC
The last of our close relatives, the Neanderthals and Denisovans, disappear, after interbreeding with modernt humans
Humans colonize the entire earth except for the Americas, Antarctica, and some Pacific Islands
Lifetimes 1,000-500: 75,000 YBP to 35,500 YBP
Fully mature language evolves
The first confirmed evidence of art occurs
Modern humans (Homo sapiens) colonize Eurasia, Australia and some Pacific islands
Lifetimes 100,000-3,000: 7,500,000 YBP to 225,000 YBP
The lineage of African great apes leading to humans, their ancestors and cousins (Hominins) splits from the lineage leading to Chimpanzees and Bonobos (genus Pan); Gorillas and Orangutans had split off earlier
Numerous Hominin genuses and species arise, a few of which are ancestral to humans (no one knows which ones)
Members of the genus Homo migrate out of Africa at various times. Homo erectus Hominins migrate into Middle East, East Asia and India about 2,000,000 YBP
Stone tools and fire are invented
Upright posture frees hands to use projectile weapons, giving Hominins a crucial advantage in hunting: attacking from a distance
Early form of language develop
Lifetime 870,000: 65,000,000 YBP
A great extinction event triggered by a comet impact results in extinction of dinosaurs (except birds), allowing the dominance of mammals
Lifetime 2,100,000: 160,000,000 YBP
Origin of flowering plants resulting in major increase in atmospheric oxygen, allowing the evolution of large terrestrial animals
Lifetime 7,200,000: 540,000,000 YBP
The “Cambrian Explosion” populates the seas with macroscopic organisms, the ancestors of all our phyla of animals
Soon thereafter, animals, plants and fungi move onto land
Lifetime 28,000,000: 2,100,000,000 YBP
The “Great Oxygen Event” occurs when enough oxygen accumulates in the atmosphere to cause a massive extinction event and the evolution of oxygen-dependent organisms
Eukaryotes, much more complex but still microscopic organisms, form by the merging of specialized prokaryotes and viruses.
Lifetime 50,000,000: 3.700,000,000 YBP
Life arises, probably in many different forms, until one chemical system becomes the standard template for all subsequent life
All life for the next 2 billion years consists of prokaryotes – bacteria and bacteria-like “archaea”, along with viruses
Lifetime 61,000,000: 4,600,000,000 YBP
The sun ignites at the center of a rotating disk of dust, ice, heavy elements and gas, in a quiet “suburb” between spiral arms of the Milky Way Galaxy
The earth, other planets, and countless larger and smaller fragment coalesce from the rotating disk
Large and small fragments bombard the earth. One the size of Mars strikes a glancing blow to the earth, and the resulting debris coagulates into our moon.
The bombardment gradually tapers off over half a billion years, until the earth cools enough for liquid water to form; it is uncertain where the water came from
Lifetime 183,000,000: 13,700,000,000 YPB
The universe expands abruptly from a tiny kernel in the “Big Bang” and the “Cosmic Inflation” that immediately followed, all within an infinitesimal fraction of a second. Or at least that what cosmologists thought last week.
After the “Dark Ages”, stars begin to form and our galaxy begins to coalesce, a process that continues to the present
Next Lifetime: 2015-2090
I see the next lifetime as a climactic episode of dramatic change in human civilization and global ecosystems. Nobody knows what the world in 2090 will be like, but it is certain that it will not remotely resemble that of 2015. It is extremely unlikely that it will be as comfortable, populous and orderly.
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 https://localwiki.org/oakland/Geodesic_Bird_Dome
The Geodesic Bird Dome is a part of the Lake Merritt Wildlife Refuge in Oakland, California. Built in  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.
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 https://en.wikipedia.org/wiki/Platonic_solid )
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.
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 https://bfi.org/about-fuller/resources/everything-i-know/session-11 , 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 https://www.google.com/patents/US2891491
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.