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Einstein’s theory of diffusion in liquids, and my extension.

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In 1905 and 1908, Einstein developed two formulations for the diffusion of a small particle in a liquid. As a side-benefit of the first derivation, he demonstrated the visible existence of molecules, a remarkable piece of work. In the second formulation, he derived the same result using non-equilibrium thermodynamics, something he seems to have developed on the spot. I’ll give a brief version of the second derivation, and will then I’ll show off my own extension. It’s one of my proudest intellectual achievements.

But first a little background to the problem. In 1827, a plant biologist, Robert Brown examined pollen under a microscope and noticed that it moved in a jerky manner. He gave this “Brownian motion” the obvious explanation: that the pollen was alive and swimming. Later, it was observed that the pollen moved faster in acetone. The obvious explanation: pollen doesn’t like acetone, and thus swims faster. But the pollen never stopped, and it was noticed that cigar smoke also swam. Was cigar smoke alive too?

Einstein’s first version of an answer, 1905, was to consider that the liquid was composed of atoms whose energy was a Boltzmann distribution with an average of E= kT in every direction where k is the Boltzmann constant, and k = R/N. That is Boltsman’s constant equals the gas constant, R, divided by Avogadro’s number, N. He was able to show that the many interactions with the molecules should cause the pollen to take a random, jerky walk as seen, and that the velocity should be faster the less viscous the solvent, or the smaller the length-scale of observation. Einstein applied the Stokes drag equation to the solute, the drag force per particle was f = -6πrvη where r is the radius of the solute particle, v is the velocity, and η is the solution viscosity. Using some math, he was able to show that the diffusivity of the solute should be D = kT/6πrη. This is called the Stokes-Einstein equation.

In 1908 a French physicist, Jean Baptiste Perrin confirmed Einstein’s predictions, winning the Nobel prize for his work. I will now show the 1908 Einstein derivation and will hope to get to my extension by the end of this post.

Consider the molar Gibbs free energy of a solvent, water say. The molar concentration of water is x and that of a very dilute solute is y. y<<1. For this nearly pure water, you can show that µ = µ° +RT ln x= µ° +RT ln (1-y) = µ° -RTy.

Now, take a derivative with respect to some linear direction, z. Normally this is considered illegal, since thermodynamic is normally understood to apply to equilibrium systems only. Still Einstein took the derivative, and claimed it was legitimate at nearly equilibrium, pseudo-equilibrium. You can calculate the force on the solvent, the force on the water generated by a concentration gradient, Fw = dµ/dz = -RT dy/dz.

Now the force on each atom of water equals -RT/N dy/dz = -kT dy/dz.

Now, let’s call f the force on each atom of solute. For dilute solutions, this force is far higher than the above, f = -kT/y dy/dz. That is, for a given concentration gradient, dy/dz, the force on each solute atom is higher than on each solvent atom in inverse proportion to the molar concentration.

For small spheres, and low velocities, the flow is laminar and the drag force, f = 6πrvη.

Now calculate the speed of each solute atom. It is proportional to the force on the atom by the same relationship as appeared above: f = 6πrvη or v = f/6πrη. Inserting our equation for f= -kT/y dy/dz, we find that the velocity of the average solute molecule,

v = -kT/6πrηy dy/dz.

Let’s say that the molar concentration of solvent is C, so that, for water, C will equal about 1/18 mols/cc. The atomic concentration of dilute solvent will then equal Cy. We find that the molar flux of material, the diffusive flux equals Cyv, or that

Molar flux (mols/cm2/s) = Cy (-kT/6πrηy dy/dz) = -kTC/6πrη dy/dz -kT/6πrη dCy/dz.

where Cy is the molar concentration of solvent per volume.

Classical engineering comes to a similar equation with a property called diffusivity. Sp that

Molar flux of y (mols y/cm2/s) = -D dCy/dz, and D is an experimentally determined constant. We thus now have a prediction for D:

D = kT/6πrη.

This again is the Stokes Einstein Equation, the same as above but derived with far less math. I was fascinated, but felt sure there was something wrong here. Macroscopic viscosity was not the same as microscopic. I just could not think of a great case where there was much difference until I realized that, in polymer solutions there was a big difference.

Polymer solutions, I reasoned had large viscosities, but a diffusing solute probably didn’t feel the liquid as anywhere near as viscous. The viscometer measured at a larger distance, more similar to that of the polymer coil entanglement length, while a small solute might dart between the polymer chains like a rabbit among trees. I applied an equation for heat transfer in a dispersion that JK Maxwell had derived,

where κeff is the modified effective thermal conductivity (or diffusivity in my case), κl and κp are the thermal conductivity of the liquid and the particles respectively, and φ is the volume fraction of particles. 

To convert this to diffusion, I replaced κl by Dl, and κp by Dp where

Dl = kT/6πrηl

and Dp = kT/6πrη.

In the above ηl is the viscosity of the pure, liquid solvent.

The chair of the department, Don Anderson didn’t believe my equation, but agreed to help test it. A student named Kit Yam ran experiments on a variety of polymer solutions, and it turned out that the equation worked really well down to high polymer concentrations, and high viscosity.

As a simple, first approximation to the above, you can take Dp = 0, since it’s much smaller than Dl and you can take Dl to equal Dl = kT/6πrηl as above. The new, first order approximation is:

D = kT/6πrηl (1 – 3φ/2).

We published in Science. That is I published along with the two colleagues who tested the idea and proved the theory right, or at least useful. The reference is Yam, K., Anderson, D., Buxbaum, R. E., Science 240 (1988) p. 330 ff. “Diffusion of Small Solutes in Polymer-Containing Solutions”. This result is one of my proudest achievements.

R.E. Buxbaum, March 20, 2024

China won’t invade Taiwan, perhaps Vietnam

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For 2000 years Chinese rulers have either fixed the problems caused by their predecessors, or become the absolute ruler who brings new problems and a war. Eighteen times over the last 2000 years, the absolute ruler has chosen is to invade Vietnam. Chairman Xi Jinping took full power over China five years ago, and as I predicted then, purged the party of all other competent leaders, including his predecessor, Hu. He now has to do something, and that’s typically a war. Xi talks like he’d like to invade Taiwan, but I believe he’ll invade Vietnam instead, as so many others did before him. Taiwan (Formosa) is separated from mainland China by 100 miles of open sea. There’s been only two successful, modern invasions; by the Qing Chinese in 1683 with the help of the Dutch fleet, and in 1895 by Japan against the 5 month old Republic of Taiwan. Vietnam is much easer to invade: you just have to walk in.

Chiang KaiShek with Roosevelt and Churchill. He took power after WWII.Taiwan is now a Republic

Here’s my brief summary of 60 years of cyclical Chinese history: In the last decade of his life, 1966-76, Mao Zedung brought a horrible cultural revolution, killing 100 million or so by violence and famine, targeting anyone who might disagree with him. He was followed by Hua Kuofeng, Hu Yaobang, and Zhao Ziyang. They removed “the Gang of Four” and brought reform, toleration, and some rapprochement with the west. Hu also returned some autonomy to Tibet. Deng Xiaoping followed, put Zhao under house arrest, removed Hua and Hu (only recently buried), absorbed Tibet, invaded Vietnam, instituted a brutal on-child policy with, forced abortions and sterilizations, and put down the Tiananmen uprising 1989 -an uprising caused by the removal of Hu. Deng was followed by Jiang Zemin and Hu Jintao who exited Vietnam, lightened the one-child policy, and brought back some measure of freedom. This lead to Xi Jinping who appears to be in the mold of Mao and Deng. He’s removed Hu publicly during a communist party meeting, has taken absolute control, arrested China’s business innovators, and (likely) caused a pandemic by unsafe research at the Wuhan bio-research facility. His response to the pandemic is worthy of Mao: he welded people into their homes. Xi now needs a war to unify the country, and talks like he’ll invade Taiwan. Xi might do it; he has increased military spending by 2.5 times, to double that of the EU (equal to US spending). His army is likely to be used somewhere soon. But where?

At first glance, it makes some sense to think he’ll invade Taiwan. He has the same justification as Deng had for Vietnam, “to punish the wayward province.” Taiwan is small, 23 million people, and very rich (GDP = $1.3T, 10-15% of China). It’s especially rich in high-tech areas that Xi seems to want, and China ruled the island (Formosa) for 212 years between 1683 and its brief independence in 1895. I expect that Xi will invade Vietnam though, for many of the same reasons that Deng did: it’s easier, and the invasion won’t destroy US trade. Vietnam will not be super easy to conquer, of course, Putin’s invasion of Ukraine looks like it will go on for years, but Xi has 100 million men of military age. Driving these men into Vietnam is a lot easier than an amphibious invasion. Vietnam now, as then, has few friends -perhaps only India -while Taiwan has many (The US and Japan, primarily), and they have naval breakwaters that would make an amphibious landing difficult. Vietnam is only about half as rich as Taiwan (GDP = $650 B), but it has two things China needs more than technology: oil and food.

The Japanese invasion of independent Taiwan, The Republic of Formosa, in 1895. It took 5 months despite lopsided advantages. Plus 7 years to pacify the population.

Japan’s invasion of the Republic of Formosa in 1895 would have to be the model for a Chinese invasion today, assuming that’s the route Xi would take. Japan used a small force of 20,000 highly trained men, and a surprise landing at two sites. The formosan army of 75,000 was hardly armed, but it still took the Japanese 5 months to defeat them. It took another 7 years to pacify the population. Taiwan today is a lot better prepared than it was, with modern missiles and drones, a well-trained army of 500,000, and an active navy. Japan, Philippines, and the US would likely try to resupply Taiwan, and would have to be blockaded to prevent delivery. This is not so easy, since the nearest Japanese island is closer to Taiwan than Taiwan is to mainland China, and the Philippines is not far either. Then there are the sanctions that would follow an invasion of Taiwan, but not an invasion of Vietnam: the freezing of assets and the closing of markets. Susch sanctions have been tough on Russia but they would be devastating to China since China imports most of its oil and half of its food, much of it by sea. Whatever Xi gets from invading Taiwan will not match these losses, and I think Xi know it.

China’s new military bases are located around Vietnam, not around Taiwan

I suspect that Xi has already concluded that Vietnam is the smart place to invade. Supporting this view is a look at his preparations. Xi has not built the landing fleet that he’d need for Taiwan, but has instead militarized Hainan Island, plus four other, newly constructed, illegal islands in the South China Sea. These islands surround Vietnam, and are well suited to cut off the country from western aid. They are too far south to be effective in an action against Taiwan. You can tell a lot about what a person will do from what he has done, and what Xi has done is to prepare for an invasion of Vietnam.

Robert Buxbaum, April 27, 2023.

Of walking sticks, canes, scepters, and wands.

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Franklin’s walking stick, willed to General Washington. Now in the Smithsonian.

Many famous people carried walking sticks Washington, Churchill, Moses, Dali. Until quite recently, it was “a thing”. Benjamin Franklin willed one, now in the Smithsonian, to George Washington, to act as a sort of scepter: “My fine crab-tree walking stick, with a gold head curiously wrought in the form of the cap of liberty, I give to my friend, and the friend of mankind, General Washington. If it were a Scepter, he has merited it, and would become it. It was a present to me from that excellent woman, Madame de Forbach, the dowager Duchess of Deux-Ponts”. A peculiarity of this particular stick is that the stick is uncommonly tall, 46 1/2″. This is too tall for casual, walking use, and it’s too fancy to use as a hiking stick. Franklin himself, used a more-normal size walking stick, 36 3/8″ tall, currently in the collection of the NY Historical Society. Washington too seems to have favored a stick of more normal length.

Washington with walking stick

Walking sticks project a sort of elegance, as well as providing personal protection. Shown below is President Andrew Jackson defending himself against an assassin using his walking stick to beat off an assassin. He went on to give souvenir walking sticks to friends and political supporters. Sticks remained a common political gift for 100 years, at least through the election of Calvin Coolidge.

Andrew Jackson defends himself.

I started making walking sticks a few years back, originally for my own use, and then for others when I noticed that many folks who needed canes didn’t carry them. It was vanity, as best I could tell: the normal, “old age” cane is relatively short, about 32″. Walking with it makes you bend over; you look old and decrepit. Some of the folks who needed canes, carried hiking sticks, I noticed, about 48″. These are too tall to provide any significant support, as the only way to grasp one was from the side. Some of my canes are shown below. They are about 36″ tall, typically with a 2″ wooden ball as a head. They look good, you stand straight, and they provides support and balance when going down stairs.

Some of my walking sticks.

I typically make my sticks of American Beech, a wood of light weight, with good strength, and a high elastic modulus of elasticity, about 1.85 x106 psi. Oak, hickory, and ash are good options, but they are denser, and thus more suited to self-defense. Wood is better than metal for many applications, IMHO, as I’ve discussed elsewhere. The mathematician Euler showed the the effective strength of a walking stick does not depend on the compressive strength but rather on elastic constant via “the Euler buckling equation”, one of many tremendously useful equations developed by Leonhard Euler (1707-1783).

For a cylindrical stick, the maximum force supported by a stick is: F = π3Er4/4L2, where F is the force, r is the radius, L is the length, and E is the elastic modulus. I typically pick a diameter of 3/4″ or 7/8″, and fit the length to the customer. For a 36″ beech stick, the buckling strength is calculated to be 221 or 409 pounds respectively. I add a rubber bottom to make it non–scuff and less slip-prone. I sometimes add a rope thong, too. Here is a video of Fred Astaire dancing with this style of stick. It’s called “a pin stick”, in case you are interested because it looks like a giant pin.

Country Irishmen are sometimes depicted with a heavy walking stick called a Shillelagh. It’s used for heavier self-defense than available with a pin-stick, and is generally seen being used as a cudgel. There are Japanese versions of self defense using a lighter, 36″ stick, called a Han-bo, as shown here. There is also the wand, as seen for example in Harry Potter. It focuses magical power. Similar to this is Moses’s staff that he used in front of Pharaoh, a combination wand and hiking stick as it’s typically pictured. It might have been repurposed for the snake-on-a-stick that protects against dark forces. Dancing with a stick, Astaire style, can drive away emotional forces, while the more normal use is elegance, and avoiding slips.

Robert Buxbaum, April 20, 2023.

Comparing Artemis SLS to Saturn V and Falcon heavy

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This week, the Artemis I, Orion capsule splashed down to general applause after circling the moon with mannequins. The launch cost $4.1 Billion, and the project, $50 Billion so far, of $93 Billion expected. Artemis II will carry people around the moon, and Artemis III is expected to land the first woman and person of color. The goal isn’t one I find inspiring, and I feel even less inspired by the technology. I see few advances in Artemis compared to the Saturn V of 50 years ago. And in several ways, it looks like a step backwards.

The graphic below compares the Artemis I SLS (Space Launch System) to the Saturn V. The SLS is 10% lighter, but the payload is lighter, too. It can carry 27 tons to the moon, while the Saturn V sent 50 tons to the moon. I’d expect more weight by now. We have carbon fiber and aramids, and they did not. Add to this that the cost per flight is higher, $4.1 B, versus $1.49 B in 2022 dollars for a Saturn V ($185 million in 1969 dollars). What’s more there was no new engine development or production, so the flight numbers are limited: Each SLS launch throws away five, space shuttle engines. When they are all gone, the project ends. We have no plans or ability to make more engines.

Comparison of Apollo Saturn V and Artemis SLS. The SLS has less lift weight and costs more per launch.

As it happens, there was a better alternative available, the Falcon heavy from SpaceX. The Falcon heavy has been flying for 5 years now, and costs only $141 million per launch, about 1/30 as much as an Artemus launch. The rocket is largely reusable, with 3D printed engines, and boosters that land on their tails. Each SLS is expensive because it’s essentially a new airplane built specially for each flight. Every part but the capsule is thrown away. Adding to the cost of SLS launches is the fuel; hydrogen, the same fuel as the space shuttle. Per energy it’s very expensive. The energy cost for the SLS boosters is high too, and the efficiency is low; each SLS booster costs $290M, more than the cost of two Falcon heavy launches. Falcon launches are cheap, in part because the engines burn kerosine, as did the Saturn V at low altitude. Beyond cost hydrogen has low thrust per flow (low momentum), and is hard to handle; hydrogen leaks caused two Artemis scrubs, and numerous Shuttle delays. I discussed the physics of rocket engines in a post seven years ago.

This graph of $/kg to low earth orbit is mostly from futureblind.com. I added the data for Artemis SLS. Saturn V and Falcon use cheaper fuel and a leaner management team.

It might be argued that Artemis SLS is an inspirational advance because it can lift an entire moon project in one shot, but the Saturn V lifted that and more, all of Skylab. Besides, there is no need to lift everything on one launch. Elon Musk has proposed lifting in two stages, sending the moon rocket and moon lander to low earth orbit with one launch, then lifting fuel and the astronauts on a second launch. Given the low cost of a Falcon heavy launch, Musk’s approach is sure to save money. It also helps develop space refueling, an important technology.

Musk’s Falcon may still reach the moon because NASA still needs a moon lander. NASA has awarded the lander contract to three companies for now, Jeff Bezos’s Blue Origin, Dynetics-Aerodyne makers of the Saturn V, and Musk’s SpaceX. If the SpaceX version wins, a modified Falcon will be sent to the moon on a Falcon heavy along with a space station. Artemis III will rendezvous with them, astronauts will descend to the moon on the lander, and will use the lander to ascend. They’ll then transfer to an Orion capsule for the return journey. NASA has also contracted with Bezos’s Blue origin for planetary, Earth observation, and exploration plans. I suspect that Musk’s lander will win, if only because of reliability. There have been 59 Falcon launches this year, all of them with safe landings. By contrast, no Blue Origin or Dynetics rocket has landed, and Blue Origin does not expect to achieve orbital velocity till 2025.

As best I can tell, the reason we’re using the Artemis SLS with its old engines is inspiration. The Artemis program director, Charlie Blackwell-Thompson is female, and an expert in space shuttle engines. Previous directors were male. Previous astronauts too were mostly male. Musk is not only male, but his products suffer from him being considered a horrible person, a toxic male, in the Tony Stark (Iron Man) mold. Even Jeff Bezos and Richard Branson are considered better, though their technology is worse. See my comparison of SpaceX, Virgin Blue, and Blue Origin.

To me, the biggest blocks to NASA’s inspirational aims, in my opinion, are the program directors who gave us the moon landing. These were two Nazi SS commanders (SS Sturmbannführers), Arthur Rudolph and Wernher Von Braun. Not only were they male and white, they were barely Americanized Nazis, elevated to their role at NASA after killing off virtually all of their 20,000, mostly Jewish, slave workers making rockets for Hitler. Here’s a song about Von Braun, by Tom Lehrer. Among those killed was Von Braun’s professor. In his autobiography, Von Braun showed no sign of regret for any of this, nor does he take blame. The slave labor camp they ran, Dora-Mittelbau, had the highest death rate of all slave labor camps, and when some workers suggested that they could work better if they were fed, the directors, Rudolph and Von Braun had 80 machine gunned to death. Still, Von Braun got us to the moon, and his inspirational comments line the walls at NASA, Kennedy. Blackwell-Thompson and Bezos are surely more inspirational, but their designs seem like dead ends. We may still have to use Musk’s SpaceX if we want a lander or a moon program after the space shuttle’s engines are used up. As Von Braun liked to point out, “Sacrifices have to be made.”

Robert Buxbaum, December 21, 2022. Here’s a bit more about Rudolph, von Braun, the Peenemünda rocket facility, and the Dora-Mittelbau slave labor camp. I may post photos of Von Braun with Hitler and Himmler in SS regalia, but feel uncomfortable doing so at the moment. I feel similarly about posting links to Von Braun’s inspirational interviews.

My hero, Peter Cooper of New York, 1791-1883.

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Peter Cooper

It’s good to have hero, someone whose approach to life, family and business you admire that you might reasonably be able to follow. As an engineer, inventor, I came to regard Peter Cooper of New York as a hero. He made his own business and was a success, in business and with his family without being crooked. This is something that is not as common as you might think. When I was in 4th grade, we got weekly assignments to read a biography and write about it. I tended to read about scientists and inventors then and after. I quickly discovered that successful inventors tended to die broke, estranged from their family and friends. Edison, Tesla, Salk, Goodyear, and Ford are examples. Tesla didn’t marry. Henry Ford’s children were messed up. Salk had a miserable marriage. Almost everyone working on the Atom Bomb had issues with the government. Most didn’t benefit financially. They died hated by the press as mass-murderers, and pursued by the FBI as potential spies. It was a sad pattern for someone who hoped to be an inventor -engineer.

The one major exception I found was Peter Cooper, an inventor, industrialist, and New York politician who was honest, and who died wealthy and liked with a good family. One result of reading about him was to conclude that some engineering areas are better than others; generally making weapons is not a path to personal success.

Tom Thumb, the blower at right is the secret to its light weight per power.

Peter Cooper was different, both in operation and outcome. Though he made some weapons (gun barrels) and inverted a remote control torpedo, these were not weapons of mass killing. Besides he but thee for “the good side” of the Civil War. And, when Cooper made an invention or a product, he made sure to have the capital available to make a profit on it too. He worked hard to make sure his products were monopolies, using a combination of patents and publicity to secure their position.

Brand management helps.

Cooper was a strong family man who made sure to own his own business, and made sure to control the sources of key materials too. He liked to invest in other businesses, but only as the controlling share-holder, or as a bond holder, believing that minor share-holders tend to be cheated. He was pro monopoly, pro trusts, and a big proponet of detailed contracts, so everyone knew where they stood. A famous invention of Cooper’s was Jello, a flavored, light version of his hide-glue, see the patent here. Besides patenting it, he sold the product with his brand, thus helping to maintain the monopoly.

Cooper was generous with donations to the poor, but not to everyone, and not with loans. And he would not sign anyone’s note as a guarantor. Borrowers tended to renege, he found, and they resent you besides. You lose your money, and lost them as a friend. He founded two free colleges, Cooper Union, and the Cooper-Limestone Institute, plus an inventor’s institute. (I got my education, free from Cooper Union.) Cooper ran these institutions in his lifetime, not waiting till he was dead to part with his money. Many do this in the vain hope that others will run the institution as they would.

Peter Cooper always sought a monopoly, or a near monopoly, patenting his own inventions, or buying the rights to others’ patents to help make it so. He believed that monopolies were good, saying they were the only sort of business that made money while allowing him to treat his workers well. If an invention would not result in a monopoly, Peter Cooper gave the rights away.

The list of inventions he didn’t patent include the instruments to test the quality of glue and steel (quality control is important), and a tide-powered ferry in New York. Perhaps his most famous unpainted invention was a lightweight, high power steam locomotive, “The Tom Thumb”, made in 1840. Innovations included beveled wheels to center the carriage on its rails, and a blower on the boiler fire, see photo above. The blower meant he could generate high-power in a small space at light weight. These are significant innovations, but Cooper did not foresee having a monopoly, so he didn’t pursue these ideas. Instead, he focussed on making rails and wire rope; he patented the process to roll steel, and the process for making coke from coal. Also on his glue/jello business. Since he made these items from dead cows and horses, he found he could also sell “foot oil” and steam-pounded leather, “Chamois”. He also pursued a telephone/ telegraph business across the Atlantic, more on that below, but only after getting monopoly rights for 50 years.

Cooper managed to stay friends with those he competed with by paying license fees for any patents he used (he tried to negotiate low rates), or buying or selling the patent rights as seemed appropriate. He also licensed his patents, and rented out buildings he didn’t need. He rented at a rate of 7% of the sale price, a metric I’ve used myself, considering rental to be like buying on loan. There is a theory of stock-buying that matches this.

The story the telegraph cable across the Atlantic is instructive to seeing how the pieces fit together. The first significant underwater cable was not laid by Cooper, by a Canadian inventor, Frederick Gisborne. It was laid in 1852 between Prince Edward Island and New Brunswick. Through personal connections, Gisborne’s company got exclusive rights for 30 years, for this and for a cable that would go to Newfoundland, but he didn’t have the money or baking to make it happen. The first cable failed, and Gisborne ran out of money and support. Only his exclusive rights remained. This is the typical story of an inventor/ engineer/businessman who has to rely on other peoples’ money and patience.

A few months after the failure, a friend of Cooper’s, Cyrus Field, convinced Cooper that good money could be made, and public good could be done, if Cooper could lay such a cable all the way to London. One thing that attracted Cooper to the project was that the cable could be made as an insulated iron-copper rope in Cooper’s own factory. Cooper, Field, and some partners (see painting below) bought Gisborne’s company, along with their exclusive rights, and formed a new company, The New York, Newfoundland & London Telegraph Company, see charter here. The founders are imagined* with a globe and a section of cable sitting on their table. Gisborne, though not shown in the painting, was a charter member, and made chief engineer. Cooper was president. He also traveled on the boat with Gisborne to lay the cable across the St. Lawrence – just to be sure he knew what was going on. This cable provided a trial for The Trans Atlantic cable.

The founding individuals to lay a transatlantic cable. Peter Cooper at left is the chairman, Cyrus Field is standing, Samuel Morse is at the back. Frederic Gisborne, a founder, does not appear in the paining. Typical.

Samuel Morse was hired as an electrician; he is pictured in the painting, but was not a charter member. Part of the problem with Morse was that he owned the patent on Morse-telegraphy, and Cooper didn’t want to pay his “exorbitant” fees. So Cooper and Field bought an alternative telegraph patent from David Hughes, a Kentucky school teacher. This telegraph system used tones instead of clicks and printed whole letters at a time. By hiring Morse, but not his patents, Cooper saved money, while retaining Morse’s friendship and expertise. The alternative could have been a nasty spat. Their telegraph company sub-licensed Hughes’s tone-method a group of western telegraph owners, “The Western Union,” who used it for many years, producing the distinctive Western Union Telegrams. With enough money in hand and credibility from the Canadian trial, the group secured 50 years monopoly rights for a telegraph line across the Atlantic. Laying the cable took many years, with semi-failed attempts in 1857, 1858, and 1865. When the eventual success came in 1866, the 50 years’ monopoly rights they’d secured meant that they made money from the start. They could treat workers fairly. Marconi would discover that Cooper wrote a good contract; his wireless telegraph required a widely different route.

I should also note that Peter Cooper was politically active: he started as a Democrat, helped form the Republican Party, bringing Lincoln to speak in NY for the first time, and ended up founding the Greenback-Labor Party, running for president as a Greenback. He was strongly for tariffs, and strongly against inflation. He said that the dollar should have the same value for all time for the same reason that the foot should have the same length and the pound the same weight. I have written in favor of tariffs off and on. They help keep manufacturing in America, and help insure that those who require French wine or German cars pay the majority of US taxes. They are also a non-violent vehicle for foreign policy.

Operating under these principles, through patents and taxed monopolies, Peter Cooper died wealthy, and liked. Liked by his workers, liked by much of the press, and by his family too, with children who turned out well. The children of rich people often turn out poorly. Carnegie’s children fought each other in court, Ford’s were miserable. Cooper’s children continued in business and politics, successfully and honorably, and in science/ engineering (Peter Coper Hewitt invented the power rectifier, sold to Westinghouse). The success of Peter Cooper’s free college, Cooper Union, influenced many of his friends to open similar institutions. Among his friends who did this were Carnegie, Pratt, Stevens, Rensselaer, and Vanderbilt. He stayed friends with them and with other inventors of the day, despite competing in business and politics. Most rich folks can not do this; they tend to develop big egos, and few principles.

Robert Buxbaum, November 30, 2022. I find the painting interesting. Why was it painted? Why is Gisborne not in it and Morse in the painting — sometimes described as vice President? The charter lists Morse as “electrician”, an employee. Chandler White, holding papers next to Cooper, was Vice President. My guess is that the painting was made to help promote the company and sell stock. They made special cigars with this image too. This essay started as a 5th grade project with my son. See a much earlier version here.

John Kennedy’s Nazi spy lover

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Photo and biographical background from Julian Wiles

Kennedy was a well-liked president with several character flaws. The most famous were his sexual dalliances. One these was with a Nazi spy, Inga Marie Arvad, “Inga Binga”. He continued with her, on and off, from his days in Naval Intelligence through his election to congress in 1946 despite being informed of her background by the navy and the FBI. When they began their relationship, Inga Marie was beautiful, 28 and married. That didn’t seem to matter, as she was beautiful, the ex-Miss Denmark, charming and instantly in love with Kennedy. She was also a close friend of Hitler, Goring, and Göebbels, and in the employ of both The Washington Times-Herald and of Axel Wenner-Gren, a suspected Nazi spy master who owned the largest private yacht in the world. It was suspected that the fuel Wenner-Gren bought for his yacht was used to refuel German U-boats in the area.

Before becoming involved with Kennedy, Ms Arvand had married Kamak Abdel Nabi, an Egyptian diplomat, and then Paul Fejos, a Hungarian film-maker. She traveled the world with Fejos, financed by Wenner-Gren, meeting, greeting, and film-making. Still married, she left Fejos in 1936 to move to the US and study journalism at Columbia University. Getting a job at the Washington Times-herald, she wrote light hearted articles based on interviews with the movers and shakers of DC, supported by $5000 checks from her friend, Wenner-Gren.

The affair with Kennedy began in the fall of 1941. Kennedy was working at the Office of Naval Intelligence, a post he’d gotten with influence from his ambassador father, “Big Joe Kennedy”. Joe was an opponent of going to war. Ms. Arvand heartily agreed. She met Jack Kennedy through his sister, Kathleen, “Kick”.

The office of Naval Intelligence had rules against adulterous relationships. Kennedy ignored them. In this case it was particularly problematic as Inga was married, Protestant, and an associate of Hitler. The navy told Kennedy to stay away, and transferred him to Charlestown with orders to stay there, “not to venture more than 50 miles”. Ms Arvad visited him there often under an assumed name, Barbara White. They stayed together, took in movies, plays, and golf. The FBI watched as Arvad was thought to be a spy. Kennedy was again told to stay clear; he did not. Eventually, Hoover intervened and got Kennedy transferred to the South Pacific despite his bad back and other health problems. Inga broke off with Kennedy though he continued to write love letters. She ignored them. Perhaps she thought Kennedy was no longer interesting, even a liability. She was trying to get a job with US overseas intelligence, the forerunner of the CIA.

Inga with husband, and two children. Note that the older, Ronald, looks like J. Kennedy. Photo from Geoffrey Gray.

When Inga didn’t get the job, she moved to Los Angeles where she continued in journalism, working for Harpers Bazar, interviewing the movers and shakers in LA and New York, generally pushing for peace. In January, 1944, she started writing to Kennedy again. He was a hero with political ambitions. She reunited with Kennedy in LA, for a private interview, published, about the sinking of his PT boat. They continued dating well into 1946, after Kennedy was elected to congress. Inga got pregnant from someone (Kennedy?) and left to marry an actor she’d been dating, Tim McCoy. Some months later, Inga gave birth to a boy who looks a lot more like Kennedy than like her husband, see photo.

While Inga no longer contacted JFK, nor JFK her, it seems that Inga was a major factor influencing Kennedy to go into politics — where he could make the world more peaceful. Inga died of Colin cancer in 1973. She only revealed her part of the affair to her eldest son, the Kennedy look-alike, near the end of her life.

The FBI’s knowledge of this story this emerged from the private files of J. Edgar Hoover, recently unsealed, reported by investigative writer, Geoffrey Gray. He speculates that Hoover used his knowledge of Kennedy’s affairs to keep from having to retire, though well past retirement age. Hover was eventually removed by Nixon, and FBI got back at Nixon by leaking as much negative as they could to the press, and more-recently helping the Democrats bug the Trump campaign, then helping spread Russian collusion documents. Also, according to the BBC they were behind the plot to kidnap MI governor Whitmer — a crime they immediately “solved”. I’m not convinced that the FBI, isn’t more of a danger than a protection. See a previous essay, “who will protect us from our protectors?”

Regarding John Kennedy, I’m less-bothered by his sexual dalliances, than by his tendency to suddenly reverse himself. Kennedy called for an attack on Cuba, then reversed while the attack was in progress, dooming the attackers. He reversed again in South Vietnam, first first supporting the government then overthrowing it, and on civil rights. Vigorous persistence, even in the face of criticism is a good trait in a president, something I liked about LBJ, Nixon, Clinton, and Reagan.

Robert Buxbaum, October 14, 2022.

A great modern artist, Duchamp becomes a great modern chess player, and returns to art.

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I’d written previously about Marcel Duchamp’s early work as a founder of the Dada school of modern art, a school that aims to say nothing about anything except about itself. Duchamp hung a urinal as art and called it “fountain.” It was comic, insulting, and engaging — an inspiration for many modern arts to follow , and much bad modern art, too — the collections of string and found objects and paintings of squares or squiggles. But the story of Duchamp is interesting. In 1925, M. Duchamp gave up on art, at least this type of art and became a chess player. As with art, he was very good at it, and became the French chess champion. Now that’s an unexpected turn.

What sort of chess did Marcel Duchamp play? Modern. Very modern. While tradition chess had focussed on the center. He developed at the sides, a strategy that was called an “Indian attack”, named (I assume) after American Indians attacking a stage-coach. Instead of attacking directly, the popular image of an Indian attack is attack from the sides, or behind trees. In chess, it involves typically a “fianchettoed bishop.” Other modern chess players of the time attacked from the side too (Réti, Alekhine) but they generally worked form one side or the other with some central presence. Duchamp worked from both, often with no center.

Position after white’s 13th move

Here is a dramatic example, a position from a game with an American great, GM George Koltanowski. It’s 13 moves in, with Duchamp, is black, generally considered the weaker side. He has fianchettoed both of his bishops, and given up the center to Koltanowski. It’s Duchamp’s turn to move/ He will win in three moves.

Notice that Koltanowsi’s bishops point outward, as a cowboys guns might point, or as from a British fighting square. Meanwhile, Duchamp’s bishops point inward, with his queen -bishop almost directly at the white king. The game proceeded as follows. 13…, Nxd5 14.Nxd7, Nxf4 15.Nxf8, Bd4, 0-1..

The full game, seen here,. It might prove instructive if you want to explore in Duchamp’s footsteps. While I play traditionally, I sometimes fianchetto, and do not find it racist that such side-attacks are called “Indian attacks.” Perhaps that’s because I’m old and used to such things, or because they very often work.

Please Touch. M. Duchamp 1947
Self-portrait, M. Duchamp, 1957 (torn paper on black velvet).

As M. Duchamp’s chess skills waned, he returned to the art world, going in the opposite direction of Dali. Duchamp’s last works are small, and simple. They are still arresting but more dream-like. Dali’s works grew bigger and busier as he got older.

That Duchamp could be both a great artist and a great chess-player suggests there is such a thing as general intelligence. It’s a touchy subject, I’ve pondered on here as intelligence appears to be inheritable.

Robert Buxbaum, September 23, 2022.

The claim that Ukrainians are Nazis is also Ukraine’s claim to statehood.

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Recently Putin claimed he was going into Ukraine to fight Nazis. Twitter makes fun of this, but also shows many pictures of these Nazis. Under the hashtag #AzovBattalion, you’ll see many pictures of white boys with swastikas and Ukraine flags (see below). Perhaps these pictures are just Russian propaganda: According to our media there are no Nazis to speak of, and besides, the president of Ukraine is a Jew. Still, the pictures look real, and based on Ukraine history, there is quite a bit reason to think they are not an aberration. Still, to the extent that they represent Ukraine, these individuals are a major basis of Ukraine’s claim for independence. They are also a good reason to leave Ukraine out of NATO, IMHO.

Let’s go back to the late days of the Tartars and the early days of the Cossacks, about 1600. There is a painting, below depicting Cossacks of those days writing a letter to The Sultan (original in the Kharkov museum). They do not seem the most savory of people, but they do seem independent and egalitarian. The letter is not written by a noble, but by a committee of pirates, and not everyone is happy about it.

Zaporozhian Cossacks write a letter to the sultan. These are the people who Putin claims should be loyal to Russia, but they have a long history of behaving otherwise. I like the scribe. A couple of people at left seem unhappy.

From 1250 to the mid 1700s, Southern Ukraine was ruled, to a greater or lesser extent, by the Crimean Tartars, a group of horse-riding Mongols who nominally served the great Khan. Moscow paid dues to them, and in 1571 the Tartar ruler,  Devlet I Giray burnt Moscow to collect his dues. The early Cossacks were Black-sea pirates, and enemies of the Tartars. Around 1600, the Cossacks and Tartars realized they had a lot in common (alcoholism, pederasty…) and formed an alliance. Mainly this was against the Poles and Jews. A famous result of this alliance was the Khmelnytsky Uprising (about 1650). Khmelnytsky was the “Hetman” (Head man?), the elected, temporary ruler for the uprising. He has become a symbol of Ukrainian independence, but he was also a brutal murderer of virtually all the Jews and Catholics. Today, he graces Ukraine’s $5 bill, and sits atop a statue in Kyiv’s central square. This elevation of Khmelnytsky is no small insult to Jews, Catholics, and civilization.

Ukrainian Republic passport, 1919.

 In 1654, via the Pereyaslav Agreement, Khmelnytsky’s Tartar-Cossacks formed an allegiance with the Tsar while retaining autonomy in Ukraine. This autonomy eroded over the years, and ended with Bolshevik rule in the early 20th century. After WWI, Ukrainians briefly tried for independence, forming the Ukraine Peoples Republic and the Ukraine Democratic republic, from 1917 to 1921. The head of the Republic was called hetman, an elected leader but also a throwback to a mass-murderer.

Stalin punished the Cossack remnant before WWIi, and when the Germans invaded in 1939, many of the remaining Ukrainians supported the Nazi invasion, and provided some of the most brutal murders of Jews; the murderers of Baba Year, for example. Putin recalls this collaboration when he calls the Ukrainians Nazis, and I suspect that he’s more right than our press will admit. These #azovbattalion pictures don’t look faked. On the other hand, the autonomy of the Ukrainians and Cossacks, and their attempts at independence provide historical backing for Ukraine’s claim to independence. Putting this another way, the more you accept that Ukraine is full of Nazi sympathizers, the more you should accept them as a distinct society from Russia.

Ukrainians of the Azov Battalion with a statue of Khmelnytsky, or some other murderer.

As an idea of how the war might go, I should mention another group of Tartar-Cossacks. These were Moslems who operated between the Don and Volga Rivers in what is known as Chechnya. Chechnya fought Russia in a long, bloody, unsuccessful struggle, that is only recently ended. Russia may win in Ukraine, but it is not likely to win easily or cheaply if Chechnya is any model.

Robert Buxbaum, Mar. 2, 2022

Low temperature hydrogen removal

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Platinum catalysts can be very effective at removing hydrogen from air. Platinum promotes the irreversible reaction of hydrogen with oxygen to make water: H2 + 1/2 O2 –> H2O, a reaction that can take off, at great rates, even at temperatures well below freezing. In the 1800s, when platinum was cheap, platinum powder was used to light town-gas, gas street lamps. In those days, street lamps were not fueled by methane, ‘natural gas’, but by ‘town gas’, a mix of hydrogen and carbon monoxide and many impurities like H2S. It was made by reacting coal and steam in a gas plant, and it is a testament to the catalytic power of Pt that it could light this town gas. These impurities are catalytic poisons. When exposed to any catalyst, including platinum, the catalyst looses it’s power to. This is especially true at low temperatures where product water condenses, and this too poisons the catalytic surface.

Nowadays, platinum is expensive and platinum catalysts are no longer made of Pt powder, but rather by coating a thin layer of Pt metal on a high surface area substrate like alumina, ceria, or activated carbon. At higher temperatures, this distribution of Pt improves the reaction rate per gram Pt. Unfortunately, at low temperatures, the substrate seems to be part of the poisoning problem. I think I’ve found a partial way around it though.

My company, REB Research, sells Pt catalysts for hydrogen removal use down to about 0°C, 32°F. For those needing lower temperature hydrogen removal, we offer a palladium-hydrocarbon getter that continues to work down to -30°C and works both in air and in the absence of air. It’s pretty good, but poisons more readily than Pt does when exposed to H2S. For years, I had wanted to develop a version of the platinum catalyst that works well down to -30°C or so, and ideally that worked both in air and without air. I got to do some of this development work during the COVID downtime year.

My current approach is to add a small amount of teflon and other hydrophobic materials. My theory is that normal Pt catalysts form water so readily that the water coats the catalytic surface and substrate pores, choking the catalyst from contact with oxygen or hydrogen. My thought of why our Pd-organic works better than Pt is that it’s part because Pd is a slower water former, and in part because the organic compounds prevent water condensation. If so, teflon + Pt should be more active than uncoated Pt catalyst. And it is so.

Think of this in terms of the  Van der Waals equation of state:{\displaystyle \left(p+{\frac {a}{V_{m}^{2}}}\right)\left(V_{m}-b\right)=RT}

where V_{m} is molar volume. The substance-specific constants a and b can be understood as an attraction force between molecules and a molecular volume respectively. Alternately, they can be calculated from the critical temperature and pressure as

{\displaystyle a={\frac {27(RT_{c})^{2}}{64p_{c}}}}{\displaystyle b={\frac {RT_{c}}{8p_{c}}}.}

Now, I’m going to assume that the effect of a hydrophobic surface near the Pt is to reduce the effective value of a. This is to say that water molecules still attract as before, but there are fewer water molecules around. I’ll assume that b remains the same. Thus the ratio of Tc and Pc remains the same but the values drop by a factor of related to the decrease in water density. If we imagine the use of enough teflon to decrease he number of water molecules by 60%, that would be enough to reduce the critical temperature by 60%. That is, from 647 K (374 °C) to 359 K, or -14°C. This might be enough to allow Pt catalysts to be used for H2 removal from the gas within a nuclear wast casket. I’m into nuclear, both because of its clean power density and its space density. As for nuclear waste, you need these caskets.

I’ve begun to test of my theory by making hydrogen removal catalyst that use both platinum and palladium along with unsaturated hydrocarbons. I find it works far better than the palladium-hydrocarbon getter, at least at room temperature. I find it works well even when the catalyst is completely soaked in water, but the real experiments are yet to come — how does this work in the cold. Originally I planned to use a freezer for these tests, but I now have a better method: wait for winter and use God’s giant freezer.

Robert E. Buxbaum October 20, 2021. I did a fuller treatment of the thermo above, a few weeks back.

Eight ways to not fix the tower of Pisa, and one that worked.

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You may know that engineers recently succeed in decreasing the tilt of the “leaning” tower of Pizza by about 1.5°, changing it from about 5.5° to about to precisely 3.98° today –high precision given that the angle varies with the season. But you may not know how that there were at least eight other engineering attempts, and most of these did nothing or made things worse. Neither is it 100% clear that current solution didn’t make things worse. What follows is my effort to learn from the failures and successes, and to speculate on the future. The original-tilted tower is something of an engineering marvel, a highly tilted, stone on stone building that has outlasted earthquakes and weathering that toppled many younger buildings that were built straight vertical, most recently the 1989 collapse of the tower of Pavia. Part of any analysis, must also speak to why this tower survived so long when others failed.

First some basics. The tower of Pisa is an 8 story bell tower for the cathedral next door. It was likely designed by engineer Bonanno Pisano who started construction in 1173. We think it’s Pisano, because he put his name on an inscription on the base, “I, who without doubt have erected this marvelous work that is above all others, am the citizen of Pisa by the name of Bonanno.” Not so humble then, more humble when the tower started to lean, I suspect. The outer diameter at the base is 15.5 m and the weight of the finished tower is 14.7 million kg, 144 million Nt. The pressure exerted on the soil is 0.76 MPa (110 psi). By basic civil engineering, it should stand straight like the walls of the cathedral.

Bonanno’s marvelous work started to sink into the soil of Pisa almost immediately, though. Then it began to tilt. The name Pisa, in Greek, means swamp, and construction, it seems, was not quite on soil, but mud. When construction began the base was likely some 2.5 m (8 feet) above sea level. While a foundation of clay, sand and sea-shells could likely have withstood the weight of the tower, the mud below could not. Pisano added length to the south columns to keep the floors somewhat level, but after three floors were complete, and the tilt continued, he stopped construction. What to do now? What would you do?

If it were me, I’d consider widening the base to distribute the force better, and perhaps add weight to the north side. Instead, Pisano gave up. He completed the third level and went to do other things. The tower stood this way for 99 years, a three-floor, non-functional stub. 

About 1272, another engineer, Giovanni di Simone, was charged with fixing the situation. His was the first fix, and it sort-of worked. He strengthened the stonework of the three original floors, widened the base so it wold distribute pressure better, and buried the base too. He then added three more floors. The tower still leaned, but not as fast. De Simone made the south-side columns slightly taller than the north to hide the tilt and allow the floors to be sort-of level. A final two stories were added about 1372, and then the first of the bells. The tower looked as it does today when Gallileo did his famous experiments, dropping balls of different size from the south of the 7th floor between 1589 and 1592.

Fortunately for the construction, the world was getting colder and the water table was dropping. While dry soil is stronger than wet, wet soil is more plastic. I suspect it was the wet soil that helped the tower survive earthquakes that toppled other, straight towers. It seems that the tilt not only slowed during this period but briefly reversed, perhaps because of the shift in center of mass, or because of changes in the sea level. Shown below is 1800 years of gauge-based sea-level measurements. Other measures give different sea-level histories, but it seems clear that man-made climate change is not the primary cause. Sea levels would continue to fall till about 1750. By 1820 the tilt had resumed and had reached 4.5°.

Sea level height history as measured by land gauges. Because of climate change (non man-made) the sea levels rise and fall. This seems to have affected the tilt of the tower. Other measures of water table height give slightly different histories, but still the sense that man change is not the main effect.

The 2nd attempt was begun in 1838. Architect, Alessandro Della Gherardesca got permission to dig around the base at the north to show off the carvings and help right the tower. Unfortunately, the tower base had sunk below the water table. Further, it seems the dirt at the base was helping keep the tower from falling. As Della Gherardesca‘s crew dug, water came spurting out of the ground and the tower tilted another few inches south. The dig was stopped and filled in, but he dig uncovered the Pisano inscription, mentioned above. What would you do now? I might go away, and that’s what was done.

The next attempt to fix the tower (fix 3) was by that self-proclaimed engineering genius, Benito Mussolini. In 1934. Mussolini had his engineers pump some 200 tons of concrete into the south of the tower base hoping to push the tower vertical and stabilize it. The result was that the tower lurched another few inches south. The project was stopped. An engineering lesson: liquids don’t make for good foundations, even when it’s liquid concrete. An unfortunate part of the lesson is that years later engineers would try to fix the tower by pumping water beneath the north end. But that’s getting ahead of myself. Perhaps Mussolini should have made tests on a model before working on the historic tower. Ditto for the more recent version.

On March 18, 1989 the Civic Tower of Pavia started shedding bricks for no obvious reason. This was a vertical tower of the same age and approximate height as the Pisa tower. It collapsed killing four people and injuring 15. No official cause has been reported. I’m going to speculate that the cause was mechanical fatigue and crumbling of the sort that I’ve noticed on the chimney of my own house. Small vibrations of the chimney cause bits of brick to be ejected. If I don’t fix it soon, my chimney will collapse. The wet soils of Pisa may have reduced the vibration damage, or perhaps the stones of Pisa were more elastic. I’ve noticed brick and stone flaking on many prominent buildings, particularly at joins in the chimney.

John Burland’s team cam up with many of the fixes here. They are all science-based, but most of the fixes made things worse.

In 1990, a committee of science and engineering experts was formed to decide upon a fix for the tower of Pisa. It was headed by Professor John Burland, CBE, DSc(Eng), FREng, FRS, NAE, FIC, FCGI. He was, at the time, chair of soil mechanics at the Imperial College, London, and had worked with Ove, Arup, and Partners. He had written many, well regarded articles, and had headed the geological aspects of the design of the Queen Elizabeth II conference center. He was, in a word, an expert, but this tower was different, in part because it was an, already standing, stone-on stone tower that the city wished should remain tilted. The tower was closed to visitors along with all businesses to the south. The bells were removed as well. This was a safety measure, and I don’t count it as a fix. It bought time to decide on a solution. This took two years of deliberation and meetings

In 1992, the committee agreed to fix no 4. The tower was braced with plastic-covered, steel cables that were attached around the second and third floors, with the cables running about 5° from the horizontal to anchor points several hundred meters to the north. The fix was horribly ugly, and messed with traffic. Perhaps the tilt was slowed, it was not stopped.

In 1993, fix number 5. This was the most exciting engineering solution to date: 600 tons of lead ingots were stacked around the base, and water was pumped beneath the north side. This was the reverse of the Mussolini’s failed solution, and the hope was that the tower would tilt north into the now-soggy soil. Unfortunately, the tower tilted further south. One of the columns cracked too, and this attempt was stopped. They were science experts, and it’s not clear why the solution didn’t work. My guess is that they pumped in the water too fast. This is likely the solution I would have proposed, though I hope I would have tested it with a scale model and would have pumped slower. Whatever. Another solution was proposed, this one even more exotic than the last.

For fix number 6, 1995, the team of experts, still overseen by Burland, decided to move the cables and add additional tension. The cables would run straight down from anchors in the base of the north side of the tower to ten underground steel anchors that were to be installed 40 meters below ground level. This would have been an invisible solution, but the anchor depth was well into the water table. So, to anchor the ground anchors, Burland’s team had liquid nitrogen injected into the ground beneath the tower, on the north side where the ground anchors were to go. What Burland did not seem to have realized is that water expands when it freezes, and if you freeze 40 meters of water the length change is significant. On the night of September 7, 1995, the tower lurched southwards by more than it had done in the entire previous year.  The team was summoned for an emergency meeting and the liquid nitrogen anchor plan was abandoned.

Tower with the two sets of lead ingots, 900 tons total, about the north side of the base. The weight of the tower is 14,700 tons.

Fix number 7: Another 300 tons of lead ingots were added to the north side as a temporary, simple fix. The fix seems to have worked stabilizing things while another approach was developed.

Fix number 8: In order to allow the removal of the ugly lead bricks another set of engineers were brought on, Roberto Cela and Michele Jamiolkowski. Using helical drills, they had holes drilled at an angle beneath the north side of the tower. Using hoses, they removed a gallon or two of dirt per day for eleven years. The effect of the lead and the dirt removal was to reduce the angle of the tower to 4.5°, the angle that had been measured in 1820. At this point the lead could be removed and tourists were allowed to re-enter. Even after the lead was removed, the angle continued to subside north. It’s now claimed to be 3.98°, and stable. This is remarkable precision for a curved tower whose tilt changes with the seasons. (An engineering joke: How may engineers does it take to change a lightbulb? 1.02).

The bells were replaced and all seemed good, but there was still the worry that the tower would start tilting again. Since water was clearly part of the problem, the British soils expert, Burland came up with fix number 9. He had a series of drainage tunnels built to keep the water from coming back. My worry is that this water removal will leave the tower vulnerable to earthquake and shedding damage, like with the Pavia tower and my chimney. We’ll have to wait for the next earthquake or windstorm to tell for sure. So far, this fix has done no harm.

Robert Buxbaum, October 9, 2020. It’s nice to learn from other folks mistakes, and embarrassments, as well as from their successes. It’s also nice to see how science really works, not with great experts providing the brilliant solution, but slowly, like stumbling in the dark. I see this with COVID-19.