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Posts Tagged ‘Goering’

The (Half-)Life of Han van Meegeren

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December 30th was the 71st anniversary of the death of Han van Meegeren, and I thought this would be an opportune time to whip up a not-so-quick post lauding the power and beauty of differential equations. (As if we need an excuse for such honorifics!) For the uninitiated, Han van Meegeren was a talented Dutch painter who suffered from a combustible fusion of realities: an insatiable desire for fame and a star on the decline; it was this desire that ultimately led him to perpetrate what some consider to be “the most dramatic art scam of the twentieth century.”

He almost got away with it.

The Pledge. The plan was simple: Forge a number of “early” Vermeer paintings that, as a collective, would serve as an organic confirmation of the more substantial and “mature” Vermeer forgeries to follow. It worked brilliantly. Abraham Bredius, the preeminent art historian of his day, adjudicated the surreptitious Han van Meegeren forgeries as Vermeer originals for a Dutch estate. He proudly published his analysis, which incidentally confirmed his pet theory that the Italians influenced Vermeer’s artistic oeuvre:

It is a wonderful moment…when [one] finds himself suddenly confronted with a hitherto unknown painting by a great master, untouched, on the original canvas, and without any restoration, just as it left the painter’s studio….Neither the beautiful signature [nor] the pointillés on the bread…is necessary to convince us that we have…the masterpiece of Johannes Vermeer of Delft….In no other picture by the great master…do we find…such a profound understanding of the Bible story—a sentiment so nobly human expressed through the medium of the highest art.

The deception, now substantiated in print by the ultimate academic authority, was complete. Van Meegeren was back on top, raking in the cash for his fraudulent paintings, and fooling the art world he now despised for failing to recognize his genius.

The Turn. Van Meegeren’s scam began to unravel after the end of WWII when authorities were tracking Nazi collaborators. An investigation discovered Van Meegeren had, through an intermediary, unwittingly sold a “Vermeer” to Goering, and as a result, he was accused of, and arrested for, treason. His defense was as simple as his scam: Admit the paintings sold to the Germans were fake. Van Meegeren’s claim was dismissed as a desperate attempt to mitigate the severer charge of treason. To prove his accusers wrong, however, van Meegeren began forging “Jesus Amongst the Doctors” in prison as proof of his skill set. But it was to no avail: van Meegeren was charged with collaborating with the enemy, and a panel of experts was introduced to examine the paintings.

What no one realized, however, was that van Meegeren had prepared for forensic scrutiny. He scratched the paint off worthless paintings from the period in order to use age-appropriate canvases and defeat X-ray analysis. He used color schemes and materials Vermeer would likely have used. He even employed Pheno formaldehyde in an attempt to mimic the rigid texture of paint that had been fossilizing since the seventeenth century. Van Meegeren was assiduous but ultimately imperfect; experts detected both the Pheno formaldehyde and some trace evidence of the color “cobalt blue,” a coeval pigment of the 1940s but wholly unknown and unavailable to Vermeer. On the basis of that evidence, van Meegeren was sentenced to one year in prison for forgery, and roughly two months later, he died of a heart attack.

The evidence presented was quite convincing in and of itself, but van Meegeren still had his doubters; some continued to believe the paintings were simply too good to be forgeries—a testament to van Meegeren’s skill. There was, however, one physical detail van Meegeren could never have addressed, a fact that would prove without question his “Vermeers” were forgeries: the rate of radioactive decay of the lead-210 and radium-226 in the paint he used. If we assumedN/dt = -\lambda Nis the change in the number of disintegrated atoms for some unit of time t with decay constant\lambdaandN(t_0)=N_0(i.e., the time at which an element begins decaying), we have the following general-solution equation

\displaystyle N(t)=N_0\exp\Big(\!\!-\!\lambda\!\int_{t_0}^t ds\Big)=N_0\,e^{-\lambda(t-t_0)}\, ,

which we can simplify asN/N_0=\exp(-\lambda(t-t_0)).Taking natural logs, definingN/N_0=1/2(i.e., the half life for radioactive decay), and solving for(t-t_0)yields(t-t_0)=(\ln 2)\lambda^{-1}.That is, we can calculate an element’s half-life by dividing the natural log of 2 by its decay constant. Unfortunately, there’s never a way to determine preciselyN_0when trying to date an object, so the above equation cannot do much to help our cause.

But all is not lost. We can use half-life values of the specific elements in question to estimate an amount of decay based on specific time frames we wish to investigate. Due to various facts about chemistry we won’t address here, we know the amount of lead-210 (half-life = 22 years) and radium-226 (half-life = 1600 years) for an authentic 300-year-old Vermeer would stand in “radioactive equilibrium,” from which we can deduce that a modern forgery will have a much higher level of lead-210 radioactivity in relation to its radium-226 content.

The Prestige. Supposey(t)is the number of grams (of white lead) at t years withy_0=t_0at production, andr(t)is the function that gives the disintigration rate of radium-226 (grams per minute) of white lead at t. Then we have the following differential equation:dy/dt = -\lambda y+r(t).Because the half-life of radium-226 is 1600 years, our estimates for a 300-year-old Vermeer will involve a pretty consistent value for r(t), which means we can replace r(t) with the constant r. Upon multiplying our differential equation on both sides by the integrating factor, we’re left withd/dt\,\,e^{\lambda t}y=re^{\lambda t}because

\displaystyle \frac{d}{dt}\,e^{\lambda t}y=\Big(e^{\lambda t}\Big)'y+y'e^{\lambda t}=\lambda e^{\lambda t}y+\frac{dy}{dt} e^{\lambda t}\,,

which is what we need whendy/dt + \lambda y=r(t).A straightforward calculation then gives us

\displaystyle e^{\lambda t}y(t) - e^{\lambda t}y_0 = r(e^{\lambda t} - e^{\lambda t_0})\lambda^{-1}\, ,

and, solving for y(t), we have

\displaystyle y(t)=\frac{r}{\lambda}\Big(1-e^{-\lambda(t-t_0)}\Big)+y_0\,e^{-\lambda(t-t_0)}

recalling thaty(t_0)=y_0and\exp(\lambda t_0-\lambda t)=\exp(-\lambda(t-t_0)).But our goal is to estimate the amount of lead-210 at the time of production in order to detect a forgery. This means we need to solve for\lambda y_0\,.Setting(t-t_0)=300in the above equation and doing some rearranging, we finally reach the form of the equation we desire:

\displaystyle \lambda y_0 = \lambda y(t) e^{300\lambda} - r(e^{300\lambda} -1)\,.

We know the decay constant for lead-210 is 22 years; thus,\lambda =\ln 2 /22and we can now calculate the exponential of e:

\displaystyle e^{300\lambda}=e^{(300/22)\ln 2}=(\exp(\ln 2))^{300/22}=2^{150/11}.

All that remains is substituting appropriate values for\lambda y(t),the disintegration rate of lead-210, and the (constant) disintegration rate r of radium-226.[1] Due to varying uranium concentrations throughout the world, a very conservative estimate for an upper bound on the disintegration rate of a modern painting was determined to be 30,000 grams/minute of white lead. Values for van Meegeren’s “Disciples at Emmaus” were determined to be\lambda y(t)=8.5andr=0.8,yielding the following calculation:

\displaystyle \lambda y_0=(8.5)2^{150/11} - 0.8(2^{150/11}-1)=98,050\,\,,

more than three times the allowable limit for an authentic painting of the seventeenth century. Clearly, van Meegeren’s “Disciples at Emmaus” was a forgery.

Differential equations 1, Abraham Bredius 0.

Footnotes:

[1] The investigation used the disintegration rate of Polonium-210 in place of lead-210 for convenience without any real loss of precision. The rates of^{210}\text{Po}and^{210}\text{Pb}equalize after only a few years.

References:

Braun, Martin. Differential Equations and Their Applications, Texts in Applied Mathematics, Springer-Verlag: New York, 1992.

Written by u220e

2019-01-29 at 1:20 am

Posted in ART, HISTORY, LAW, MATHEMATICS, SCIENCE

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