Mathematics in the detection of art
forgeries: The Johannes Vermeer –
Han van Meegeren example
By: Iordana Avgodopoulou, Daniel
Biagioni, Laura Booth, David Collinge,
Andreas Hadjigeorgiou, Tham Mbung,
James Veall
Year 1 students, Group E,
Dept. of Mathematics, Univ. of
Portsmouth, UK
Group E tutor: Dr. Athena Makroglou
Maths and Art Festival 2004
Dept. of Mathematics, Univ. of
Portsmouth, UK
18 - 21 October, 2004.
On Wed. 20 October 2004: Exhibition of
students’ work (Posters)
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1. Introduction-History
Han Van Meegeren was a Dutch painter
and famous art forger. At the end of World
War II the Allies came across the Nazis’ hidden works of art they had stolen or bought
from people of other countries. According to Burghes and Borrie (1981, p. 79)
amongst the art found was a painting (‘Woman
Taken in Adultery’) by Johannes Vermeer
a famous Dutch painter who died in 1975.
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Figure 1: ‘Woman Taken in Adultery’ Ref:
http://www.tnunn.ndo.co.uk/vm-pics.htm
It was traced as sold to Goering by Hans
van Meegeren who was arrested for collaborating with the enemy in May 1945.
When he could not explain the origins of
the Vermeer painting, on July 12 1945, van
Meegeren announced that he had forged
this painting and also several more (‘Disciples at Emmaus’ was another one).
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Figure 2: Disciples at Emmaus
Ref:
http://www.tnunn.ndo.co.uk/vm-pics.htm
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At first the authorities did not believe him
as the painting was so well done. Not only
was the style and skill executed perfectly,
but the painting was done on 17th century canvas. In order to prove his case, he
started forging a known Vermeer painting
‘Jesus Amongst the Doctors’. The story
continues with a panel was formed which
concluded that the paintings were indeed
forgeries. Van Meegeren was sentenced to
one year in prison on October 12, 1947
(died by heart attack on Dec. 30, 1947).
Despite the panel’s conclusion, many people continued to believe that ‘Disciples at
Emmaus’ was original Vermeer. It was not
until 1967 when scientists at the Univ. of
Carnegie Mellon proved that the painting
was indeed a forgery using Mathematics.
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Pictures of paintings by Vermeer can be
found for example at:
http://www.mystudios.com/vermeer/index.html.
Here are two samples from the same reference:
Figure 3: The little street, 1657-58, by
Vermeer
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Figure 4: The astronomer, 1668 by
Vermeer
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There are many other famous art forgers
just like van Meegeren, so detecting fake
art is an important thing to do now.
Next section will look at how art forgeries
are detected using mathematical techniques.
We will be using differential equations to
look at the radioactive decay of a substance
that is in paint, which is how the art was
proved as fake. The main references are:
Burghes and Borrie (1981), Keisch etal (1967),
Keish (1968).
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2. Mathematical models
(Burghes and Borrie (1981))
White lead is a radioactive substance and
it is used in painting. It has a half life
of 22 years. It is manufactured from ores
that contain a number of elements, one
of which is Radium 226 (Ra226 ) with a
half life of 1600 years and which decays
to P b210 . While still in the ore, P b210 and
Ra226 are in a ‘radioactive equilibrium’ (i.e.
that amount of radium decaying to P b210 is
equal to the amount of P b210 disintegrating
per unit of time).
When the white lead pigment is manufactured, most of the radium is removed, but
very small quantities remain. P b210 decays until a new equilibrium with the small
amount of Radium is reached.
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Notation:
y(t): amount of P b210 /gm of ordinary lead
at time t
r(t): number of disintegrations of Ra226 /
minute gm of ordinary lead.
λ: the decay constant for P b210 .
y(t0 ) = y0, t0: the time of manufacture.
The differential equation relating y(t) and
r(t) with λ, is:
dy
= −λy + r(t), y(t0 ) = y0.
dt
The solution of this differential equation for
r constant is:
r
r
y(t) = + (y0 − )e−λ(t−t0).
λ
λ
The original amount of P b210 was in a state
of balance with most of the radium in the
ore from which the pigment was manufactured, therefore, λy0 = R, where R represents the number of disintegrations of
Ra226 per minute per gm of ordinary lead
and its value is known to be less than 200.
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If we suppose we have a painting which was
created 300 years ago and was not a forgery
then we would have t − t0 = 300 and by
using the equation above we can get an
estimation of λy0 as follows:
λy0 = λye300λ − r[e300λ − 1].
and λ = 3.151 × 10−2 . If the number λy0
is in the range 0-200 then we expect the
painting not to be a forgery but if larger
than 200 the painting is indeed a forgery.
For the painting ‘Disciples at Emmaus’ it
was found that
λy0 = 98, 147.
Thus, this number is too large for a original painting of the 17th century, which
means that this painting is a forgery with
no doubt.
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REFERENCES
Burghes, D. N., Borrie, M. S. (1981), Modelling with Differential Equations, Halsted
Press.
Keisch, B. (1968), Dating Works of Art
through their natural radioactivity: Improvements and Applications, Science, 160, 413–
415.
Keisch, B., Feller, R. L., Levine, A. S.,
Edwards, P. R. (1967), Dating and Authenticating Works of Art by Measurement
of Natural Alpha Emitters, Science, 155,
1238–1241.
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REFERENCES cont.
http://www.tnunn.ndo.co.uk/vm-pics.htm
http://www.vectorsite.net/tzcon.html#m4 (article about Han van Meegeren)
http://www.mystudios.com/vermeer/index.html
http://www.wordiq.com/definition/Art_forgery#
Famous_forgeries
http://www.wordiq.com/definition/Johannes_
Vermeer
http://essentialvermeer.20m.com
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