11 September 2010 | NewScientist | 37

Imaginary numbers

WHEN Paul Kammerer shot himself on an Austrian hillside in 1926, he seemed destined to be remembered only as a scientific fraudster who had faked his results to prove a controversial theory. In fact, he might well have glimpsed epigenetics, influential changes in gene activity that do not involve alterations to the DNA sequence.

Kammerer was infamous for his experiments on the midwife toad, Alytes obstetricans (pictured above), an unusual amphibian which mates and raises its eggs on dry land. By keeping toads in unusually hot, dry conditions, he drove them to breed and raise their eggs in water. Only a few eggs hatched, but the offspring of these aquatic unions also bred in water. Kammerer claimed this as proof of Lamarckian inheritance – the idea (now known to be wrong) that traits acquired during an individual’s lifetime can be passed on to its offspring.

In August 1926, Kammerer was condemned as a fraud in the pages of Nature (vol 118, p 518). He killed himself six weeks later. The sad story was largely forgotten until 1971, when Arthur Koestler published a book claiming that the biologist’s experiments may have been tampered with by a Nazi sympathiser . Kammerer was a socialist who was planning to build an institute in the Soviet Union, which would have made him a target of Vienna’s burgeoning Nazi movement.

Then, last year, biologist Alex Vargas of the University of Chile in Santiago re-examined Kammerer’s work. He was not a fraud, Vargas suggests, but had inadvertently discovered epigenetics (Journal of Experimental Zoology B, vol 312, p 667). “Kammerer had the right approach,” says Vargas, who hopes that the toad experiments will one day be repeated.

We now know that the patterns of inheritance of the kind Kammerer claimed to have seen can be due to epigenetics. This process is central to molecular biology, and numerous drugs based on it are in development. It would have been discovered regardless of Kammerer – but perhaps we would not still be waiting for those drugs if he had been taken seriously. Graham Lawton

Epigenetics

came up with imaginary numbers, even negative numbers were treated with deep suspicion. Though they were difficult beasts, Cardano pressed ahead. At one point, Cardano even wrote that they were “useless”, but it is clear that he found them intriguing as well as frustrating. “Cardano wrote a formal expression for complex numbers, he could add and multiply them, but he could not give them any practical or geometrical sense,” says Artur Ekert of the University of Oxford.

Rafael Bombelli built on Cardano’s work in the 1560s, but imaginary numbers were not taken seriously until mathematicians found links between them and constants such as π and e. In the 18th century, Leonhard

Euler showed that e raised to the power i × π equals -1 (where i is the square root of -1). Now imaginary numbers are indispensable.

It seems fitting that their role in quantum theory is to explain the most bizarre aspect of the theory: that quantum objects such as atoms and electrons can exist in two or more places at once. Physicists and philosophers still argue over what this means, but it is clear that the mathematics only works when it includes a complex number known as a “probability amplitude”.

Without imaginary numbers, you won’t get an answer that reflects the reality of the physical world. And you won’t get an iPod either. Michael Brooks

na

sa/s

pl

Ha

ns

D D

oss

enba

cH/a

rDea

100911_F_Unlikely_Sci_Idea.indd 37 1/9/10 17:07:00