A mathematical challenge.

In 1929 in Göttingen, a challenge to express any whole number using the number 2 precisely four times, and using only well-known mathematical symbols, was introduced. 

The first few numbers are easy:

1 = (2 + 2)/(2 + 2),

2 = (2/2) + (2/2),

3 = (2 x 2) - (2/2),

4 = 2 + 2 + 2 - 2.

The game became much more difficult even for Göttingen’s finest mathematical minds. Hundreds of hours were spent playing the game with higher and higher numbers - until Paul Dirac found a simple and general formula enabling any number to be expressed using four 2s, entirely within the rules. He had rendered the game pointless. 

Dirac’s solution relies on a basic property of logarithms: 

and is

where the number of radicals is exactly n square roots. 

One may think that Dirac killed the game using only three 2s. Each symbol in the formula is very common in mathematics, so Dirac’s solution is still within the rules of the game. 

Content inspired by The Strangest Man by Graham Farmelo 

Gorget Taumi

18th Century


This is a very rare and well-preserved masterpiece of the Cook/Forster Collection. The taumi is an almost complete specimen with all the ornamental components present. Only at the upper left side of the neckline are some of the feathers missing which form the wreath around the mother-of-pearl discs mounted there. In addition to comparable pieces it consists of a pale plant fiber material intertwined with a coconut fiber string, attached at the upper edge of the neckline on both sides to a 2 cm high additional frill made of diagonally plaited light and reddish-brown coconut fibers. The lattice-work consisting of cane slips and the coconut fiber weave over it, are firmly tied together, as are the tufts of white dog’s hair, up to 10 cm long and protruding outward from the lattice-work. On the front, three rows of shark’s teeth and three strips with pigeon feathers - cut back to a length of 3 cm - are attached to the coconut fiber weave. Two diagonally plaited strips, form the inside border of the neckline. The outside strip is plaited from reddish-brown coconut fibers, while the light-brown material of the inside strip cannot be identified as coconut fiber with any certainty. Three mother-of-pearl discs, each with a diameter of 3 cm, are mounted on each of the two upper edges of the collar, as is a wreath of feathers surrounding them. In its completeness, this object best corresponds to the gorget a “kind of priest” (more likely to be a warlord) wears in one of Parkinson’s illustrations from the first voyage of Cook. (The VCM)

University of Göttingen via the VCM

Is Big Business scared of being side-stepped by renewable energy?

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There was a nice blog on the Mother Nature Network (MNN) a few weeks ago about the renewables revolution currently sweeping through rural Germany.  The country has been ahead of the game for a while now, but with Germany abandoning its nuclear programme in the wake of the Fukushima disaster, and with rising costs of foreign oil, its residents are finding innovative ways to ditch big company dependence and even government dependence altogether.

The small 750-person village of Jühnde was the first to do so as part of a large-scale experiment by the University of Göttingen, when in 2005 it began using local liquid manure and crops to produce its own heat and electricity.

Since then a further 70 small towns have ended their dependence on big oil companies and offshore energy developers by obtaining fuel from wood chips, crops and manure to heat their homes.  Any excess electricity is sold back to the Grid.  Another 14 towns and villages across Germany are now in the process of converting their energy dependence from the local power grid to self-styled wind and solar power.

The MNN article was keen to point out that this kind of energy revolution is subversive.  It is decentralising power and telling government, large greedy energy companies and of course, bureaucracy to piss off.  Course, I dread to think how much bureaucracy there’d be in getting a scheme like this set up in the UK, but surely it MUST be possible.  Do we give out subsidies for that kind of thing?  Surely we do, given the billions of subsidies given to windfarms?

MNN goes on to say that the revolutionary nature of such energy schemes should naturally appeal to American citizens:

“What’s surprising is that this line of cleantech advocacy – which is basically Tea Party rhetoric minus the tricornered hats, Glenn Beckian misinformation and palpable xenophobia – hasn’t caught on in the United States.

What could be a greater force for individual liberty, after all, than to sieze direct control of the appartus of energy production?  What kind of twisted definition of small-government conservatism finds itself advocating for bigger and more centralized energy bureaucracy?  Drill-baby-drilling in northern Alaska, fracking for gas in central Pennsylvania, mountaintop-removal mining and “clean coal” research – sorry, small-government champions, but these are inherently centralized, large-scale, ultra-bureaucratic operations”

Good points.  Personally I see the large onshore windfarm development currently ravaging our countryside as part of that same problem.  Greedy, hollow, underhanded and uncaring.  Local communities should be getting those enormous windfarm subsidies rather than the huge companies and the government should be helping them to create and source their power locally.  Where possible of course, because obviously it gets more difficult the bigger your town gets, and is especially difficult for cities.  But certainly the ‘by the community, for the community’ model could easily be applied to many of our rural towns and villages.

Anyway, an interesting article from MNN if you want to read all of it.

A final point to consider of course, is the question about how much space biofuel crops take up.  It’s grand that anyone can grow their own biofuels locally, but is this at the expense of losing agricultural land for growing food?  As oil prices start driving food prices up and localism takes hold, we may try to source our resources from closer to home.  There needs to be a balance between food and fuel production in that event, and we should be trying to use whatever waste products we can before we start sacrificing food production for fuel……as that does nothing to end the worrying practice of growing our food in other people’s countries.  But that’s another story…..

PS - I know George Bush has gone…..but I couldn’t resist the cartoon :)

Watch on

Barbara - Göttingen

In 1964 Barbara was invited by Gunther Klein, director of the Jungen Theater in Göttingen for a performance. At first she refused, with her Jewish background in her mind, but not knowing why, the next day she agreed. The one performance became eight and there she wrote the first simple version of her famous “Göttingen”, which she completed in Paris. A song symbolic dedicated to children in Paris and Göttingen and wishing there will be no more bloodshed and hatred, referring to the war. A love affair she discovered unexpectedly in herself for the people in Göttingen.

This simple song is said to have contributed more to German-French reconciliation than any speech of a politician. On the 40-years-jubilee of the Elysée agreement ex-chancellor Schröder quoted from this text in his official speech in the Château de Versailles.

Tranlated lyrics

Germany Update Pt 1

Arrived in Frankfurt. Actually awake some how. Had some trouble getting to sleep on the plane but once I took some benadryl and started listening to some Balance and Ruin, it knocked me right out for four hours and I woke up to us landing.

Glad I booked the 12:56 train to Gottingen and not later. Was really worried about missing it but I have plenty of time.

I got to order coffee and water entirely in German! (By that I mean I asked for coffee and water in German and was able to respond correctly when the barista asked me what size), but I think I used up most of my German with that.

Once I get to Goettingen and to my dorm, I will probably explore a bit. I really hope I can find that delicious crepe place from last summer.

Tomorrow is my talk at Max Planck Institute for Dynamics and Self-organization, which I need to fix up a bit. Then Saturday is more fun in Goettingen. And then off to Dresden for the workshop and my poster presentation.

I should be okay with everything, even if I am completely by myself.

I am nervous but…. also excited! Worst case scenario, I don’t make it to Goettingen or Dresden and I am stuck in Frankfurt for two weeks and need to find a hostel and then fly home. That would suck yes but I at least made it here. (Sorry I need to reassure myself to stay calm).

Unknown Woman in Mathematics: Emmy Noether

German mathematician Emmy Noether made fundamental and lasting contributions to the field of abstract algebra. Noether was widely recognized for her creativity and her ability to make connections between abstract concepts. In addition, she led the mathematical community as a whole to adopt a more axiomatic approach.

Noether’s mathematical contributions are even more impressive in light of the prejudice she encountered throughout her career as a woman in mathematics in early 20th century Germany. After completing her PhD in 1907, Noether had trouble finding an academic position because most German universities had a policy against hiring women. Without a job, she worked on her research and gave some lectures for her father, Max Noether, an algebrist at the University of Erlangen. During World War I, Noether applied for a position as a Privatdozent (roughly equivalent to an assistant professor) at the University of Gottingen. However, the philosophers and historians on the faculty opposed her because of her sex. They argued, “What will our soldiers think when they return to the university and find that they are expected to learn at the foot of a woman?” 

Nonetheless, Noether had the strong support of David Hilbert, one of the leading mathematicians of the time, who asserted at a faculty meeting, “I do not see that the sex of the candidate is an argument against her admission as a Privatdozent. After all, we are a university and not a bathing establishment.” Even with Hilbert on her side, Noether was not given the position. Instead, she taught courses that were listed under Hilbert’s name. Finally, in 1922, Gottingen gave her a symbolic as an adjunct professor. The position carried no regular salary and no small duties, but it paid her a small amount to give lectures on abstract algebra.

On the occasion of Noether’s death, Albert Einstein wrote in a letter to the New York Times:

In the judgment of the most competent living mathematicians, Noether was the most significant creative mathematical genius thus far produced since the higher education of women began. In the realm of algebra in which the most gifted mathematicians have been for centuries, she discovered methods which have proved of enormous importance in the development of the present day younger generation of mathematicians.

Excerpt from Number Theory: A Lively Introduction with Proofs, Applications and Stories

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