# How mathematician Hendrik Lenstra completed an unfinished artwork by Escher

Twenty years onwards, in a review of an Escher exhibition in Italy, Nature Physics writes about it again: Leiden mathematicians helping Escher out. A reconstruction of how emeritus professor Hendrik Lenstra tracked down a 44-year-old problem.

## An ordinary flight

It is the spring of 2000 when mathematician Hendrik Lenstra boards a white Continental Airlines plane in America. Lenstra is a professor both in Leiden and at the University of California and travels back and forth regularly. Upon entering, a flight attendant in a deep dark blue suit greets him kindly, after which he takes his seat and buckles his seatbelt around his waist.

A few hours later, as the plane sits at an altitude of 10 kilometres above East America, a moment follows that many will recognise: feeling bored, Lenstra snatches the airline magazine out of the seat in front of him and starts flicking through it. At that moment, he does not yet suspect that this choice will give his future self an avalanche of publicity.

## A reunion with Escher

‘In the magazine I read a piece about the Dutch artist M.C. Escher, which is accompanied by a picture of his lithograph the Print Gallery,’ the mathematician outlines the situation twenty years later. ‘I knew Escher and his prints from my days at the gymnasium, from Pythagoras, a maths magazine for young people. It was always accompanied by a comprehensible explanation for secondary school students, written by science populariser Bruno Ernst. When I went to study mathematics, I thought I had grown out of it. Anyway, while I was on the plane, I suddenly look at the lithograph with completely different eyes. It is clear to me that there is a mathematical theory hidden inside of it, and I wonder which one it is. But sitting in the plane, I can't figure it out...’

A picture of the artwork Print Gallery at the Escher exposition in Italy:

## Escher in trouble

The lithograph that grabs Lenstra's attentions is a print by Escher from 1956. In the artwork, a man in a gallery is looking at a print of a coastal city, with the gallery and the city merging into one another – the man appears to be part of the artwork he is looking at. However, this is not the most striking feature of the print. In the centre sits a conspicuous white circle with Escher’s signature. According to some stories, Escher wants to create a ‘circular expansion without beginning or end’. But it is not entirely clear why he leaves the centre open. Over the years, many have been fascinated by the incomplete print. In his book Gödel, Escher, Bach: An Eternal Golden Braid, the American writer Douglas Hofstadter explains the missing middle section with theories of the German philosopher and logician Kurt Gödel. Despite all the attention, no one manages to fill the mysterious gap. Until Lenstra, on that conscious journey back to the Netherlands, faces the print.

## Lenstra gets a clue from an old acquaintance

After a long flight, the professor gets off the aircraft at Schiphol and returns home. When he arrives, he walks to the bookcase and picks up a 116-page dusty paperback called The Magic Mirror of M.C. Escher. ‘A book in which Bruno Ernst had bundled his Pythagoras contributions,' says Lenstra. ‘In the book, Ernst explains how Escher had made the print, and also shows working drawings by Escher, very insightful. In addition, one of Ernst’s remarks puts me on the trail of the so-called "conformal transformations". These are transformations that leave the form unchanged on a small scale. The theory stems from the mid-nineteenth century and can be largely attributed to the German mathematician Riemann.’

But Lenstra wonders what kind of conformal transformation is involved exactly. 'From the working drawings it becomes clear to me that, in principle, the print should repeat itself inwards, just like the depiction on a Droste tin. If one makes the picture on a Droste tin about seven times as big in all directions, one sees the same picture.’ But what factor must be used to blow up Escher's print, and turn over what angle, to see the same print? ‘A specific question I can deal with,’ he thinks to himself.

## The back of an envelope

Lenstra rolls up his sleeves and goes to work. A simple calculation, on the back of an envelope, gives him a precise formula for the desired transformation. ‘In this way, I fully expose the mathematical theory behind the print, in fact in no time at all,’ says Lenstra. ‘Those are always beautiful moments, when you suddenly have the feeling that you understand something right down to its core!’ Yet he remains modest: ‘This is actually nineteenth-century mathematics. I therefore see this project more as popularisation than as scientific research.’

The math is cracked, but the job isn't done yet. ‘We still have to make a real print in which the repetition is visible,’ says the emeritus professor. And that involves both programming and artistic work. Fellow mathematician Bart de Smit, now Vice Dean of the Faculty of Science, decides to take on this part as project leader. Eventually, master’s student Joost Batenburg, now also a professor in Leiden, writes a computer program to calculate the endless repetition of the drawing. Artists Hans Richter and Jacqueline Hofstra then take care of the interpretation of the print. And behold, after 50 years, the artwork the Print Gallery has finally been completed.

## A timeless character

Another twenty years later, Lenstra’s name appears in Nature Physics. In a review of an Escher exhibition in Trieste, Italy, the writer mentions that Lenstra managed to complete the work of art. The review shows how timeless the effort of Lenstra and his colleagues is. Lenstra is not surprised that their work is mentioned in the magazine in 2020. 'Firstly, every pure mathematician is used to the timeless character of his work. And secondly, I still receive invitations to lecture about it. People can't get enough of it, and frankly, I can’t either.’

### Biography

Hendrik Lenstra (1949) became professor at the University of Amsterdam at the age of 28. In 1998, Lenstra was awarded the Spinoza Prize, the highest scientific award in the Netherlands. In 2007, the Royal Netherlands Academy of Arts and Sciences (KNAW) appointed him as Academy Professor. Lenstra is well known for the development of the Lenstra algorithm and the LLL algorithm. The latter he developed with his brother Arjen. Brothers Andries and Jan Karel Lenstra are also mathematicians.

## Animation: Spanish artists animate the Leiden efforts

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Text: Bryce Benda