Some more memories from Friesland. Binary Calculator.

A couple more images from our trip to Friesland. Simon’s binary calculator:

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Doing math at a restaurant where we were celebrating his friend’s birthday:

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Ramanujan-converging

Today Simon learned that it was Euler who first came up with the idea that the infinite sum 1 + 2 + 3 + 4 +… converges to -1/12. Simon explained to me the other day that there are several ways of looking at an infinite sum. One way is looking at its partial sums and summing those up. Another way is averaging partial sums and see what their average converges to (or what the average of their averages converges to). “That’s called Cesaro Summation and it’s good for closely related sums like 1 – 1 + 1 – 1 + 1… but not for 1 + 2 + 3 + 4 +…”, Simon explained. “Then there is Ramanujan Summation – a Calculus way of looking at infinite sums using derivatives and gamma, etc. That is the only way 1 + 2 + 3 + 4 +… converges to -1/12. All possible infinite sums converge if you use Ramanujan Summation.”

“Simon, you don’t trust Ramanujan Summation, do you?” I asked.

“No. Only an infinitely small section of infinite sums converge using the standard method. Converging means it settles down. That’s what we call a fixed point. If an infinite sum doesn’t converge, it can either explode to infinity or it can have more than one fixed point or do something else weird like that. Sums that are not convergent are called divergent.”

“And Ramanujan had none of those, no divergent sums?”

“Yeah, he really made it to the extreme! It’s an infinitely large extreme. All infinite sums Ramanujan-converge.”

Geometric Definition of e

The idea comes from a video by Mathologer. Simon sketches a geometric definition of the Euler’s number (e) using integrals. He messed up a little with the integral notation, but corrected it later (after we stopped filming). Please see the photos below:

The rest of the trip to London

Simon loved the Science Museum, even though he did not get to see the Klein Bottles from the museum’s permanent collection (none of them was on display). He particularly enjoyed the math and information age spaces. The Original Tour was a success, too – giggling at all the jokes on the English audio guide, he was bubbling with joy that he could follow everything and was actively studying the map, together with Dad. The only thing Simon really hated to tears was The Tower.

The Maisie Day

For Simon and me, this book (“Infinite Lives of Maisie Day” by Christopher Edge) has probably been one of our most profound experiences of the year. We read it together, sometimes, giggling with joy as we recognized Simon’s favorite topics interwoven in the plot (like that the main character also dreams of proving the Riemann hypothesis), and sometimes tears choking our throats as we went through the sad and scary bits of the story. And what a trip down the memory lane last night, at the Royal Institution in London, where we attended a lecture about the science behind “Infinite Lives of Maisie Day”! As Simon proudly told one of the lecturers (University College London’s cosmologist Dr Andrew Pontzen) after the show, he even predicted something important in the book. Simon recognized that Maisie turned into a mirror image of herself after she had traveled around the Mobius-shaped universe, just as depicted in Escher’s “the impossible staircase” painting . “But that’s only possible if you’re flat, a 2D object! So it’s not correct in the book, but they probably put that in to make it simpler,” Simon laughed. “You’re absolutely right! Keep doing science!” the cosmologist told him. @Ri_Science