Mirror, mirror, on the wall, which is the fairest rectangle of them all?

Described by the 17th-century astronomer Johannes Kepler as a ‘precious jewel’, the so-called ‘golden ratio’ is an irrational number whose physical manifestation is prevalent in nature and has been employed by artists, architects and designers for its aesthetic allure. It has been argued, however, that the ratio’s occurrence in the natural world has been exaggerated, and that its ‘beauty’ can be chalked up to its pleasing simplicity. This brief animation explores the ratio through its most famous manifestation, the ‘golden rectangle’, whose very name begs the question whether some shapes are inherently more beautiful than others.

After 30 years of solitude, Peter forms an unlikely friendship with a fellow loner

‘I had left my flock, and Ben had left his.’

After taking a walk through a remote Welsh valley, Peter committed himself to a life there, and disconnected from the outside world. In doing so, he found a solitary inner peace – a peace he maintained for nearly three decades, until, one day, he stumbled upon a lamb that had been left for dead. Finding kinship with the fellow ‘dropout’, Peter took the abandoned creature home and named him Ben. The short Peter and Ben (2007) by the UK filmmaker Pinny Grylls captures the duo’s relationship three years after their chance meeting, as Peter attempts to return Ben to the wild. With a melancholic piano score and sweeping views of the Welsh countryside, her touching film lends a lyrical beauty to this tale of unlikely connection and camaraderie between outsiders.

Trawling for secrets in haunting films recovered from the bottom of the sea

The British ocean liner RMS Lusitania embarked on its infamous final voyage from New York to Liverpool on 1 May 1915. Six days later, torpedoed by German U-boats off the southern coast of Ireland, the ship sank in less than 20 minutes, killing 1,198 passengers and crew, and setting the US on the path to join the fight against Germany in the First World War. One of the most luxurious ocean liners of its time, the Lusitania was equipped with what was then a novelty – an onboard movie theatre.

In Sunken Films, the US artist and filmmaker Bill Morrison uses archival footage to unspool the stories of the sinking of this luxury liner, its incendiary movie reels, as well as other films about or from shipwrecks. One early clip was salvaged from the sunken Lusitania in a 1982 expedition; another mysterious film, featuring the Russian revolutionary Vladimir Lenin in 1919-20 with his cat, was recovered from a fishing net off the Danish coast in 1976. By trawling for memories in deep-sea shipwrecks, Morrison offers haunting glimpses into early-20th century film and world history.

Is a red apple proof that all ravens are black? A paradox of scientific logic

Can we learn anything about what makes a raven by looking only at apples? The German-born logician Carl Gustav Hempel (1905-97) thought that, using the inductive logic that scientists rely on to prove or disprove hypotheses, you ought to be able to – but in such a way that clashes mightily with human intuition. This peculiar ripple in reasoning, which became known as ‘the raven paradox’ due to the example Hempel used to elucidate it, goes as follows:

1. All ravens are black
2. If something is not black it is not a raven
3. The fact that my pet raven is black supports the hypothesis that all ravens are black
4. The fact that my apple is red also supports the hypothesis that all ravens are black

The sequence appears to break down somewhere between the third and fourth claims. And yet, upon examination, inductive logic tells us that claim four does indeed follow. In this brief animation, Marc Lange, a professor of philosophy at the University of North Carolina at Chapel Hill, dissects why Hempel’s claim seems to hold to reason, even as it cuts against our intuitions in a way that seems unresolvable.

Lockdown is a way of life for the US asylum-seekers living in churches

While much of the world was adjusting to lockdown and socialising via screens, life went on more or less the same for Vicky Chavez and her two young daughters. For more than two years, they have been unable to leave the First Unitarian Church of Salt Lake City in Utah, their makeshift home. After Chavez was ordered to deport to her native Honduras, which she fled to escape an abusive relationship, the church provided her refuge from US Immigration and Customs Enforcement (ICE) – which doesn’t pursue raids at places of worship – while her case is being fought in court. In her moving short Shelter in Place, the US director Kelsie Moore captures the family’s precarious life in limbo, which includes Chavez’s regular video calls with other asylum-seekers living in churches around the country.