A study in self-reference After Diego Velázquez

The painting is in the painting.

Paint a smaller room inside this one. Repeat, then hang the mirror to make the room repeat itself.

The calculation · $3!$

See the code

The ! means multiply down to 1: 3! = 3 × 2 × 1 = 6.

// A function with a hole where its own// name should go.const open = self => n =>  n === 0 ? 1 : n * self(n - 1); // No name. So: a copy. A copy of the copy.const fact = blank; fact(3)

fact (3)

The canvas is blank. The King asks for $3!$ . Paint the room onto the canvas, and paint the canvas in that room too.

Try the numbers 3, 5, 9, then an unknown number.

Fixed-point combinators

It needs its own name.
Give it a mirror.

In the lambda calculus nothing has a name, so a function cannot call itself. It can only be handed things. Hand it a copy of itself and it can recurse once. Hand it a copy of the copy and it can recurse twice. That is Velázquez’s problem exactly: the picture holds the room, the room holds the picture, and somebody has to paint every level by hand.

Y = λ f . (λ x . f (x x)) (λ x . f (x x))

The Y combinator is the mirror. $Y f = f (Y f)$ : it hands $f$ to itself as deep as the argument needs, without a single name. $Y f$ is a fixed point of $f$ , a value $f$ returns unchanged, which is why the painting of the painting is the painting.

You already trust fixed points. A layout engine iterates until nothing moves. A type checker widens until the types stop changing. A build is done when running it again changes nothing. The startup accelerator borrowed the name for the shape of the idea: a program that runs programs.

If you stepped into the mirror, that was your browser’s own camera. The picture never left this page.

Diego Velázquez’s Las Meninas: the Infanta Margarita stands in a tall dim room with her maids, a dog and a dwarf, while the painter looks out from behind a huge canvas and a small mirror on the back wall reflects the King and Queen. — Diego Velázquez, Las Meninas, 1656. The painter looks at you. The mirror shows whoever stands where you stand.Museo del Prado

It needs its own name.Give it a mirror.

It needs its own name.
Give it a mirror.