### Computing primes with CSS

Thursday 29 September 2016

I've been working on a redesign of this site, so doing more CSS, finally internalizing Sass, etc. During my reading, the nth-child pseudo-class caught my eye. It's oddly specific, providing syntax like "p:nth-child(4n+3)" to select every fourth paragraph, starting with the third. It isn't an arbitrary expression, it has to be of the form An+B, where A and B are integers, possibly negative. An element is selected if it is the An+B'th child of its parent, for some value of n ≥ 0.

It struck me that this is just enough computational power to compute primes with a Sieve of Eratosthenes, so I whipped up an demonstration (see it live here):

`<html>`

<head>

<style>

/* A stupid pet trick by Ned Batchelder @nedbat */

html { max-width: 40rem; }

span { display: inline-block; width: 2em; text-align: right; }

span:nth-child(2n+4),

span:nth-child(3n+6),

span:nth-child(4n+8),

span:nth-child(5n+10),

span:nth-child(6n+12),

span:nth-child(7n+14),

...

span:nth-child(30n+60),

span:nth-child(31n+62),

span:nth-child(32n+64),

span:first-child { display: none; }

</style>

</head>

<body>

<p>Primes:</p>

<div>

<span>1</span>

<span>2</span>

<span>3</span>

<span>4</span>

...

<span>996</span>

<span>997</span>

<span>998</span>

<span>999</span>

</div>

</body>

</html>

The code has only linear sequences of numbers. There are spans for 1 through 999, the candidate numbers. These are arranged so that the number N is the Nth child of their containing div. The CSS has nth-child styles for 2 through 32, the possible factors.

The Sieve will hide numbers that are determined not to be primes with a "display: none" rule. A first-child selector hides 1, which is typical, seems like you always have to treat 1 specially when looking for primes. The other selectors for the display:none rule select the multiples of each number in turn. "nth-child(2n+4)" will hide elements 4, 6, 8, and so on. "nth-child(3n+6)" will hide 6, 9, 12, and so on.

So CSS has two features that together are just enough to implement the Sieve. The nth-child selector accomplishes the marking of factors. The overlapped application of separate rules implements the multiple passes, one for each factor.

Of course, I didn't write this file by hand, I wrote a Python program to do it. It's pretty simple, I won't clog up this post with the whole thing. But, it was my first use of a new feature in Python 3.6: f-strings. The loop that writes the nth-child selectors looks like this:

`for i in range(2, 33):`

print(f"span:nth-child({i}n+{2*i}),")

The f"" string has curly-bracketed expressions in it which are evaluated in the current scope. This string in Python 3.6:

`f"span:nth-child({i}n+{2*i})"`

is equivalent to this in previous Pythons:

`"span:nth-child({i}n+{i2})".format(i=i, i2=2*i)`

It felt really natural to use this new feature, and really convenient.