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Sample a wave and you lose the difference between it and every wave a multiple of the sampling rate away — not approximately, exactly: the numbers you record are the same numbers. A wheel turning 23 times a second filmed at 24 frames looks like one slow turn backwards, and a 30 kHz tone recorded at 44.1 kHz comes back as an audible 14.1 kHz. The fold is at half the rate and there is nothing on the far side of it.

Take the powers of two and look only at the leading digit. It is a 1 three times in ten and a 9 once in twenty-two, and that is not a quirk of two: it follows from log base ten of two being irrational, and it fails exactly when that number is not. The powers of ten start with a 1 every single time, and the powers of the square root of ten use two digits and no others.

A hanging chain is a catenary and a suspension bridge cable is a parabola, and they are not the same curve. Over a hundred-metre span sagging ten metres they part by 32 millimetres — a hand's width, at one place a little past a third of the way out from the middle, and nowhere else. The difference is the fourth term of a series, which is why every drawing of a chain you have ever seen was near enough right.

A comparison answers one bit, so k of them tell apart at most 2 to the k orders — and five things can be in 120 orders, which six questions cannot separate. The bound is log2(n!), it is reached exactly up to eleven things, and at twelve it stops being reachable: the counting says 29 comparisons and 30 are needed.

Look at the previous letter and the Gettysburg Address falls from 4.12 bits a character to 2.99. Look at the previous four and it falls to 0.35, which sounds like a triumph and is a model that has memorised the text: the table you would have to send costs 11,466 bits, and the address is 11,664 bits as plain bytes. The same letters shuffled tell you which part of the fall was structure and which was memory.

Every number on this wall is a double: 52 bits of fraction, and a gap that doubles with the magnitude. Between 1 and 2 there are 4,503,599,627,370,496 of them and between 1024 and 2048 there are exactly as many, which is why half of all doubles lie between minus one and one — and why a clock holding the seconds since the Big Bang can only tell you the time to the nearest minute.

Buses every ten minutes, and the average rider lands in a gap of twenty. Your friends have more friends than you do. The college's average class is 32 and the average student sits in 73. Sixty out and twenty back averages thirty, not forty. Four famous surprises, one fraction: mean times one plus the coefficient of variation squared.

A CSS pixel is not a length, it is an angle: one ninety-sixth of an inch held at arm's length, 0.0213 degrees. A 20/20 letter is five arcminutes, which is 8.73 mm on a chart at six metres and 3.9 reference pixels on a screen. Work out what a pixel really subtends on the screens people own, held where people hold them, and the phone — the one this wall spent a day worrying about — turns out to be the most generous of them all.