Accidentally mathematical songs

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Girls & Boys by Blur

The chorus of Girls & Boys by Britpop band Blur goes:

Girls who are boys
Who like boys to be girls
Who do boys like they’re girls
Who do girls like they’re boys.
Always should be someone you really love.
If you take the sequence of ‘girls’ (G) and ‘boys’ (B), you get: GBBGBGGB. This follows the Thue–Morse sequence.

The Thue–Morse sequence is represented as a string of binary digits. The sequence can be extended by taking the Boolean complement (take the opposite to each position) of the current string and adding this to the end of the string. We start with 0, and add the Boolean complement (1) to obtain the string 01. Now we have 01, and add the Boolean complement (10) to get 0110. Repeating this a few times, we get the sequence 0110100110010110…

In fact, you could take any string of ones and zeros from the Thue–Morse sequence, and it won’t appear again until after the string finishes. For example, take the string 1101 which appears near the start. The next time this string appears is in position 14, without overlapping with the original string:

0110100110010110100101100…In fact, you could take any series of ones and zeros in the Thue–Morse sequence, and it too won’t overlap with the same series later in the sequence.

To prove this in a hand-wavy way*, suppose that the Thue–Morse sequence is overlap-free until a string $A$ that contains an overlap appears (for example, $A$ could be 10101 as it contains overlapping 101s). But by the nature of the sequence, the Boolean complement of $A$ (01010 in our example) must have already appeared in the sequence, which means there must have been an earlier overlap. This is a contradiction, so the Thue–Morse sequence must be overlap free. QED.

Replacing 0 and 1 with G and B, we see that the start of the Thue–Morse sequence is the same as the Girls & Boys sequence. As such, we can extend the chorus:

Girls who are boys
Who like boys to be girls
Who do boys like they’re girls
Who do girls like they’re boys
Who need boys like they’re girls
Who need girls like they’re boys
Who have girls who are boys
Who have boys who are girls
Who choose boys who see girls
Who choose girls who see boys
Who meet girls who like boys
Who meet boys who like girls
Who verb girls who verb boys
Who verb boys who verb girls
…
Always should be someone you really love.
* Proving this fully is a little harder than this. Can you spot which bit of the proof needs a bit more work?

Falling to Pieces by Faith No More

In 1990, American rock band Faith No More, released the single Falling to Pieces, from their album The Real Thing. Within the song is the verse:

From the bottom, it looks like a steep incline,
From the top, another downhill slope of mine,
But I know, the equilibrium’s there.
Which makes me think of the intermediate value theorem.

The intermediate value theorem states: ‘If $f$ is a continuous function whose domain contains the interval $[a,b]$ then it takes any given value between $f(a)$ and $f(b)$ at some point within the interval.’ A corollary of this theorem is that if you have the same conditions, then there must exist $c\in[a, b]$ such that $f(c)=(f(a)+f(b))/2$.

But how does this relate to the song? Well, if we take ‘the bottom’ to be a point at $x=a$ and ‘the top’ to be a point at $x=b$, then we can define a function between them. As stated, to use the intermediate value theorem we need the function to be continuous; since it is a slope, I think it is fair to say that the surface rarely breaks—you never have a hill where suddenly halfway up you find a huge gap in the floor!

The corollary of the intermediate value theorem tells us that there must be a point exactly halfway up the vertical cross section. In other words, ‘the equilibrium’ is there; exactly as Faith No More’s conjecture stated.