[Verse 1]
Started with the reals, ordered and serene
Every number placed precisely in between
But Hamilton dreamed wilder, craved rotation's dance
So we built the complex plane, gave geometry a chance
[Chorus]
Double up the dimension, sacrifice a trait
Real to complex loses order at the gate
Complex to quaternions, commuting slips away
Quaternions to octonions, associative decay
Sedenions steal alternativity
Hurwitz drew the boundary, this far and no further free
[Verse 2]
Eighteen forty-three, carved on Dublin's stone
Hamilton discovered what rotation truly owns
i, j, k, quaternions spin the world around
Graphics cards and spacecraft use what he unwound
[Chorus]
Double up the dimension, sacrifice a trait
Real to complex loses order at the gate
Complex to quaternions, commuting slips away
Quaternions to octonions, associative decay
Sedenions steal alternativity
Hurwitz drew the boundary, this far and no further free
[Bridge]
Zero divisors creep in when sixteen dimensions bloom
Structure crumbles piece by piece, mathematics meets its doom
Not because we cannot build, but because the cost runs steep
Every doubling claims a law that made the system neat
[Verse 3]
Commutativity vanished when we left the complex field
Associativity abandoned, quaternion rules must yield
Alternative behavior fades when octonions expand
Sedenions bring chaos to what once was planned
[Final Chorus]
Double up the dimension, sacrifice a trait
Construction halts not from failure, but from structural weight
Each expansion costs us something mathematicians hold dear
Till zero divisors whisper the endgame is here
[Outro]
Hurwitz theorem stands guard at dimension's precipice
Structure is the currency, and we've spent all of it