[Verse 1]
In Königsberg they pondered long
Seven bridges crossed the Pregel strong
Four districts split by flowing water
Could you walk them all, each bridge but once?
The townsfolk tried but always failed
Some path would leave them trapped or jailed
Until Euler cast his brilliant eye
Not on the bridges, but on why
[Chorus]
Don't count the steps or measure distance
Look for patterns in existence
Every vertex tells its tale
Odd degree makes crossing fail
V minus E plus F equals two
Topology sees what's really true
Strip away what eyes can see
Find the hidden structure's key
[Verse 2]
He drew each landmass as a dot
Connected where the bridges crossed
Four vertices with edges traced
The geometric flesh erased
Count the bridges touching ground
In each district, look around
Three here, five there, three again
Three more - all odd, now see the pain
[Chorus]
Don't count the steps or measure distance
Look for patterns in existence
Every vertex tells its tale
Odd degree makes crossing fail
V minus E plus F equals two
Topology sees what's really true
Strip away what eyes can see
Find the hidden structure's key
[Bridge]
If you enter, you must exit
Every crossing needs its pair
Start and finish break this rule
But four odd vertices aren't fair
This invariant reveals the truth
No coordinates required
Connectivity decides the fate
Of paths that can't be acquired
[Verse 3]
From bridges to the sphere above
Polyhedra show the same love
Vertices and edges, faces counted
Euler's formula surmounted
Genus grows, the formula bends
Two minus twice the surface trends
Sheaves and chains, Betti's dance
Algebraic topology's advance
[Outro]
When possibility seems unclear
Let invariants make it disappear
The structure holds the answer tight
Beyond what meets the naked sight