Gabriel’s horn is the curve y = 1/x (from x = 1 onward) spun around the x-axis. You can fill it with exactly π units of paint, but its surface is infinite, so you can never paint it. Why?
Slice the horn into thin disks.
Each disk’s AREA is π/x². That shrinks like 1, 1/4, 1/9… fast enough to add up: the volume is exactly π.
Each disk’s RIM is 2π/x. That only shrinks like 1, 1/2, 1/3… the harmonic series, which never stops growing. So the surface area is infinite.
The catch: filling it does coat the inside, but far down the tail the horn is narrower than an atom (with a 1 m mouth, 10^11 m in it’s 2×10^-11 m wide). Any coat of real thickness on the outside needs at least π·t² of paint per unit of length, forever: infinite paint.
Did the horn fool you? Tell me below.
Sources: Gabriel’s horn and Harmonic series (Wikipedia). Every number here was checked with SymPy.
#Shorts #math #calculus #gabrielshorn
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