Breaking a magnet

I used to think a magnet was two pieces stuck together: a north end and a south end. So I took a magnetised needle ten centimetres long and snapped it in the middle, to get one north on its own and one south on its own.

I thought a magnet was two pieces stuck together. I snapped a $10$ cm needle in the middle to get a lone north $N$ and a lone south $S$.

Needle $L=10\text{cm}$; snap: $1\to 2$ magnets.

I tested the halves with a compass. Each half had a north end and a south end. Two magnets, not two poles; four poles in all.

Compass test: each half has an $N$ and an $S$. $2$ magnets, not $2$ poles: $2\times 2=4$ poles.

Each half has $N$ and $S$; $2\times 2=4$ poles.

I snapped each half again: four magnets and eight poles. Again: eight magnets and sixteen poles. Every piece always had both ends. Each round of halving doubles the number of magnets, and the north poles and the south poles stay equal in number.

Halve again: $4$ magnets, $8$ poles. Again: $8$ magnets, $16$ poles. After $k$ rounds: $2^k$ magnets, $2^{k+1}$ poles, and $N=S=2^k$.

After $k$ rounds: $2^k$ magnets, $2^{k+1}$ poles, $N=S=2^k$.

If I could go ten rounds, I would have one thousand and twenty-four pieces, each shorter than a tenth of a millimetre, and each still with a north and a south. I never found a lone pole. No one ever has.

$k=10$: $2^{10}=1024$ pieces, each $100/1024\approx 0.098$ mm, which is less than $0.1$ mm, and each still has $N$ and $S$. A lone pole: never found.

$k=10$: $2^{10}=1024$; $L/1024\approx 0.098\text{mm}$; lone poles: $0$.