‘Fire’ is more of a system than a single thing. You need fuel, heat, and oxygen, but when we think of ‘fuel’ the thing that burns isn’t actually the piece of wax or the stick of wood. It’s tiny pieces of wax or wood that have *pyrolyzed* into tiny particles that float in the air. That stream of particles is called ‘smoke’.
When you see a candle flame, the visible part you see emitting light is *not* where any burning occurs. The burning occurs on the tips and edges of the flame.
The visible flame is actually smoke – smoke heated up so much that it glows red-hot, just like metal coming out of a forge. But it doesn’t burn because there’s no oxygen inside the flame. As the smoke flows upward and outward it eventually comes into contact with oxygen. As soon as the super-hot atomized fuel touches oxygen, it burns, becoming water vapor and invisible gasses like CO2, and emitting a lot of heat. This heat is sent out in all directions, including back towards the oxygen-free bubble and towards the fuel source. This produces more smoke, and heats up that smoke so it glows red-hot, which will eventually float out to find some oxygen and do some burning of its own, perpetuating the cycle.
So to answer your question, the candle is *always* creating smoke. It’s just also heating up and burning that smoke, in order to create and heat up more smoke to burn. When you extinguish the candle, you’ve interrupted the cycle. The fuel source still has heat, and is still pyrolyzing, generating a stream of smoke until it cools down. But the smoke is no longer being heated up, so it doesn’t glow, and it doesn’t burn.
This is why the [this cool party trick](https://www.youtube.com/watch?v=cPJLw7Xkmzk) works. The candle is relit from the stream of smoke, because lighting the smoke, and lighting the candle is basically the same thing. You heat up the smoke that’s mixed in with oxygen, it burns instantly, the burning heats up more smoke/oxygen mix below it, and that combusts, heating up more below it, and it follows this cycle all the way back to the base of the candle, where it takes root heating up the fuel source to get more smoke to burn.
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