Shadows occur when an object obstructs light emitted from a light source.
Caustics occur when light passes through two transparent materials with different densities. The difference in density determines how much the light will be bent. You can see this effect in the bottom of a pool, where water has a different density to the air.
Incandescence occurs when hot objects release som of their thermal energy as photons. It starts emitting infrared photons at temperatures of 800°C and will emit higher frequencies with higher temperatures above that.
In a flame, all these effects can take place. When you burn a candle, tiny particles of wax will reach high enough temperatures to glow yellow with incandescence, they will also block the light from other light sources to form a shadow, and finally, the hot air riding from the candle has a lower density than the cold air surrounding it, creating a caustic effect for any light passing through the air.
In some flames, the effects can be missing, for example alcohol fires can look completely transparent on a bright day, and can be very dangerous because of it.
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