So, fireworks come it two parts.
Patterns, and colors/brightness.
For patterns, this comes from how the firework is assembled.
Say you have a ball firework. That firework has first the launching charge on it, the charge that explodes first and sends it flying up into the air.
Next it has a secondary charge. After the laud info charge there is a ball of explosives flying through the air, outside the ball there are dozens or hundreds of little pockets of colored charge. Each colored charge causes a specific element or mineral to burn, producing colored light. For example, copper will produce green light when “burnt”.
So, this secondary charge is surrounded on all sides by pockets of copper, so when it explodes it sends copper flying in all directions, creating a big, green, ball firework.
Now if instead of a ball, you just wanted to make a circle, you would only pack those copper pockets around the rim of the secondary charge, not completely around it like a ball.
So in that case when the secondary ring explodes, it just sends out a circle of shots.
Now that is the very basic ELI5 explanation of firework shapes.
If you want to know how some fireworks look like, sat, flags, that is a combination of special packing by the pyrotechnics crew, and setting multiple fireworks off at the exact same time, to trick the audience into thinking it was just one big firework.
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