a lovely capretto sand witch, light on the mayo.

Either that or she can sell you potions.

Greatest of all time witches

Because it operates based on complicated physics and maths that scholars understood widely at the time.

It provides better air circulation and looks really cool.

Keeps cool, looks cool.

+3 magic accuracy and defence

Yes to everyone here :)

They feel cool to wear, are cool to look at or visualise, and make you a more powerful Wizard by building up your confidence.

https://www.mathsisfun.com/geometry/circle-theorems.html

They're actually an advanced form of this special easy to visualise maths all wizards should be using.

The extra mental challenge would be to visualise how things like 3D cone sections overlayed, and be able to use this to explain things about your own logic in your head.

Other weird hats have powerful properties too, like beanies. They're shaped like half a standard Orb, and their maths involves much more constant and unusual data and energy leakages than standard cone hats.

I would start by trying to explain to yourself how conic sections work and if they tell you anything about the real world.

Real life, fully physical, conic sections storing information actually look almost exactly the same as German enigma machines. They can be used for trying to encode advanced mathematical encryption in your head, to be shared with other advanced wizards whose thoughts you want to align with. Please don't hide huge amounts of forum info though, lol.