It’s the Archchancellor’s hat, don’t you understand? It’s worn by the head of all wizards, well, on the head of the head of all wizards, no, metaphorically it’s worn by all wizards, potentially, anyway, and it’s what every wizard aspires to, it’s the symbol of organized magic, it’s the pointy tip of the profession, it’s a symbol, it’s what it means to all wizards…
