There is no one origin of pointed birthday hats. In fact, this distinctive headgear has been around at the very least since 2800 BC, when ancient Egyptian pharaohs donned it. It is unknown if pointed hats were adopted from Egypt or if they simply began systematically appearing around the world. Whatever the case, it's safe to say that not only are cylindrical birthday hats fun and cheap to make, they continue the proud tradition of many different cultures.
The Ancient Egyptians
Only ancient Egyptian nobility were allowed to wear tall pointed hats as a symbol of their elevated status above their subjects. These hats were ancient Egyptian crowns, passed down pharaoh to pharaoh, and the different designs made individual pharaohs identifiable to their subjects.
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Medieval Jewish Traditions
Some Jewish people in Europe started wearing cylindrical hats secured around the chin to distinguish themselves in medieval Christian Europe. These hats were higher and more pointed than the traditional Jewish Kippah worn today. While most didn't partake in this tradition, certain sects, like the Ashkenazi Jews (Albert Einstein's ancestors), were required to wear them.
Witches and Wizards
The historical association between paganism and pointed hats is one brought on by minority persecution. Because Jewish people in medieval Europe donned pointed hats, pagans were viewed as part of the same "unwelcome minority." It is also sometimes believed that witches and wizards wore pointed hats because the hats were representations of the cone of power. When pagan rituals were performed, witches and wizards would gather in circular formations and draw energy from above and from the center of the circle, thereby creating a conical shape.
Princesses and Noblewomen
Interestingly, while Jewish women and practicing pagans would not wear pointed hats, Catholic princesses and noblewomen in Medieval Europe sometimes would. These would be intricately decorated with long veils, precious jewels and even intricate butterfly and flower detailing.
The Point of Pointed Hats on Birthdays
The point of cylindrical hats seems to be to elevate the wearer to the status of royalty. While in most places nowadays, we don't have royal families running around in traditional garments, the act of celebrating royalty (which a person becomes on their birthday) transpires on their birthday.