Expired: e ^ (i * π) = -1
Tired: e ^ (i * τ) = 1
Inspired: i ^ i is a real number (!!)

Think of "a^b" as "start at 1. Take the process that 'grows' the number slowly until it reaches a, and repeat it b times"

Think of "i times" as "once, but the process is rotated 90 degrees" (assume we're working on the complex plane)

Here's how I thought of it algebraically

x = i ^ i
x = e^( (ln i) * i )

What is (ln i)? Let (ln i) = Q
e^Q = i
e^(i*Q/i) = cos Q/i + i sin Q/i
i = cos (Q/i) + i sin (Q/i)
i = cos(pi/2) + i sin(pi/2) thus Q/i = pi/2
Q = pi/2*i

x = e^( Q * i )
x = e^( pi / 2 * i * i)
x = e^( pi / 2 * -1)
x = ~0.2079

Recommend this classic “unusual” book about this topic: Visual Complex Analysis , by Dr. Tristan Needham.

Vitalik thought you would like to know I bought some eth yesterday.