5 ms·

Euler's formula is "e^ix = cos(x) + i sin(x)", which arguably encodes the "trick" of thinking of e^ix being a rotation. Once you understand this relationship,

Euler's formula is "e^ix = cos(x) + i sin(x)", which arguably encodes the "trick" of thinking of e^ix being a rotation. Once you understand this relationship, the special case of x=pi becomes obvious and boring. If you arrived at that special case without first hitting the more general understanding, than I can see calling it "beautiful" because it is so suggestive of something deeper going on. Of course, when we present this identity as the most "beautiful formula", it is often to people who don't have any understanding of what it means to take something to a complex power, so the correct response should be to look at you funny.

jacobolus 9y ago
Personally I would take θ = log(x + yi) / i – where x is a scalar and yi is a bivector, with x² + y² = 1 – to be the definition of angle measure θ of the rotation x + yi, though usually I like to think of this as θi, a bivector-valued quantity. In that formulation, the structure of rotations is already clear, and the better question might be why the (principal branch of the) natural logarithm of some bivector i is (π/2)i. That starts us on thinking about what the logarithm really means, and asking about conformally mapping a punctured plane to an infinite two-ended cylinder, etc. The reason this formula of Euler’s seems like a mystery is that we teach angle measure to students as the canonical way to represent a rotation which makes it logically primary in their minds, when (in my opinion) it should be a derived quantity. Rotation is a naturally multiplicative kind of operation, a linear transformation of a vector space. If we need to represent a rotation of 2-dimensional vectors by a single parameter, I generally prefer the stereographic projection yi/(1 + x) (“half-angle tangent”) instead of the angle measure (logarithm). Depends a bit on the particular use case though.

jacobolus 9y ago

Personally I would take θ = log(x + yi) / i – where x is a scalar and yi is a bivector, with x² + y² = 1 – to be the definition of angle measure θ of the rotation x + yi, though usually I like to think of this as θi, a bivector-valued quantity. In that formulation, the structure of rotations is already clear, and the better question might be why the (principal branch of the) natural logarithm of some bivector i is (π/2)i. That starts us on thinking about what the logarithm really means, and asking about conformally mapping a punctured plane to an infinite two-ended cylinder, etc. The reason this formula of Euler’s seems like a mystery is that we teach angle measure to students as the canonical way to represent a rotation which makes it logically primary in their minds, when (in my opinion) it should be a derived quantity. Rotation is a naturally multiplicative kind of operation, a linear transformation of a vector space. If we need to represent a rotation of 2-dimensional vectors by a single parameter, I generally prefer the stereographic projection yi/(1 + x) (“half-angle tangent”) instead of the angle measure (logarithm). Depends a bit on the particular use case though.