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Euler’s formula says (in one interpretation) that a half-turn rotation in the plane is the same as a reflection through a point, a statement which is obvious to

Euler’s formula says (in one interpretation) that a half-turn rotation in the plane is the same as a reflection through a point, a statement which is obvious to children who think about it for a bit. It’s phrased in a cute way, but “most beautiful formula of all time” etc. is really a stretch. The tricky part is thinking of rotations as a type of “number” with a multiplicative structure, in which case the [additive] angle measure becomes the logarithm of the rotation, letting us write our half-turn rotation using the fancy-looking exp(πi) instead of ½⥀ or rot(180°) or whatever. If you asked someone whether ⤺² = ⥀ (or similar) was the most beautiful formula in mathematics they would probably laugh or look at you funny.

eru 9y ago
It is beautiful that this interpretation maps (complex) numbers and geometry so well---even if the formula itself doesn't look so beautiful and surprising anymore.
- jacobolus 9y ago
  Personally I take “similarity transformations which fix a point in the plane” (or if you like, “amplitwists of planar vectors”) to be the definition of complex numbers, from which a Cartesian coordinate representation (i.e. breaking them into scalar + bivector parts) can easily be derived. In this view the notion of “[mapping] complex numbers to geometry” is just tautological. Maybe better would be to say that Euler’s formula shows how classical trigonometry (angle measure &c.) is what you get when you apply logarithmic thinking to Euclidean vector geometry.
  - eru 9y ago
    If you see Geometry as flowing from something like Euclid's axioms, and (complex) numbers as flowing from a long line of advances in essentially bookkeeping/accounting, it is quite nice that they have such a neat synthesis.
    - jacobolus 9y ago
      A whole lot of the original bookkeeping/accounting was about adjudicating land boundary disputes and calculating land taxes based on area, or predicting/explaining astronomical events. In fact, the origins of both trigonometry and logarithms are in astronomy, though I’ll admit that the natural logarithm per se came from Bernoulli’s study of compound interest. But yes, the connection between geometry and arithmetic is something people have been noticing for at least 5000 years, and is certainly important and very interesting. I just think this particular expression of that connection is oversold as being uniquely insightful or beautiful.
      - eru 9y ago
        Sure. And if anything the more general equation e^(iphi)=cos phi + i cos phi is much more useful: it allowed me to forget all the special rules we learned in school about how to manipulate the trigonometric functions and just use my knowledge of exponentials instead.
gizmo686 9y ago
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.
deleted 9y ago
[deleted]
ouid 9y ago
is there anything interesting to you in the notion that cos(x)+isin(x) is the unique analytic continuation of e^x in the complex plane? It's not just a rotation by definition, I don't think.
- deleted 9y ago
  [deleted]
- jacobolus 9y ago
  If you think about it for a bit, it’s clear that this must be so. A circle’s circumference is proportional to its radius, so if you want to have a conformal map between a rectangular grid in one space and a polar-coordinate grid in another space, the spacing of the polar-coordinate grid lines in the radial direction must necessarily be proportional to the radius so that they can match the radius-dependent spacing in the tangential direction. Anyhow, there are many interesting features of the structures here. Both rotation and logarithms are very interesting and important both in mathematics and in science/engineering. I just don’t think Euler’s identity in particular is worthy of so much attention.
  - ouid 9y ago
    Sure, but I think most people believe that cos(x)+isin(x)=e^ix because that's how it is defined, in which case e^(i\pi)=-1 isn't even a result. It is at least the tiniest bit more interesting than that.
alexvoda 9y ago
I love the way you wrote that? Why do mathematicians have a predisposition to overload symbols? I hope in the future, mathematical notation will evolve to make full use of Unicode and of names with more than 1 letter.
JadeNB 9y ago
> Euler’s formula says (in one interpretation) that a half-turn rotation in the plane is the same as a reflection through a point, a statement which is obvious to children who think about it for a bit. I guess by "reflection through a point" you mean "the composition of two reflections through orthogonal axes through the point", right? Otherwise this cannot possibly be true, since a rotation is orientation-preserving but a reflection is orientation-reversing.
- jacobolus 9y ago
  If you reflect every point in the plane across a given “axis” point, you get a half-turn rotation. Reflecting across a point is also the same as reflecting twice across two perpendicular lines which intersect at that point.
  - JadeNB 9y ago
    > If you reflect every point in the plane across a given “axis” point, you get a half-turn rotation. > Reflecting across a point is also the same as reflecting twice across two perpendicular lines which intersect at that point. My claim was rather that "reflecting across a point" has no meaning by itself—in the strict, mathematical sense according to which a reflection has a fixed hyperplane https://en.wikipedia.org/wiki/Reflection_(mathematics) https://en.wikipedia.org/wiki/Reflection_(mathematics) —unless you use one of these two statements as the definition. (Probably you'd better use the second, because the first doesn't generalise to higher dimensions.)

eru 9y ago

It is beautiful that this interpretation maps (complex) numbers and geometry so well---even if the formula itself doesn't look so beautiful and surprising anymore.

jacobolus 9y ago
Personally I take “similarity transformations which fix a point in the plane” (or if you like, “amplitwists of planar vectors”) to be the definition of complex numbers, from which a Cartesian coordinate representation (i.e. breaking them into scalar + bivector parts) can easily be derived. In this view the notion of “[mapping] complex numbers to geometry” is just tautological. Maybe better would be to say that Euler’s formula shows how classical trigonometry (angle measure &c.) is what you get when you apply logarithmic thinking to Euclidean vector geometry.
- eru 9y ago
  If you see Geometry as flowing from something like Euclid's axioms, and (complex) numbers as flowing from a long line of advances in essentially bookkeeping/accounting, it is quite nice that they have such a neat synthesis.
  - jacobolus 9y ago
    A whole lot of the original bookkeeping/accounting was about adjudicating land boundary disputes and calculating land taxes based on area, or predicting/explaining astronomical events. In fact, the origins of both trigonometry and logarithms are in astronomy, though I’ll admit that the natural logarithm per se came from Bernoulli’s study of compound interest. But yes, the connection between geometry and arithmetic is something people have been noticing for at least 5000 years, and is certainly important and very interesting. I just think this particular expression of that connection is oversold as being uniquely insightful or beautiful.
    - eru 9y ago
      Sure. And if anything the more general equation e^(iphi)=cos phi + i cos phi is much more useful: it allowed me to forget all the special rules we learned in school about how to manipulate the trigonometric functions and just use my knowledge of exponentials instead.

gizmo686 9y ago

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.

deleted 9y ago

[deleted]

ouid 9y ago

is there anything interesting to you in the notion that cos(x)+isin(x) is the unique analytic continuation of e^x in the complex plane? It's not just a rotation by definition, I don't think.

deleted 9y ago
[deleted]
jacobolus 9y ago
If you think about it for a bit, it’s clear that this must be so. A circle’s circumference is proportional to its radius, so if you want to have a conformal map between a rectangular grid in one space and a polar-coordinate grid in another space, the spacing of the polar-coordinate grid lines in the radial direction must necessarily be proportional to the radius so that they can match the radius-dependent spacing in the tangential direction. Anyhow, there are many interesting features of the structures here. Both rotation and logarithms are very interesting and important both in mathematics and in science/engineering. I just don’t think Euler’s identity in particular is worthy of so much attention.
- ouid 9y ago
  Sure, but I think most people believe that cos(x)+isin(x)=e^ix because that's how it is defined, in which case e^(i\pi)=-1 isn't even a result. It is at least the tiniest bit more interesting than that.

alexvoda 9y ago

I love the way you wrote that? Why do mathematicians have a predisposition to overload symbols? I hope in the future, mathematical notation will evolve to make full use of Unicode and of names with more than 1 letter.

JadeNB 9y ago

> Euler’s formula says (in one interpretation) that a half-turn rotation in the plane is the same as a reflection through a point, a statement which is obvious to children who think about it for a bit. I guess by "reflection through a point" you mean "the composition of two reflections through orthogonal axes through the point", right? Otherwise this cannot possibly be true, since a rotation is orientation-preserving but a reflection is orientation-reversing.

jacobolus 9y ago
If you reflect every point in the plane across a given “axis” point, you get a half-turn rotation. Reflecting across a point is also the same as reflecting twice across two perpendicular lines which intersect at that point.
- JadeNB 9y ago
  > If you reflect every point in the plane across a given “axis” point, you get a half-turn rotation. > Reflecting across a point is also the same as reflecting twice across two perpendicular lines which intersect at that point. My claim was rather that "reflecting across a point" has no meaning by itself—in the strict, mathematical sense according to which a reflection has a fixed hyperplane https://en.wikipedia.org/wiki/Reflection_(mathematics) https://en.wikipedia.org/wiki/Reflection_(mathematics) —unless you use one of these two statements as the definition. (Probably you'd better use the second, because the first doesn't generalise to higher dimensions.)