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0 pointsjacobolus2y ago0 comments

The reason it is confusing is because an angle measure is a kind of logarithm of a rotation, and logarithms (sort of) have a unit: the base.

The appropriate canonical representation of a rotation is a unit-magnitude complex number z = exp iθ = cos θ + i sin θ, which has a planar orientation (whatever plane i is taken to represent; if you want to represent a 3D rotation you can replace i with an arbitrary unit bivector) but is unitless.

Such a rotation z can be thought of as the ratio of two vectors of the same magnitude: z = u / v satisfies zv = u, i.e. is the object by which you can multiply v on the left to obtain u. Whatever original units your vectors u and v had gets divided away.

This is similar to the way the "ten" in "scale by ten" is unitless, but if you take the logarithm you get "scale by 10 decibels" or "go up by 3 octaves and 3.9 semitones", which have the base of the logarithm as a kind of unit.

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Misdicorl2y ago

I think I fundamentally disagree with you. Angles do not have a "base" any more than meters do (being embedded inside some metric space could be considered a base I suppose).

But you seem to be drawing a distinction between meters and angles in your analogy where I assert none exists. The base of a number system only affects representations.

This is not true for divisions of lengths. 1 meter divided by 2 meters is 0.5 as a number. But it is only 0.5 radians under (1ish) specific arrangements of those lengths in a particular metric space

jacobolusOP2y ago

The logarithmic base for an angle is something like "degrees", "radians" or "turns".

This is analogous to the way a scalar logarithm can have a base of "octaves" (doublings), "decibels", or "powers of the golden ratio" (as found in the Zometool construction toy). Or pick your favorite other logarithmic system.

Both are "units" in a certain sense, but neither one is quite the same kind of "unit" as light years or foot–pounds or amperes.

> 1 meter divided by 2 meters is 0.5 as a number. But it is only 0.5 radians under (1ish) specific arrangements of those lengths in a particular metric space

Just as 1 meter straight ahead divided by 2 meters straight ahead is the unitless scalar number 0.5, we can likewise treat angles (i.e. rotations) as ratios: 1 meter straight ahead divided by 1 meter to the right has the unitless bivector-valued ratio i, oriented like the ground you are standing on. You can multiply this bivector by some other coplanar vector to rotate it a quarter turn. For example, you can multiply it by the vector «3 inches due North» to get the new vector «3 inches due West»; notice how the units do not change because our bivector i is unitless.

Misdicorl2y ago

I understand your analogy, but I reject it's validity. Degrees/radians/turns map to meters/feet/angstrom. Decibels and octaves are true "number" multipliers. You could for instance talk about degrees in octaves or in dB if you like. It's just not particularly useful for the domain

Edit: another example difference. I can't measure an octave or dB. I can measure a degree

Edit2: we've reached reply limit but I concede you can measure a decibel. Point about dB degrees still stands though

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Misdicorl2y ago

I think I fundamentally disagree with you. Angles do not have a "base" any more than meters do (being embedded inside some metric space could be considered a base I suppose).

But you seem to be drawing a distinction between meters and angles in your analogy where I assert none exists. The base of a number system only affects representations.

This is not true for divisions of lengths. 1 meter divided by 2 meters is 0.5 as a number. But it is only 0.5 radians under (1ish) specific arrangements of those lengths in a particular metric space

jacobolusOP2y ago

The logarithmic base for an angle is something like "degrees", "radians" or "turns".

Both are "units" in a certain sense, but neither one is quite the same kind of "unit" as light years or foot–pounds or amperes.

> 1 meter divided by 2 meters is 0.5 as a number. But it is only 0.5 radians under (1ish) specific arrangements of those lengths in a particular metric space

Misdicorl2y ago

Edit: another example difference. I can't measure an octave or dB. I can measure a degree

Edit2: we've reached reply limit but I concede you can measure a decibel. Point about dB degrees still stands though

1 more reply

j / k navigate · click thread line to collapse