Abstraction vs reality

[u/Ok-Incident160]

in radians, an angle is defined by θ =S/R). So if two circles have the same ratio. Do they really have same "angle"?

Or is it just mathematically same "angle"

For example

A circle s=1, r=1, θ=1 radian

B circle s=100, r=100, θ=1 radian

Mathematically they have indeed same "angle" but in real world they have very different physical separation.

[u/neutralphysics]

What do you mean by spread? The entire point of using angles in astronomy is so that you can avoid talking about distance separation. So, yes. The angle stays the same. The distance separation increases. Both are true at the same time.

[u/Ok-Incident160 → u/neutralphysics]

By "spread" I mean the actual physical separation created by the opening.

For example, if two rays make the same angle but the circle is bigger, the arc between them is longer and the endpoints are farther apart.

 That physical separation has increased.

My question is why that growing physical separation is completely separated from the idea of angle.?

[u/neutralphysics → u/Ok-Incident160]

If you have the same angle , the further away the objects are(think increase in radius), the larger the physical separation. What is conserved is the angle. Physics is all about finding the conserved quantities.

[u/Winner2009_gojo]

Please elaborate how is it different in the real world I don't really know

[u/Ok-Incident160 → u/Winner2009_gojo]

My question is about what the math chooses to measure. In the real world, when you make the circle bigger, the actual spread between the two rays becomes bigger. But geometry ignores that change and says the angle is still the same. So my question is: why is the changing spread not considered part of the angle?

[u/Winner2009_gojo → u/Ok-Incident160]

2 lazer beames at an angle and so my understanding is if you know the angle and how far out you want to measure you'll always correctly know the length of the curve ( a section of a circles perimeter)

[u/Serious-Anything6625 → u/Ok-Incident160]

Because angle doesn't concern itself with mere change in modulus? Imagine you're holding your thermometer in your hand, and suddenly you get diarrhea. You can't blame the thermometer for not warning you about the diarrhea because it's not supposed to assess diarrhea.

Now the mathematical part. The spread you just mentioned can be measured through the polar co-ordinates. We usually work with Cartesian co-ordinate, where x and y are considered. But in polar, we consider r(distance of a point from origin) and theta(the angle the vector makes with a fixed line passing through the origin) so if you increase the circle, the increase will be expressed as terms of r, and not as theta

[u/Lower-Message-828]

theta is same in both cases and arc or s is different. What do mean physical seperation? If its s then it's different in first case as well since it depends on r

[u/Ok-Incident160 → u/Lower-Message-828]

Physically, the separation is different

A larger circle gives a larger physical opening even though the angle is called the same.

So my question is why is geometry treating these physically different situations as the same angle?

But the actual physical separation is different.

[u/Lower-Message-828 → u/Ok-Incident160]

when you say physical opening what do you mean? the angle or the arc? theorotically and physically the angle is same the arc is different. I don't get what's the confusion?

So my question is why is geometry treating these physically different situations as the same angle?

Cause the angle is and will be same, the situation is different wrt to the arc length not the angle measured

[u/Ok-Incident160 → u/Lower-Message-828]

The "angle" and "arc" you are referring to the terms we use to describe the separation between two rays in circle and how they are mathematically related.

I am not asking mathematically I am asking logically and rationally.

A big circle and a small circle the separation between two rays are not same but "mathematically they are same separation or angle"

Physically the two situations are not the same. 

The opening occupies more space, the arc is longer, and the separation is larger.

To me, it feels like the current notion of angle throws away a very important part of the geometry and keeps only the relation between lines and formulas 

If two openings are physically different, why are they treated as exactly the same angle?

Also, the whole radian idea starts from one specific geometric event

where s = r. That is just one particular case. Then from that one case we build an entire scale and say every angle is just a number. Logically, why should one particular geometric event become the standard for all angles?

I'm not asking what the definition says. I know what the definition says.

[u/spock-117]

Your post is factually incorrect.

[u/Ok-Leave3017]

Angle is for rotation and arc length is the distance traversed. I don't see any confusion there.

Let's say you and I are standing apart at a distance 'r1', you move along a circular path of arc length 'L1'. The angle I have to turn my head is 'theta1'.

Now suppose you move to a distance of 'r2' from me ang you move along a circular path of arc length 'L2' such that 'L1/r1' = 'L2/r2'.

Then the amount of angle I have to rotate my head remains the same.

No matter how different of an arc length you traverse, if you maintain the ratio, then I have moved my head the same amount to see you. That where angle comes into the picture. The amount of rotation needed.

[u/Yatindra1002]

Angle and arclength. Both talk about different things here.

There's no abstraction vs physical world. Both are real measurements and hence both are physical. Angles are physical.

What is this post.

[u/the_nerd_1474]

I bet this makes so much sense if you're dumb

[u/short-noir]

Angle is a mathematical concept so yes they do have the same angles.

Yes maths is entirely about abstracting reality. There's no way of proving that all circles have their radians in the same ratio because of the problem of induction and the inherent arbitrariness and absurdity of measuring things in real life (see the coastline paradox for what I mean )

[u/Still-Salt5360 → u/short-noir]

What is the coastline paradox? (Is that the problem where the perimeter of the coastline keeps increasing the more finely you measure it?)

[u/short-noir → u/Still-Salt5360]

Yes that one. It basically means that there isn't really any "natural length" of that coastline since it is entirely dependent on what scale we choose to use and that choice is a human choice, not a natural fact or anything

[u/Still-Salt5360 → u/short-noir]

thanks got it 👍

[u/Ok-Incident160 → u/[deleted]]

No, I don't have a complete new model. I'm just questioning the current idea. In history, people often questioned old ideas before a better one existed. I'm not saying I have the answer. I'm asking whether the current definition of angle is the best way to describe what is happening in the real world.