What is inverse of the radius, if we think mathematically, ignoring Newton’s explanation as a varying force? The same with inverse of the radius square, does it have a geometric meaning?

Yes, they do have natural geometric meanings—just in different contexts than one might first expect.

I was reading about curvature of circles: “A smaller circle (small ) is very “curvy” (high curvature).” If so, doesn’t this contradict the fact that all circles are similar? Can you clarify the confusion?

You’ve raised an interesting geometric question about circles, curvature, and similarity.

While all circles are indeed similar in the geometric sense (they all have the same shape and can be transformed into one another through scaling), their curvature values differ based on their size.

The curvature of a circle is mathematically defined as κ = 1/r, where r is the radius. This means:

• A circle with radius 1 has curvature \(\kappa = 1\)
  
• A circle with radius 2 has curvature \(\kappa = 0.5\)
  
• A circle with radius 10 has curvature \(\kappa = 0.1\)
  

So smaller circles do have higher curvature values, which matches your quoted statement.

The key insight is that similarity in geometry preserves angles and proportions, but not absolute measurements like curvature. When you scale a circle to make it larger, you’re changing its curvature even though you’re preserving its shape.

This is actually consistent with our intuitive understanding - a tiny circle appears to bend more sharply at each point than a large circle, even though they’re mathematically similar shapes.

Does that help resolve the confusion?

“The key insight is that similarity in geometry preserves angles and proportions, but not absolute measurements like curvature.” Ok, this clarifies it for me. But I also have a vague intuition that, since in nature large and small are relative quantities, this may effect curvature too. Seen from one astronomical unit the earth may be a dot but for us its surface is almost flat (low curvature). Can you clarify this intution as well?

Your intuition is spot-on! Perception of curvature is indeed relative to scale and perspective.

Think about standing on Earth’s surface. To us, the Earth appears nearly flat (very low curvature) because we’re experiencing it at a human scale relative to Earth’s massive radius (\(\sim 6,371\) km). Our local experience is of a curvature so slight it’s imperceptible without instruments or traveling long distances.

Yet from space, at a distance of one astronomical unit, Earth appears as a small circle or dot with a much higher perceived curvature. If we could somehow “feel” the Earth’s surface from that distance, it would seem highly curved.

This relates to an important concept in differential geometry: curvature is an intrinsic property with an absolute mathematical value (1/radius for circles), but our perception and experience of curvature is relative to our scale of observation.

We see this principle across many scales:

• A bacterium crawling on a grain of sand might experience that grain as having very low curvature
  
• The same grain of sand viewed under a microscope appears highly curved to us
  
• Our galaxy’s spiral arms have curvature that’s imperceptible to us locally but visible when viewed from a larger conceptual scale
  

This relationship between scale and perceived curvature is fundamental to how we interpret geometry across different frames of reference, from quantum scales to cosmic ones.

So in this case, we must state curvature always by comparing two radii. When we say “small circle” or “large circle” we implicitly assume that we are taking as unit the human scale. Given two radii \(R > r\), we must state \(1/R < 1/r\) to talk about curvature. But \(1/R\) or \(1/r\) alone will not make sense. Can you comment on this issue of the relativity of curvature?

You’re exploring an important nuance here about curvature measurement and relativity.

While mathematically, curvature is defined absolutely as \(\kappa = 1/r\), you’re right that our understanding of “high” or “low” curvature often implicitly depends on comparison or context.

The mathematical value \(1/r\) does make sense on its own - it’s a well-defined quantity with physical meaning. A circle with radius \(1\) meter has curvature \(1 \; 1/\text{m}\), which tells us precisely how quickly the curve turns per unit distance traveled along it.

However, our interpretation of these values as “high” or “low” curvature introduces relativity. This happens in two ways:

1. Comparison between objects: When we say a circle has “higher curvature” than another, we’re making a relative judgment between two different curvature values (\(1/r > 1/R\) when \(r < R\)).
   
2. Scale of observation: As you noted, human scale often serves as an implicit reference. A curve with radius 1 km appears nearly flat to us, while a curve with radius \(1\) mm appears sharply bent, even though mathematically both have precise curvature values.
   

This relates to dimensional analysis - curvature has units (typically 1/m), meaning it’s inherently measured against some reference scale. When we say something has “high curvature” without explicit comparison, we’re often making an implicit comparison to our perceptual reference frame or the context of the problem.

In general relativity, this becomes even more profound, as curvature becomes relative to the observer’s reference frame and the distribution of energy-momentum in spacetime.

Would you like to explore how this relativity of curvature extends to more complex geometric or physical situations?

If curvature is defined with radius how do we measure the curvature of a straight line? Intuitively the straight line should have zero curvature. In fact a straight line does not have a radius. How do we state mathematically the curvature of a straight line?

You’re exactly right - a straight line has zero curvature, which aligns with our intuition.

Mathematically, we can approach this in several ways:

1. Limit approach: Consider a circle with radius \(r\). As r approaches infinity, the circle becomes closer and closer to a straight line. Since curvature \(\kappa = 1/r\), as \(r \rightarrow \infty, \kappa \rightarrow 0\). Thus, a straight line has curvature zero.
   

2. Calculus definition: For a general curve, curvature is defined using calculus. For a parametric curve \((x(t), y(t))\), the curvature is:
   

   \[ \kappa = \frac{|x'y'' - y'x''|}{(x'^2 + y'^2)^{(3/2)}} \]
   

   For a straight line like \(y = mx + b\), the second derivatives \((x''\) and \(y''\)) are zero, making the numerator zero, so \(\kappa = 0\).
   

3. Radius of curvature: A straight line can be thought of as having an infinite radius of curvature. Since \(\kappa = 1/r\), when \(r = \infty, \kappa = 0\).
   
4. Geometric interpretation: Curvature measures how quickly a curve deviates from its tangent line. Since a straight line never deviates from itself (it is its own tangent line everywhere), its curvature is zero.
   

This zero curvature of straight lines is a fundamental concept in differential geometry and extends naturally to surfaces and higher-dimensional spaces, where flat planes and spaces also have zero curvature.

“A straight line can be thought of as having an infinite radius of curvature.” Arguments involving the concept of infinity always sound like sophistry to me, so the argument with tangent makes more sense to me. But essentially, if we strictly uphold our definition of curvature as \(1/r\), we must admit that there are no straight lines. All “straight” lines are scaled curved lines. You need to introduce play on words on the word radius to claim the existence of “straight” lines. What is your comments on this view?

You’ve raised a fascinating philosophical point about how we mathematically conceptualize straightness and curvature.

Your skepticism about invoking infinity is well-founded. The statement “a straight line has infinite radius of curvature” does have an air of mathematical sleight-of-hand to it, as infinity isn’t a number in the conventional sense but a limit concept.

The tangent-based definition is indeed more precise: curvature measures deviation from the tangent line. For a straight line, there is no deviation from its tangent (it is its own tangent everywhere), so the curvature is exactly zero - without needing to invoke infinity.

Regarding your view that “there are no straight lines” if we strictly adhere to \(\kappa = 1/r\) - this highlights a limitation in starting with circles as our foundational objects. If we begin with the formula \(\kappa = 1/r\) derived from circles, then yes, we seem forced to describe straight lines through the awkward notion of “circles with infinite radius.”

But in more fundamental [?] approaches to differential geometry, we don’t start with circles and radii. Instead, we define curvature directly in terms of how quickly a curve turns - the rate of change of the tangent vector as you move along the curve. This definition naturally accommodates both curved and straight lines without any conceptual gymnastics.

In this view, straightness isn’t the absence of a well-defined property (radius) but rather the presence of a specific measurable property (zero rate of change in direction).

This is similar to how we conceptualize zero in mathematics - not as the absence of quantity, but as a specific quantity with well-defined properties.

Make sense. But I wish you didn’t mention the can of worms called “zero” in mathemathics :) In case of zero I believe and uphold the Euclidean concept of zero. So let’s discuss zero some other time. But I have this last question about curvature: When you choose a standard unit like meter you say curvature is absolute. But I don’t see any difference between standard units or non-standard units. I can choose an arbitrary R as my unit and measure all other given radii with it. So, the fact that meter is a standard unit does not make the measurement absolute. But I can see that, regardless of the unit chosen, curvature may be an absolute quantity.

You’ve made an excellent point about units and absoluteness in measuring curvature.

You’re right that the choice of a standard unit like the meter doesn’t inherently make curvature more “absolute.” Whether we measure in meters, feet, or any arbitrary unit \(R\), the numerical value of curvature will change, but the physical reality it describes remains constant.

This touches on an important concept in physics and mathematics: dimensionful versus dimensionless quantities. Curvature (\(\kappa = 1/r\)) has dimensions of \([\text{length}]^1\). When we change our unit of length, the numerical value changes, but what it represents physically doesn’t.

For example:

• A circle with radius 1 meter has curvature 1 m\(^{-1}\)
  
• The same circle measured with radius 100 centimeters has curvature 0.01 cm\(^{-1}\)
  

Both describe exactly the same physical curve with the same “bendiness,” just expressed in different units.

What makes curvature “absolute” in a deeper sense is that it describes an intrinsic geometric property that doesn’t depend on how we embed or view the curve. Two observers using different units will disagree on the numerical value, but agree on the physical reality being measured.

This is similar to how we might disagree on whether a room is \(20\) feet long or \(6.1\) meters long, but we’re describing the same physical distance.