Ideal and Real Geometry of Convex Edges

[ToddS]

For being so obsessed with definitions, I thought you would understand the mathematical definition of an arc. Yes, it can be defined by three measurements.

"Arc" usually refers to a circle, the equation of which can be determined from three points. Mathematically, "arc" refers to any curve and the special case of a circle is strictly called a "circular arc." Obviously you cannot determine the equation of an arbitrary arc from three points. You can approximate an arbitrary (smoothly varying) curve by a circular arc over some range from three points.

 

[ToddS]

Good points and you explained that in a very clear manner Todd, thank-you.

But that's why I specified between "real world" and "calculus" because I am thinking more along the lines of real world use. For example. If I get a convex edged knife in, and I want to determine the angle at the apex, I can simply coat w/ a sharpie and hit with my 1k grit stones on my wicked edge until one swipe removes all of the marker at the apex. This is what I would determine to be the apex angle.
...

An important fact about convex edges is that when you grind on a flat stone the contact area is extremely small and therefore the pressure is extremely high (pressure=force/area). Since metal removal rate increases with pressure, you are flattening the bevel near the apex with only one pass.

 

[edgemaster]

When I knock the edge off 2 intersecting surfaces I use the terms CHAMFER, BEVEL and RADIUS, not CONVEX.

If we exchange BEVEL for CONVEX in all preceeding discussions, what happens?

Maybe for another thread..

in this corner the simple, yet fantastic V edge
and in this corner, the multibevel with micro radius edge

Thanks for this discussion, reminds me why I like math and applications.

 

[bpeezer]

"Arc" usually refers to a circle, the equation of which can be determined from three points. Mathematically, "arc" refers to any curve and the special case of a circle is strictly called a "circular arc." Obviously you cannot determine the equation of an arbitrary arc from three points. You can approximate an arbitrary (smoothly varying) curve by a circular arc over some range from three points.

My mistake, I guess I was over generalizing! Yes, I was referring strictly to a circular arc.

 

[chiral.grolim]

"Arc" usually refers to a circle, the equation of which can be determined from three points. Mathematically, "arc" refers to any curve and the special case of a circle is strictly called a "circular arc." Obviously you cannot determine the equation of an arbitrary arc from three points. You can approximate an arbitrary (smoothly varying) curve by a circular arc over some range from three points.

My mistake, I guess I was over generalizing! Yes, I was referring strictly to a circular arc.

Thank you, ToddS, for helping bpeezer understand :thumbup:

If you already KNOW the nature of your arc - i.e. strictly circular, or specifically quadratic, etc. or if you already know the radius or the equation itself that can be used to define it - THEN you know precisely how many points you need to determine the geometry of that arc.

But you do NOT have that information, only a few measurements taken to a specific level of precision (you'll note the range that Gillette lists with each of their measurements). The variety of arcs that could fit those data points is infinite, you don't have enough information to "determine" the arc mathematically. Whatever you come up with is a guess, though it might be a very good one.

We also know that the apex dose not end as the intersection of two arcs, rather it rounds over to a blunt point with a tangent at 180 degrees.

But even if we pretend that the edge geometry can be determined mathematically and does not end in a blunt point, why would we use "tangent" to characterize the geometry anyway? Again, the slope of the tangent is only applicable to a single dimensionless point, NOT a bevel. The angle between the intersecting tangents of each arc describes exactly the space between what? Between those tangents, decidedly NOT between ANY part of the arcing bevels.

Again, the angle of intersection of the arcs does not describe the space between the arcs, it describes the space between the tangents. It is only applicable to the arcs at a single point with NO SPACE, NO BETWEEN, NO THICKNESS, NO DEPTH.

I object to those who think they understand the math but do not understand the concept of "limits" appropriately.
Let's work with the well characterized example of f(x)=x^2, the derivative of which is f'(x)=2x. In the presentation f'(x)=2x we are missing some information that is absolutely fundamental to the concept of a derivative, namely "lim" over "h->0". What those pieces of information explain is that the derivative approximates a value without ever reaching it by taking a "limit".

The function f(x)=2x describes THE line that is "tangent" to f(x)=x^2, but f(x)=2x is absolutely NOT interchangeable with f'(x)=2x. Do those who think they understand calculus understand that, or do they disagree? If they agree, do they understand why they are not interchangeable?

 

[chiral.grolim]

That straight line would be the tangent, 90° to the circumference of the circle from which we define the arc. If we have the value of the circumference, we can plot any point along that arc.

What is circumference, how do you measure it? Also, we decidedly do NOT "define the arc" from the circumference of a circle. That assumption is false.

EDit to add: Also, it's much easier to measure the length of the radius from which to estimate the circumference, since we measure in straight lines.

And again, the line of the tangent does NOT describe anything more than a single dimensionless point at the apex - there is no "line" on the surface of the curved bevel that matches the angle of the tangent, the curve only approximates the behavior of the line. How closely does it approximate the line? As closely as the secant lines are taken to match it. That's how finding a tangent - i.e. finding the angle of an arc - works, through inscribed secant lines.

Nothing arbitrary about the centerline, and it isn't theoretical either. When folks say "I ground a V bevel at 15° per side" what exactly do you think they are referencing? 15° to what? If we're working on a true chisel edge, the flat side effectively becomes the centerline, 0°, a 0° primary grind is 0° to what? An 8° primary is 8° to what.

The "centerline" is that which bisects the angle of the bevels, i.e. "central". I find it odd to declare the face of one bevel as the "centerline" (e.g. chisel blade), what is it "central" to?? It's neither center of the spine nor center of the edge... No wonder you have trouble with "convex" and "concave".

And in your drawing, the tangents for the convex path do not match the V bevel of the red triangle that you describe as the "only way to measure an angle". That's fine, but it should have a relationship to the arc, in your diagram it doesn't - you aren't measuring the intersect angle of the arc in any way shape or form with the triangle, no matter how imperfect.

Actually, I am measuring it in the ONLY way possible, in the way described by the mathematics used to define it, used to find the tangent = the LEAST "imperfect" way. You note that the secant lines don't match the tangents exactly - GOOD, they shouldn't, otherwise you'd know I divided by zero to get my answer AND you'd know that the angle doesn't apply to the shape of my bevel beyond that single point = irrelevant to describing the shape of the bevel.

...your argument does not hold water except through torturous interpretations of what is "convex" in this instance, and in the real world of contouring steel surfaces, it holds no water at all.

I think, or at least hope, that you'll come to recognize the opposite. "Convex" is exactly as I have described it, curving out from a surface; and "concave" is curving in from a surface, e.g. a "hollow". When we remove steel to grind a bevel, we ALWAYS remove steel, but how much and where makes a difference. If we allow the surface to arc outward between two points (carving around a center point), that is "convex". If we cut the arc flat between those same two points, that is "flat", and if we cut (or scoop) deeper between those same two points, that is "concave". That is the "real world of contouring".

Edit to add: In the "real world of contouring" there are many places where the "convex" hills were carved out by nature over time, just as flat plains and valleys were carved. In cutting statues and figures from stone or a glass lens or a wooden bowl or spoon, material is always being removed but in such manner that we form shapes - some with a flat surface, some which arc above the flat (convex), and some which arc below the flat (concave). This is how the words are used, how they have been used for millennia.

 

[bpeezer]

The function f(x)=2x describes THE line that is "tangent" to f(x)=x^2, but f(x)=2x is absolutely NOT interchangeable with f'(x)=2x. Do those who think they understand calculus understand that, or do they disagree? If they agree, do they understand why they are not interchangeable?

Nope, I disagree. With the right differentiable function, a derivative can ACCURATELY produce another differentiable function. If taking a derivative were not accurate, then how could we take 2nd, 3rd, etc. level derivatives?

Let's look at another example

The derivative of e^x is, excitingly enough, e^x. If derivatives were not accurate, then each time we take the derivative it will be slightly off. Let's say the derivative of e^x is slightly different than e^x. If we take the derivative of something slightly different than e^x, then that derivative will be more different than the first one. Let's say we go to the tenth derivative of e^x...I took the liberty of plugging it into Wolfram Alpha.

Surprise...It's e^x!!! Not almost e^x, but e^x. If we took the billionth derivative of e^x, it will still be e^x. Because derivatives ARE accurate and CAN produce real functions.

 

[HeavyHanded]

I suspect that much of the confusion in these convex vs straight bevel discussions can be summarized with the following sketch. The analogy of taking a shortcut at the intersection of two sidewalks was offered by HeavyHanded in an earlier thread.

Imagine you are walking along a sidewalk, approaching a corner, and decide to take a shortcut; leaving the sidewalk at point A and re-entering the adjacent sidewalk at point B. While the convex (red) path is a shortcut to walking to the very corner, the straight line is shorter still.

Knocking the shoulder off of a bevel with a convex curve (blue line) thins the blade, but a flat (secondary) bevel between the start and end points (green) of that convex curve is thinner still. The key point is that the bevel is convex (curves outward) relative to the straight line connecting the start and end points (the secant line). Comparing those two lines (bevels) the convex is thicker, as it must be by the definition of convex (curves outward).

Todd, I understand the undercutting of the convex in the sidewalk analogy, but if you make that straight line it no longer joins or leaves the sidewalk along the original angle. That is what I've been saying, the straight line can undercut, but ultimately it will have to become a FFG with a much reduced final angle. Carried on the sidewalk analogy, one can keep taking a longer and longer arc and undercutting it with a straight line until both the arc and straight line are virtually indistinguishable and both paths have you pushing through hedges and climbing over the neighbor's car to see who can stay off the sidewalk the longest!

In the second diagram, the added bevel transition is well on the way to an *arced* edge (in honor of semantics, when everyone that's been following this thread knows exactly what we're referring to by now I will stop referring to it as a "convex" for the duration). In fact if one was a touch sloppy making that extra green bevel they would have accomplished the *arced* edge anyway.

All this talk about the edge being unable to measure if it is only a tangent really amounts to a smokescreen. While my microscope is optical, it still is reasonably powerful - as such I can use the depth of field and a leveled stage to determine with a reasonable amount of accuracy the final edge angle. Only on the most poorly prepared edges does it curve down into an unidentifiable end, most carry the final angle long enough for a value to be closely estimated. So most of the conversation from my POV is based on material behind the edge. As long as a transition exists with a meeting of two identifiable planes, it can be replaced with an arc at that transition and produce a profile with less meat.

As a simple way of looking at it, the area of a polygon is always going to shrink if you replace one of the straight line intersections with an arc. The cross-section of a cutting tool is no different from any other polygon, just a matter of how the lines are arranged.

 

[HeavyHanded]

What is circumference, how do you measure it? Also, we decidedly do NOT "define the arc" from the circumference of a circle. That assumption is false.

EDit to add: Also, it's much easier to measure the length of the radius from which to estimate the circumference, since we measure in straight lines.

And again, the line of the tangent does NOT describe anything more than a single dimensionless point at the apex - there is no "line" on the surface of the curved bevel that matches the angle of the tangent, the curve only approximates the behavior of the line. How closely does it approximate the line? As closely as the secant lines are taken to match it. That's how finding a tangent - i.e. finding the angle of an arc - works, through inscribed secant lines.

My mistake, I meant to say radius. If we have the tangent, the center and the radius, all the points on the circle can be plotted.

The "centerline" is that which bisects the angle of the bevels, i.e. "central". I find it odd to declare the face of one bevel as the "centerline" (e.g. chisel blade), what is it "central" to?? It's neither center of the spine nor center of the edge... No wonder you have trouble with "convex" and "concave".

Not all tools have "bevels" plural.
Then what are folks referring to when they say they have a plane blade or chisel ground at 28° - 28° to what? If you asked them what degree the flat side was ground to, they'd likely dismiss you out of hand, same as if you asked them to what they were referencing the 28°.

The flat (or center) is what we reference the position of the cutting tool to the abrasive surface with which we grind our edge, or any other part of the cutting tool. Also one could just as easily and accurately (in the case of chisel ground tools) use any part of the cutting tool that was parallel ( 0° for all intents) to the centerline.

Actually, I am measuring it in the ONLY way possible, in the way described by the mathematics used to define it, used to find the tangent = the LEAST "imperfect" way. You note that the secant lines don't match the tangents exactly - GOOD, they shouldn't, otherwise you'd know I divided by zero to get my answer AND you'd know that the angle doesn't apply to the shape of my bevel beyond that single point = irrelevant to describing the shape of the bevel.

When grinding steel it is virtually impossible to grind down into some theoretical non existence - the edge terminates at a point that can be determined by physical examination if nothing else.

I think, or at least hope, that you'll come to recognize the opposite. "Convex" is exactly as I have described it, curving out from a surface; and "concave" is curving in from a surface, e.g. a "hollow". When we remove steel to grind a bevel, we ALWAYS remove steel, but how much and where makes a difference. If we allow the surface to arc outward between two points (carving around a center point), that is "convex". If we cut the arc flat between those same two points, that is "flat", and if we cut (or scoop) deeper between those same two points, that is "concave". That is the "real world of contouring".

Edit to add: In the "real world of contouring" there are many places where the "convex" hills were carved out by nature over time, just as flat plains and valleys were carved. In cutting statues and figures from stone or a glass lens or a wooden bowl or spoon, material is always being removed but in such manner that we form shapes - some with a flat surface, some which arc above the flat (convex), and some which arc below the flat (concave). This is how the words are used, how they have been used for millennia.

The Latin simply means arched or vaulted and does not describe "arched above or outside of" anything. Likewise the term "convex" when applied to knife edges does not automatically reference "convex" to the hypothetical V bevel that doesn't approximate much of anything, when one can just as easily and more accurately reference it to the centerline of the cutting shape and at least come up with an approximation that jibes with real world application.

My participation on this thread is drawing to a close. I believe I've made my point as best I can and anytime someone wants to send me a cutting tool with crisp transitions at the bevel intersections (no FFG), I'll be happy to thin it out and send it back (I'll cover the return postage) with the original cutting angle preserved and less meat behind the edge.

When I have to respond to line by line responses with my own line by line responses instead of carrying on a back and forth dialog in a normal fashion, I find the conversation too tedious to continue. My apologies.

Martin

 

[ToddS]

Todd, I understand the undercutting of the convex in the sidewalk analogy, but if you make that straight line it no longer joins or leaves the sidewalk along the original angle. That is what I've been saying, the straight line can undercut, but ultimately it will have to become a FFG with a much reduced final angle. Carried on the sidewalk analogy, one can keep taking a longer and longer arc and undercutting it with a straight line until both the arc and straight line are virtually indistinguishable and both paths have you pushing through hedges and climbing over the neighbor's car to see who can stay off the sidewalk the longest!
.....

To be clear, I do not doubt the fact that the contouring you perform reduces the thickness and improves the cutting efficacy relative to the initial geometry. We agree that you are producing a convex arc and by definition that must be thicker than the straight line that undercuts (marked in bold above) - that line is a secant line. I believe your preference for the arc over the straight line is to maintain a smoothly varying angle, essentially to produce an aerodynamic shape.

 

[chiral.grolim]

Let's work with the well characterized example of f(x)=x^2, the derivative of which is f'(x)=2x. In the presentation f'(x)=2x we are missing some information that is absolutely fundamental to the concept of a derivative, namely "lim" over "h->0". What those pieces of information explain is that the derivative approximates a value without ever reaching it by taking a "limit".

The function f(x)=2x describes THE line that is "tangent" to f(x)=x^2, but f(x)=2x is absolutely NOT interchangeable with f'(x)=2x. Do those who think they understand calculus understand that, or do they disagree? If they agree, do they understand why they are not interchangeable?

Nope, I disagree. With the right differentiable function, a derivative can ACCURATELY produce another differentiable function.

The existence of a derivative to a function was not in contention. You missed the entire point and again the problem seems to be that you do not understand the definition of "derivative" though it has been posted again and again. Do you know what a "premise" is?

Thank you for bringing up 'e'. What is the value of 'e'?

'e' = lim (1+h)^(1/h)
...... h->0

'e' is not a value at all, it is a limit that can be approximated by some value (similar to Pi). What is the exact value of f(x) = e^x where x=1? Where x=2? Are you able to determine the exact value of ANY point on that curve other than x=0? Why not?

Because any value for 'e' is an approximation all by itself. Oops.

Nevertheless, let's take the derivative of e^x (which is, granted, a neat situation), which amounts to getting an approximation of the slope at approximate points on the curve = approximations based on earlier approximations Sheesh.

http://www.wyzant.com/resources/lessons/math/calculus/derivative_proofs/e_to_the_x

Do you see that part about "lim" and "h->0" ? That is an indelible and necessary part of the formula - you can write it different ways (e.g. dy/dx or f'(x)) but it must always be part of the formula. That is the premise upon which the rest of the calculus is based!! Without that part of the formula, the rest is impossible. That part of the formula tells you that 'h' never equals zero, it only ever approaches it. As such, 'h' can never be eliminated from the equation, and any equation generated that assumes 'h=0', like f'(x) = 2x, is an approximation of the value at the point specified because the definition clearly states that 'h' does NOT equal zero or reduction of the formula from its original form would be impossible.

Understand, nowhere does a curve cease to be a curve = nowhere are there 2 adjacent points available to form a straight line that has a slope = the derivative gives the equation of the line with the slope that the curve most closely approximates at a given point.

The equation f(x) = e^x is not a derivative, it is not in derivative form, it does not have the notation for limits in the formula, there is no "lim" and "h->0" restriction in the premise. Just as f(x)=2x defines the same line that is the derivative of f(x)=x^2, it is not the derivative of f(x)=x^2 without the notation indicating it as such. There is no 'lim' and 'h->0' in f(x)=2x just as there isn't in f(x)=e^x, but there IS in the formula f'(x)=2x and f'(x)=e^x. The latter equations are derivatives because they are limits approximated by the equations they present, the former are not because no limit is predicated. If the derivative is NOT a limit formula, it is NOT a derivative. Limits are part of the definition.

If derivatives were not accurate, then each time we take the derivative it will be slightly off. Let's say the derivative of e^x is slightly different than e^x. If we take the derivative of something slightly different than e^x, then that derivative will be more different than the first one.

I'm not sure where you got this notion.

Each derivative includes the notation 'lim' over 'h->0' (forgive the switch from 'a' to 'x', i am pulling these images from wikipedia, didn't find one for the 3rd derivative or after, it gets cumbersome quickly):

See the limit notation? Without it, you wouldn't have a derivative at all, as that notation is part of the definition of "derivative".
As 'h' goes to zero, what would lead you to think that any difference in approximation from first derivative to second would be "slightly different"? Could you give me an example? Your assertion seems unfounded...

 

[chiral.grolim]

My mistake, I meant to say radius. If we have the tangent, the center and the radius, all the points on the circle can be plotted.

Plotting the points does not give us the circumference, there are infinite points. To get the circumference, we need to multiply the radius by 2*Pi, and we do not have an exact value for Pi, only an approximation, so any value for the circumference will be an approximation.

Not that it matters, not every arc is circular so trying to describe your convex that way requires knowledge you may not have (or be able to prove) in the first place.

Not all tools have "bevels" plural.
Then what are folks referring to when they say they have a plane blade or chisel ground at 28° - 28° to what? If you asked them what degree the flat side was ground to, they'd likely dismiss you out of hand, same as if you asked them to what they were referencing the 28°.

The flat (or center) is what we reference the position of the cutting tool to the abrasive surface with which we grind our edge, or any other part of the cutting tool. Also one could just as easily and accurately (in the case of chisel ground tools) use any part of the cutting tool that was parallel ( 0° for all intents) to the centerline.

By 28° you mean "inclusive", correct? You cut a bevel that was 28° relative to the other side of the angle? When i tell you I want a blade with a cutting edge of 28°, do you know if i want it cut asymmetrically or symmetrically?
If i say "14° per side" you know i want it symmetrical about the center-line of the blade itself. If i say "28°" that does not specify if i want it all on one side or symmetrical about the centerline. In either case, a "0° bevel" is not a "centerline" by any understanding, it is the "flat" as you said.

When grinding steel it is virtually impossible to grind down into some theoretical non existence - the edge terminates at a point that can be determined by physical examination if nothing else.

Yup, which is why every edge is 180-degrees at the apex = flat blunt. :thumbup:

The Latin simply means arched or vaulted and does not describe "arched above or outside of" anything.

No, it means arched outward. Wow, I hope you are alone in that misunderstanding.

...one can just as easily and more accurately reference it to the centerline of the cutting shape and at least come up with an approximation that jibes with real world application...

Yeah, because the "geometry" of a dimensionless-ly thin centerline approximates real world application.

My participation on this thread is drawing to a close. I believe I've made my point as best I can and anytime someone wants to send me a cutting tool with crisp transitions at the bevel intersections (no FFG), I'll be happy to thin it out and send it back (I'll cover the return postage) with the original cutting angle preserved and less meat behind the edge.

I'm glad you stipulated "no FFG" :thumbup:
So you'll round down the bevel shoulders without touching the edge :thumbup:

 

[bpeezer]

snip

In the case of taking the derivative of x^2, the limit is 2x. That's why the limit notation is important...it allows us to determine the derivative to be exactly 2x.

 

[Lagrangian]

Hi everyone,

I don't think a knife forum is the right place to debate calculus and analysis. So I'll be mostly pretty brief. Partly because I doubt there is must interest here. And partly because I'm not that interested; mostly I wrote posts to this thread for the sake of others. I would like to believe that I have gotten my points across, and any of those of you who disagree, that's fine. That said, here are a few points from me. Possibly, this will be the last post I make in this thread? Maybe. I probably will respond to non-trollish questions, if any.


-----------------------------------------------------
Intro
-----------------------------------------------------

Below, I would like to make four points. Briefly, the five points are:

(A) A general description of my background. This is mostly to demonstrate that I have experienced the consensus on what is "conventional math" when talking about calculus.

(B) Some technical discussions on limits.

(C) Some non-rigorous "reduction to absurdity" to show how some misunderstandings of limits/calculus do not make sense.

(D) An example of a limit. Namely, that the number 0.999999...=1. (The decimal with infinitely repeating 9's.) This equation is able to be understood at increasing levels of rigor which cover undergraduate calculus, to undergraduate analysis, to graduate mathematics. If you understand this elementary example very well, at your specific level, then you have a decent grasp of what a limit is (at the level of rigor that is appropriate for you).

(E) I would like to explain why we "care" about the angle of the apex for a convex knife. This is more of an "engineering" and "practical scientist" discussion, and is not meant to be mathematical or rigorous.

In the following posts, I will go into these points in more detail. Feel free to skip the "mathy" parts, namely (B),(C), and (D). In that case, just read the first and last, (A) and (E).

 

[Lagrangian]

-----------------------------------------------------
(A) Personal background
-----------------------------------------------------

As someone who has undergraduate and graduate degrees, completed a post-doc, worked in industry on developing CAD software, and published maybe a dozen peer-reviewed papers, I have literally met thousands of technical people. Many of which are more inventive, technically more brilliant, and more experienced than myself. Among all the technical people I've known (engineers, mathematicians, scientists, etc.) not a single one of them has said that f'(x)=2x is "an approximation" to the derivative of f(x)=x^2. (I have now found only one.)

I'm pretty comfortable with my own technical understanding, so I really see no point in debating this technical point, except that I like to teach and learn things from the general community here at bladeforums.com. But I doubt this thread holds much continued interest for anyone at bladeforums.com. So, I have little interest in continuing this technical debate. (On the other hand, I would love to talk more about how blade shape affects cutting, both theoretically (speculatively?), and experimentally.

 

[Lagrangian]

-----------------------------------------------------
(B) Some technical discussion on limits.
-----------------------------------------------------

We have some disagreements about what a limit is, how it is defined, and what it means.

This knife forum, is not the place for technical discussions on calculus, so I'm not going to do things like define a limit using epsilon-delta (or epsilon-n) notation, etc. Instead, I'm going to tell a pseudo-story, namely the "story" we are taught in math class about how the real numbers were constructed, and conceptually what is happening in calculus. This is only a pseudo-story, because it is not historically accurate; instead it is a story that helps students understand deeper and deeper levels of rigor.

Along the way, I hope you get some idea (or flavor) of what it means for something to be "mathematically well defined."

Very often in mathematics, we have some cool-thing that we understand very well for simple cases. And then we have this idea, that our cool-thing should also apply to more complicated cases. But, we don't know how to apply our cool-thing to more complicated cases. In fact, initially, we may not even know how to define what our cool-thing *is* in the more complicated cases. What do we do then?

To answer this question, let's start on our pseudo-story for how the real numbers are constructed.

We start with the natural numbers, namely N={1,2,3,4,...} This is our simple case. And our cool-thing is arithmetic. So we understand very well, things like how to add natural numbers (2+2=4), subtract (4-3=1), multiply, and divide. That's fine.

But the one day, someone invents "zero." Oh my goodness, what is this?! Does zero really exist? What does it even mean? When it was invented, zero was very controversial. Is it a number, or just the absence of any number? Maybe it is only a convenient "theoretical" construction, but is not real.

Well, soon enough, mathematicians defined formal rules for how zero behaves. Since we are interested in arithmetic, all we have to do is *define* how to perform arithmetic with zero. So we *define* elementary axioms like:

For any natural number N, we have:
(1) N+0=N
(2) 0+N=N
(3) N*0=0
(4) 0*N=0
(5) N/0 is undefined.
(6) 0/0 is undefined
(7) etc.

These rules became so insanely useful that once understood, mathematicians stopped debating what zero "means" or whether it "really exists." Instead, zero became a totally intuitive and mundane concept. It was "obvious" to invent it. At that point, the whole numbers were created: W={0,1,2,3,4,...}.

Let's stop and reflect for a second. Some new thing (zero) is "invented" in mathematics, and at first is controversial. But then, it's mathematical behavior (what it can do, what it can't do) is rigorously defined. Not only that, it's mathematical behavior is show to be *consistent* with what was previously known. So for example, in arithmetic, we have the law of commutation for addition: for any two numbers A and B, we have A+B = B+A. So in our construction of zero, we have to have things like 0+N = N+0.

So now, we can get an idea of what it means for something to be mathematically well defined:
First, that something must have a complete set of formal rules of how it behaves mathematically. Some rules might be that certain behaviors are undefined, or are disallowed.
Second, that something must have behaviors that are proven to be mathematically consistent with earlier mathematics.
Third, sometimes (but not always) that something is proven to be unique (if uniqueness is important).

*Roughly speaking*, the above three qualities are used as a test for when something is mathematically well-defined or not. As you can imagine, the number zero became mathematically well-defined. (There are technical notions of what "well defined" actually means, but there's no need to go into that here!)

Now, one can debate all one wants about whether the new something actually "exists" or not. Whatever that means. Does the number one actually "exist" in mathematics? Or is it only theoretical? Or what?

For most mathematicians, such philosophical questions are more or less unanswerable and possibly pointless. Instead they say,"Well, it is mathematically well defined. So let's move on. Let's see what it can do!" And if it turns out to be very cool or very useful, then it enters the mathematical mainstream, and people stop questioning whether it "really exists" or not.

Let's continue our pseudo-story, but more briskly:

After whole numbers, people wanted to subtract things. For societies with money, they wanted to represent "negative" things, like if you owe someone money. So they invented a new thing: negative numbers. And they defined mathematically precise rules about how to do arithmetic on negative numbers. And the behavior of negative numbers is consistent with earlier mathematics, namely, the previous understandings of arithmetic. For example, commutivity of addition is preserved: A+B=B+A regardless of whether A or B are negative numbers, zero, or positive.

And so, negative numbers became well-defined. And debate about whether negative numbers "actually exist" faded away. They were simply too useful and too cool not to be accepted.

Next, fractions were constructed. And as you can guess, fractions (also known as rational numbers) became well defined. And they too, were so insanely useful and cool, that no one debates whether fractions "exist." Instead, we just accept and use them. Not only that, they are well-defined, so we know their behavior will not "get us into trouble" with some weird inconsistency.

This is the point where most calculus classes start: we have the rational numbers. But then we start doing more and more algebra. And we might wonder about things like square roots. There are easy cases, of course, like sqrt(4)=2. But then, what is sqrt(2)? Well, in the rational numbers, we're not allowed to take the square root of two! Darn.

But then it was realized that the square root of two could be made to be mathematically well-defined. But how? The answer to this question is actually non-trivial. (If you think it is trivial, you have probably never taken a class in analysis. Analysis is the study of limits and continuity.)

So how can we make sqrt(2) to be mathematically well-defined?? Well, at this point, I am going to make a break in our mathematical pseudo-story, because it is pointless to get too technical in a knife forum. But briefly, we can consider a sequence of rational numbers:
1
1.4
1.41
1.414
1.4142
1.41421
etc.
This sequence of rational numbers converges to a limit. Notice that every member of the sequence is a *rational number*. However, the *limit* is not a rational number. As mathematicians thought this over, they realized the following construction would make irrational numbers well defined:

Every real number is either:
(1) A rational number.
(2) An irrational number, which is the *limit* of some convergent sequence of rational numbers.

Without going into the details, mathematicians found that with additional work, this idea could be made mathematically precise: The behavior of irrational numbers was precisely defined *and* the behavior or irrational numbers was consistent with rational numbers.

Now you can debate if sqrt(2) actually exists or not. Is it just an "approximation"? Or is it just "theoretical"? If you ask any conventional and modern mathematician he believes sqrt(2) exists and is exact.

Suppose you believe sqrt(2) is "just an approximation." Then what? Find out in the next section.

But before that, let me just point out that after real numbers became well-defined, mathematicians invented imaginary numbers. Do imaginary numbers "really exist"? Doesn't matter: mathematicians have shown imaginary numbers are well defined. And imaginary numbers are so insanely useful and cool, we just take them for granted. In fact, imaginary numbers are very mundane in physics; they feel completely natural in Fourier Analysis, harmonic motion, quantum mechanics, the Special Theory of Relativity, etc. So as far as physicists are concerned, they are super-duper useful for describing the real world. So the philosophical debate is more or less pointless.

 

[Lagrangian]

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(C) Reduction to absurdity (reducto ad absurdum)
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For the sake of argument, let's say we believe that sqrt(2) is "just an approximation." What does this mean? Well, I can imagine two possibilities:
(1) There exists a *more accurate* number than sqrt(2).
(2) Maybe our notion of sqrt(2) is actually some "fuzzy idea." Maybe by "approximation" we mean an idea which is only approximately defined, or maybe only partially defined. In this case, we can only talk about "sqrt(2)" in quotation marks, because we don't even really know what we *mean* by it.

Well, let's consider possibility (1). The real number sqrt(2) is defined to be a "number" such that sqrt(2)^2=2. Suppose there was another number S such that S^2=2, and in fact, S is the *more accurate* solution to this equation. Well, we can represent S by its difference from sqrt(2). Namely S = sqrt(2)+d for some number d which is not zero. (By assumption, S is different from sqrt(2), otherwise this argument is pointless.) Well, what happens if we plug S into S^2=2?

(sqrt(2)+d)^2 = sqrt(2)^2 + 2*sqrt(2)d + d^2 = 2
2*sqrt(2)d + d^=0 (Because sqrt(2)^2=2 by definition.)
(2*sqrt(2)+1)*d = 0

Oops! This can only be true of d=0. But if d is zero, then S=sqrt(2). What we have actually done is show that sqrt(2) is *unique*. That is, there can only be a single (positive) number that is the square root of 2. To assume there exists some other number that is a "more accurate" (positive) solution to S^2=2 is absurd (leads to a mathematical contradiction).

(Of course, in the above, we also have (-sqrt(2))^2=2. But let me ignore the negative solution for the sake of brevity.)


Hmm. Well, we just considered the possibility that there exists a *more accurate* number that sqrt(2), but that didn't go so well. So let's consider possibility (2), namely that sqrt(2) is a "fuzzy idea" and only "approximately" or "partially" defined. What would this mean? Well, it would mean that there exist mathematical equations or questions that either don't make sense, are undefined, or don't have a well-defined answer. Something like that. But mathematicians have proven that sqrt(2) is *well defined*. That means that any arithmetic operation we wish to perform, can be done with sqrt(2). That means sqrt(2) has a complete set of behaviors like any rational number. So in fact (without going into the details), sqrt(2) can be treated as "just another number", and we will not get into any trouble.

There is no mathematician who says sqrt(2) is an approximation. However, it *is* the result of a limit. And that limit is *well defined* and also *unique*. So sqrt(2) is not only a limit, it is also well-defined, exact, and unique.

Similarly, no mathematician says that pi is an approximation. Or that the mathematical constant e is an approximation. Instead, everyone says that pi and e are exact, well-defined, and unique. They "exist" as much as any other rational or irrational number.

 

[Lagrangian]

-----------------------------------------------------
(D) Example of 0.9999999.... = 1
-----------------------------------------------------

We can consider the following sequence:
0
0.9
0.99
0.999
0.9999
0.99999
etc.

This sequence has a well-defined limit. What is that limit? That limit is the definition of the number 0.99999999... (infinitely repeating 9's).

It can be proven that 0.99999...=1. How is this possible?

Well, there are many proofs of this at different levels of rigor.
But for brevity, let's consider proof by contradiction.

Suppose 0.99999... is not equal to 1.
Then we can subtract them and get a non-zero number

1 - 0.99999... = d where d is not zero.

Re-arranging this equation, we get:

1-d = 0.99999...

For the sake of brevity, let's assume d>0. (The case where d<0 is proven exactly the same way.)

But now we have a problem. We can consider the following sequence of numbers:
1-0 = 1
1-0.9 = 0.1
1-0.99 = 0.01
1-0.999 = 0.001
1-0.9999 = 0.0001
etc.

At *some* point in the sequence, 1-0.9999...9 = 0.000...01 we will have enough zeros, that 0.000...01 is smaller than d. For example, suppose d in decimal form is 0.00001834. So d has five leading zeros. So if we look at 1-0.9999999 which has seven nines, we find that we get 0.0000001. That means that the distance from 0.9999999 to the number 1 is *smaller* than d. This is a contradiction! Because we assumed that d *is* the distance from 0.99999... to the number 1. But our sequence of numbers gets closer to 1 than d.

In other words, there is *no gap* between 0.99999... and the number 1. Therefore, 0.9999...=1.

If you don't buy this "proof" I don't blame you. First of all, I did not explain it very well. Second of all, I didn't consider the case where d<0 (but that is equally easy to prove).

If you like, you can read half a dozen mathematical proofs that 0.9999...=1 here:
http://en.wikipedia.org/wiki/0.999...

The result is a fact that is not obvious to many: Namely, that in our base-10 representation (or any base), not every real number has a unique representation. For example, both 1 and 0.9999... represent the same number.

Just to summarize:
Limits are well defined and exact (when they exist). Derivatives are limits, and so are well-defined and exact. So when we say f'(x)=2x is the derivative of f(x)=x^2, we are stating a mathematical fact that is well-defined and exact. There is no "approximation" in this statement nor any "fuzziness" of idea.

( Side note: Prof. N. J. Wildberger has a non-conventional take on calculus. He actually thinks that calculus has some "fuzziness" difficulties. However, he is in the minority of mathematicians and does not represent the conventional understanding of calculus. I do like Prof. Wildberger, and I think his points have some merit. If you want to learn about his perspective, you can watch some of his YouTube videos here:
https://www.youtube.com/watch?v=fCZ8jJCVinU
https://www.youtube.com/watch?v=fXdFGbuAoF0 )

 

[Lagrangian]

-----------------------------------------------------
(E) Why do we care about the apex angle of a convex edge?
-----------------------------------------------------

Okay, this section is about "engineering" and "practical science." What do we do as *technical* scientists and engineers? More or less, we form *mathematical models* about how the real-world behaves. We then test those mathematical models by doing experiments. Often, we find our models are flawed. But, then, that is interesting! We go study the problem in greater detail, and form an improved mathematical model. If the improved model works better, then we have made scientific progress, and our understanding has increased. (Okay, this is an over-simplified description of how science actually works. But let's just accept it for the sake of discussion. Otherwise, we will go down the infinite rabbit hole that is philosophy of science.)

Let's look at an example. In the ideal gas law, we say:

PV = nkT

where P is pressure, V is volume, n is the number of molecules, k is Boltzmann's constant, and T is temperature (in Kelvin).

The ideal gas law is awesome! Super insanely useful for gases. Well, *most* of the time it is. It is *clearly* an approximation. Just consider what happens if you compress the gas so much that the molecules start to touch each other. At that point, the gas becomes as incompressible as a solid! Clearly the ideal gas law is flawed. Another example: What happens if T gets so cold that the gas condenses into a liquid, then what?! Also an obvious flaw.

Therefore, PV = nkT is *wrong*.

But, not all hope is lost of course. The ideal gas law is an extremely *useful* approximation. In most cases, it is *highly accurate*. And in those cases where it is wrong, we need to learn and study about *additional effects* that are occurring. (Like molecules touching each other, or condensation into a liquid.) Those *additional effects* are things we learn about and study.

Just because the ideal gas law is *wrong*, that doesn't mean we don't use it to study the world. Sometimes we still use a modified version of the ideal gas law when studying highly compressed gasses. That is, gasses where they behave almost like the ideal gas law, but with a small correction added in because the molecules spend a significant amount of time touching each other. Depending on the situation, we might have something like

PV = nkT + (correction factor due to non-ideal gas).


Okay fine, we have a simple mathematical model (ideal gas law) that is approximately correct *and* the errors in the ideal gas law are *very interesting* to study.

I would argue that this is *exactly* the same case when we talk about V-edges on knives. At a macroscopic scale, we can describe the knife shape as a perfect V-edge (ie: a dihedral angle). But this is only an approximation! We all know from ToddS's amazing electron micrographs that real knife edges do not form a perfect V-edge. They have an *overall* shape that is a V-edge, but then we have to consider some small corrections: The surface has very tiny scratches and bumps. The shape near the apex rounds off. Etc. In fact, the deviation from a perfect V-edge is what fascinates us. It is exactly why ToddS's micrographs are so stunning.


Okay then. Enough about V-edges. What about convex edges?

In convex edges, we have something similar. If you look at a convex knife on a macroscopic scale, you see a gentle curve on both sides. That is, the radii of curvature are macroscopic. That is, if you took the cross section of the knife and approximated it with a series of circular arcs, you would find that the circles have large (macroscopic) radii. That is, on a macroscopic scale, the radii of curvature are on the order of centimeters or millimeters. So, this means we can *mathematically model* the cross section as a pair of gentle curves that are differentiable. (For example, circular arcs aren't too bad, but if you want, you could use low-degree polynomials and/or splines, NURBS, etc.) And in this model, we have a well-defined apex angle (namely, the angle between the two tangent lines at the point of intersection).
http://en.wikipedia.org/wiki/Angle#Angles_between_curves

So let us study this mathematical model (just as we might study the ideal gas law, or ideal V-edges). Because the radii of curvature are macroscopic, we might expect that on microscopic scales that the apex looks similar to a V-edge. And indeed, if we study this idealized model, we see that this is the case. I tried to show this in the two diagrams I made.

But, we know that this idealized model is wrong. As you get near the apex, electron micrographs show scratches and bumps on the surface, and in fact, the radii of curvature start to get very small. That is, the radii of curvature near the apex shrinks to a microscopic scale (ie: on the order of 0.1 microns).

So what do we have?

Both V-edges and convex-edges have idealized geometry on the macroscopic scale. For V-edges, our macroscopic model is a perfect V angle (dihedral angle) with a definite included angle. For convex-edges, our macroscopic model is the intersection of two gently curving arcs that form an apex that has a definite included angle.

However, both models are wrong. Both V-edges and convex-edges have very complex geometry near their apex (where "near" means on the order of microns and smaller). In both cases, scratches and bumps become important, and the radii of curvature goes down to around 0.1 microns.

So why do we care?

Macroscopically, both V-edges and convex-edges have well-defined apex angles. And if we go to an intermediate length scale (say, 0.2 millimeters) which is neither macroscopic (centimeters, millimeters) nor microscopic (microns, 0.1 microns), we see that V-edges and convex edges are actually rather similar. At these intermediate length-scales, we see that a convex edge actually looks very similar to a V-edge. If I showed you an image of a knife edge which only showed, say, the 0.2mm to the apex, I think you would have a hard time saying if the knife was sharpened as a V-edge or a convex edge. And this is simply the main point I wished to make.

Could I be wrong? Sure! But now we can go look at experimental data. And if I am wrong, then the errors are *very interesting* to study! That would mean that convex edges differ greatly from V edges at intermediate length scales. For now, I would bet that they are similar. But as you can see, I have now made a hypothesis that is falsifiable, and that I think would be interesting to study.

-----------------------------------------------------

Sincerely,
--Lagrangian

 

[Pondoro2310]

-----------------------------------------------------
(A) Personal background
-----------------------------------------------------

As someone who has undergraduate and graduate degrees, completed a post-doc, worked in industry on developing CAD software, and published maybe a dozen peer-reviewed papers, I have literally met thousands of technical people. Many of which are more inventive, technically more brilliant, and more experienced than myself. Among all the technical people I've known (engineers, mathematicians, scientists, etc.) not a single one of them has said that f'(x)=2x is "an approximation" to the derivative of f(x)=x^2. (I have now found only one.)

I'm pretty comfortable with my own technical understanding, so I really see no point in debating this technical point, except that I like to teach and learn things from the general community here at bladeforums.com. But I doubt this thread holds much continued interest for anyone at bladeforums.com. So, I have little interest in continuing this technical debate. (On the other hand, I would love to talk more about how blade shape affects cutting, both theoretically (speculatively?), and experimentally.

I agree. The derivative of X^2 is exactly equal to 2X.