hollow vs flat grinding thoughts

[Jan]

   When honing a hollow ground chisel  on a bench stone, the bevel angle is being increased if the bevel is contacting the stone at the tip and the heel of the bevel. So honing (which is working to form a flat bevel) a hollow grind causes a more blunt angle. Interestingly, there will be more metal left supporting the tip now than there was supporting the hollow grind you just honed. You have just formed a micro-bevel because the heel of the bevel is holding the angle of the stone at a greater number of degrees than that of the hollow grind. There is a nifty way to micro-bevel your chisel or plane blades if you are so inclined.

Yes exactly, Elden! 

Assuming we hollow ground a chisel with 25 degrees using grindstone with the diameter 250 mm (10"), then honing on a bench stone will create micro-bevel.

The angle of this micro-bevel depends on the thickness of the blade in the following way:

for tool thickness tt = 4 mm (0.16") the micro-bevel angle will be 25 + 2 = 27 degrees,
for tool thickness tt = 8 mm (0.31") the micro-bevel angle will be 25 + 3.8 = 28.8 degrees,
for tool thickness tt = 16 mm (0.63") the micro-bevel angle will be 25 + 6.9 = 31.9 degrees.

Jan

[Rob]

Hi Folks

Well you've certainly been "riddling this" and no mistake.

Despite the brilliantly documented and geometrically perfect diagrams I think you're all missing one salient point which I cant prove mathematically but just seems intuitively correct to me.

Its that you're not comparing apples with apples when you measure the angles in the way you're discussing.  To my thinking, the angle formed by the hollow grind method should be measured using the chord that would be formed as if it were straight ground (like Hermans theoretical line drawn with the carpenters sliding bevel tool).

Better yet think of two identical triangles where the hypotenuse is the chisel bevel.  One is hollow ground and one is straight ground.  Now overlay one triangle on top of the other.  All the angles remain exactly the same (if we're comparing apples with apples).  All the corners exactly line up.  The only difference between those two triangles is that one still has a straight hypotenuse while the other has a concave one ie the hollow ground one.  If you were to calculate the area inside those two triangles the hollow ground one would have slightly less area than the straight edged one.  And thus, it would have less molecules with which to support the edge, thus it would be weaker.  Even though this is clearly angels dancing on the head of a pin.  But in theory, the edge has less support.

I believe the primary confusing fulcrum in all of this is the notion that each angle must still measure 25 degrees at the tip at the end of the grind.  If that set of variables is true then of course the edge will have more steel because its not "legitimately" a directly comparative test against its rival.

Only when you measure the chord (ie where my theoretical triangles occupy identical footprints) does the discrepancy become clear.

[jeffs55]

That is exactly what I was screaming all the time. Thank you for saying what I wanted to say.

[Jan]

Hi Folks

Well you've certainly been "riddling this" and no mistake.

Despite the brilliantly documented and geometrically perfect diagrams I think you're all missing one salient point which I cant prove mathematically but just seems intuitively correct to me.

Its that you're not comparing apples with apples when you measure the angles in the way you're discussing.  To my thinking, the angle formed by the hollow grind method should be measured using the chord that would be formed as if it were straight ground (like Hermans theoretical line drawn with the carpenters sliding bevel tool).

Better yet think of two identical triangles where the hypotenuse is the chisel bevel.  One is hollow ground and one is straight ground.  Now overlay one triangle on top of the other.  All the angles remain exactly the same (if we're comparing apples with apples).  All the corners exactly line up.  The only difference between those two triangles is that one still has a straight hypotenuse while the other has a concave one ie the hollow ground one.  If you were not to calculate the area inside those two triangles the hollow ground one would have slightly less area than the straight edged one.  And thus, it would have less molecules with which to support the edge, thus it would be weaker.  Even though this is clearly angels dancing on the head of a pin.  But in theory, the edge has less support.

I believe the primary confusing fulcrum in all of this is the notion that each angle must still measure 25 degrees at the tip at the end of the grind.  If that set of variables is true then of course the edge will have more steel because its not "legitimately" a directly comparative test against its rival.

Only when you measure the chord (ie where my theoretical triangles occupy identical footprints) does the discrepancy become clear.

Rob, it is nice to have you here! 
Your thread "riddle me this", started last year, focused my thoughts to this interesting subject.

The most brilliant, for me, is your formulation in reply #13: ,,So if one fixes the angle at the very tip of the edge of both hollow and straight grinds and then compares them, the hollow grind will have more metal BECAUSE the chord will be at a different angle than the straight ground bevel.  If the chord of both are at 25 then the tip of the hollow will actually be less and one might argue that the edge is weaker."

I am a little bit confused by your recent post in this thread. 

For chisel properties, to my thinking, mainly the angle at the cutting edge is important, not the angle of the chord, which corresponds to the angle in the mid point of the chisel thickness.

I am really afraid, that Mr. Euclid of Alexandria did not even left here a tiny spot for your dancing angles.


Happy Easter!  Jan

[Herman Trivilino]

The only difference between those two triangles is that one still has a straight hypotenuse while the other has a concave one ie the hollow ground one.

There's a difference in the way the tools will perform, though. The one that's hollow ground forms a smaller angle at the tip than the one that's ground flat. As Jan has calculated, it's only a couple of degrees, but that's enough to notice in some circumstances, depending on the wood hardness and type of work you're doing. It can also make a difference in the durability of the edge.

On the other hand, if you compare two chisels that are ground to the same angle at the tip, the one that's hollow ground will have more steel remaining on the tool and thus may be stronger, which was the original issue.

[Ken S]

What does this Mr. Euclid know? We all know the earth is flat. Why shouldn't our chisel bevels be flat, also?

Back to my original post, why not just add three degrees to the bevel angle for a typical chisel?

Interesting responses, guys.

Stig, this topic has raised an interesting issue. A clarifying sentence or two in the next revision of the handbook would be useful. (what the ang
emaster is actually measuring)

Ken

[Ken S]

On a more serious note..... Jan, I have always associated Euclid and his geometry with ancient Greece. I realize the ancient Greeks established trading post cities throughout the Mediterranean world. As I remember, Alexandria was noted for an outstanding library which burned. I did not realize than Euclid lived there. Interesting.

For the record, I do know that the ancient Greeks realized the earth was round, as did well educated persons in Columbus' time. With this knowledge, perhaps they also favored hollow grinding?

Ken

[Rob]

The only difference between those two triangles is that one still has a straight hypotenuse while the other has a concave one ie the hollow ground one.

There's a difference in the way the tools will perform, though. The one that's hollow ground forms a smaller angle at the tip than the one that's ground flat. As Jan has calculated, it's only a couple of degrees, but that's enough to notice in some circumstances, depending on the wood hardness and type of work you're doing. It can also make a difference in the durability of the edge.

On the other hand, if you compare two chisels that are ground to the same angle at the tip, the one that's hollow ground will have more steel remaining on the tool and thus may be stronger, which was the original issue.

That's JUST IT RIGHT there though Herman :-)  This is such a gossamer thin, subtle difference.  The original issue was an attempt to resolve the assertion that "a hollow ground bevel has a weaker edge than a straight ground bevel"  There was no mention of the tip angles being identical.  The original discussion evolved on a different forum and it was simply an argument where one party asserted hollow ground was weaker and one party asserted the opposite.

It's Mr Euclid and all the other mathematical positions taken that have decided that in order to answer the question then we MUST fix the angle measurement at the tips.  In fact we MUST NOT in order to satisfy the criteria of a fair test in the context of the original question.

Now please remember everyone that this question was never phrased in context of a practical working reality.  It was in theory only, we all know that because the actual hollow ground is so small that it's real difference is negligible. 

To cycle back to the original question:  Is a hollow ground bevel weaker at the edge than a straight ground one?  To my mind (which I fully accept is entirely warped) implicit in that statement is a tacet assumption which states "test must be conducted where conditions are equal for both bevels undergoing the test to remove spurious variables from the conclusion"

In my mind the only test that meets the ciriteria "conditions are identical" is the one where the two triangles are overlapping and identical ie all angles are equal with 25 degrees at the bevel.  Then the grinds take place and the hollow ground triangle now has a crescent moon "nibble" out of it whereas the straight ground one has uniformly less metal across the chord but the tip is still 25 degrees. The hollow ground bevel has a tip that is n degrees less then 25 owing to the metal removed by the arc of the grinder's stone.  n is clearly a very small number indeed but it is still measurable and therefore the ONLY conclusion that can be drawn for the purposes of answering the original question is that in a test where conditions are identical a hollow ground edge will be weaker attributable to the marginally lower mass of metal supporting its edge when compared to a straight grind.

The very MOMENT someone arbitrarily makes the decision that the test has to be conducted with both FINISHED grind tip angles being 25 degrees, the essence of the original question is lost because you're now conducting a different experiment.

Now, the one practical reality it does throw up which Ken alluded to is that the angle master is in essence faulty because the resultant angle you actually grind when you dial in 25 degrees is actually different when you're done.  Does that matter when you're planing?  Not sure, it doesn't to me but then I'm not doing battle with curly maple etc.

But I think for the purposes of answering the original question it's important we draw the distinction between the test conditions because having re-read all this, it seems to me we've allowed the allure and seductive nature of a binary proposition to sucker us into thinking the tip angles must have perfect mathematical symmetry at the end of the process.  How simple life might be if such things were always possible :-)

Hark.....do I hear dancing angels.......

[Jan]

The only difference between those two triangles is that one still has a straight hypotenuse while the other has a concave one ie the hollow ground one.

There's a difference in the way the tools will perform, though. The one that's hollow ground forms a smaller angle at the tip than the one that's ground flat. As Jan has calculated, it's only a couple of degrees, but that's enough to notice in some circumstances, depending on the wood hardness and type of work you're doing. It can also make a difference in the durability of the edge.

On the other hand, if you compare two chisels that are ground to the same angle at the tip, the one that's hollow ground will have more steel remaining on the tool and thus may be stronger, which was the original issue.

That's JUST IT RIGHT there though Herman :-)  This is such a gossamer thin, subtle difference.  The original issue was an attempt to resolve the assertion that "a hollow ground bevel has a weaker edge than a straight ground bevel"  There was no mention of the tip angles being identical.  The original discussion evolved on a different forum and it was simply an argument where one party asserted hollow ground was weaker and one party asserted the opposite.

It's Mr Euclid and all the other mathematical positions taken that have decided that in order to answer the question then we MUST fix the angle measurement at the tips.  In fact we MUST NOT in order to satisfy the criteria of a fair test in the context of the original question.

Now please remember everyone that this question was never phrased in context of a practical working reality.  It was in theory only, we all know that because the actual hollow ground is so small that it's real difference is negligible. 

To cycle back to the original question:  Is a hollow ground bevel weaker at the edge than a straight ground one?  To my mind (which I fully accept is entirely warped) implicit in that statement is a tacet assumption which states "test must be conducted where conditions are equal for both bevels undergoing the test to remove spurious variables from the conclusion"

In my mind the only test that meets the ciriteria "conditions are identical" is the one where the two triangles are overlapping and identical ie all angles are equal with 25 degrees at the bevel.  Then the grinds take place and the hollow ground triangle now has a crescent moon "nibble" out of it whereas the straight ground one has uniformly less metal across the chord but the tip is still 25 degrees. The hollow ground bevel has a tip that is n degrees less then 25 owing to the metal removed by the arc of the grinder's stone.  n is clearly a very small number indeed but it is still measurable and therefore the ONLY conclusion that can be drawn for the purposes of answering the original question is that in a test where conditions are identical a hollow ground edge will be weaker attributable to the marginally lower mass of metal supporting its edge when compared to a straight grind.

The very MOMENT someone arbitrarily makes the decision that the test has to be conducted with both FINISHED grind tip angles being 25 degrees, the essence of the original question is lost because you're now conducting a different experiment.

Now, the one practical reality it does throw up which Ken alluded to is that the angle master is in essence faulty because the resultant angle you actually grind when you dial in 25 degrees is actually different when you're done.  Does that matter when you're planing?  Not sure, it doesn't to me but then I'm not doing battle with curly maple etc.

But I think for the purposes of answering the original question it's important we draw the distinction between the test conditions because having re-read all this, it seems to me we've allowed the allure and seductive nature of a binary proposition to sucker us into thinking the tip angles must have perfect mathematical symmetry at the end of the process.  How simple life might be if such things were always possible :-)

Hark.....do I hear dancing angels.......

Rob, your post reassured me that you understand the subject to the finest details. Without any sophisticated drawings and calculations. Hut off, you have perfect imagination! 

All the disputation with you here is about the criteria of a fair test in context of your original question. I am afraid, that your posts may sow seeds of distrust, whether Herman's explanations and my drawings presented here are correct or not. I do not believe it is your intention.

Details are important because "God is in the detail". But also "the devil is in the detail". So, if you hear dancing angels, be careful what kind of angels it is. Good angels function mainly as God's messengers. 

Jan

[Jan]

On a more serious note..... Jan, I have always associated Euclid and his geometry with ancient Greece. I realize the ancient Greeks established trading post cities throughout the Mediterranean world. As I remember, Alexandria was noted for an outstanding library which burned. I did not realize than Euclid lived there. Interesting.

For the record, I do know that the ancient Greeks realized the earth was round, as did well educated persons in Columbus' time. With this knowledge, perhaps they also favored hollow grinding?

Ken

Euclid lived mainly in Alexandria in Egypt. Alexandria was the intellectual and cultural center of the ancient world. The Library of Alexandria had some 700 000 scrolls and books. Euclid is called "Father of geometry". "Our" geometry was fully described in his Elements.
He was surely aware that the Earth is spherical.
I suppose, it would be a pleasure for him to follow our hollow grinding discussion. 
Jan

[Herman Trivilino]

The original issue was an attempt to resolve the assertion that "a hollow ground bevel has a weaker edge than a straight ground bevel"  There was no mention of the tip angles being identical. 

If the angles aren't the same then the comparison makes no sense.

A 10 degree bevel will be weaker than a 45 degree bevel, regardless of whether they're ground flat or hollow ground.

If you want to make a meaningful comparison it has to be done for equal angles. If the angles aren't equal the comparison is meaningless.

[Rob]

I absolutely 100% appreciate the position you've adopted with the statement that the comparison is meaningless unless you fix the bevel tip angles....but.....it's wrong :-) (And you know I love you so don't take that the wrong way).

Actually, that's a bit strong, it's not wrong at all.  It's the inevitable assumption that must be made if mathematical symmetry is to be maintained. And it certainly has a lot going for it doesn't it.  It "feels" right.  To any normal rational mind it is such a seductive assumption to make because you get a nice clean binary solution where the answer is yes or no.  Clean, neat, lovely.  But also wrong unfortunately, at least in the context of the original question and of course in my humble opinion.

If I give you some background maybe you can all click into my wavelength on this, or just tell me I'm an idiot which is infinitely more likely :-)

I was having a debate on another forum ages ago with another chap about this very subject because someone had mentioned it was "common knowledge" that a hollow ground bevel has a weaker edge than a straight ground one.  We then followed pretty much identical lines to what's happened here but it never resolved and normal life resumed.  It's been taken much much further here with the grateful addition of elegantly drawn technical specifications which prove beyond doubt that in the event the angles at the tip remain the same then there is more metal behind the edge of the hollow ground bevel.  Lets be clear, nobody is refuting that conclusion to that set of variables.

But we appear to have gone so far down the channel of the assumption that the tip angles must be fixed that we've either forgotten or ignored the original question.  In order to answer this correctly, it is my view that we need to re-evaluate the original question because we've missed the point.

Herman, you argue that a none fixed angle question is meaningless.  I disagree, it has meaning in the role of attempting to dispel the urban myth about weakness of grind one way or another.  Given that was the original basis for discussion, I think that we should value that meaning appropriately.

So, lets re-state that rather un-technical, none mathematical, almost entirely free from specification and dimension question:  Is a hollow ground bevel weaker at the edge than a straight ground one?  Can someone point out where in that question we are instructed to fix the tips of the bevel angles please?  Maybe I'm a bit bonkers but I cant see it myself.  To my way of thinking, in order to satisfy the question ie value it's meaning appropriately we must take two theoretical chisels and simply apply the two different grinding techniques and then measure the results.  To be fair to the original question, there is no way we can mess with those chisels in any way.  It's almost like they need to be just fixed in space and the grinding media brought to them with no movement of the chisels at all.  Only then are the test conditions met only then is the test a fair comparison.  Now does anybody dispute that those conditions are reasonable and fair?  So when we do that, inevitably, of course, perfect sense....the chords are what remain the same because the chisels haven't moved at all.  The chords have remained identical at the start of the grind but at the end of the grind, one has slightly less metal all the way down the bevel (straight ground) whereas one has a crescent moon ground away, which undermines the tip and "bingo" weakens the cutting edge.

So now we have two competing methods for testing the theory.  Model one, the above where the chords are fixed, the hollow grind loses.  Model two, the tip angles are fixed, hollow grind wins.

I think what's been troubling me all along is the insistence on model two being the right model for the problem. That doesn't make sense to me.  If we're to satisfy the original question context then what makes sense to me is a test where the chisels are ground in conditions where everything about them is fixed.  In model two we actually rotate the chisel about it's tip when we put it on the hollow grinder to satisfy mathematical symmetry and yet we don't do that for the straight grinder!!!!! When did changing the variables to satisfy a desired outcome become proper science???  On the contrary, we fix all the variables and then change ONLY ONE and measure the results to determine the effect of that variable.  The only variable under test here is the method of grinding, straight or hollow. What model two does is introduce a variable called rotating the tip angle....but only on the hollow grinder!!!  Sorry chaps, that's not science, that's not objective, that's subjective, that's the operator frigging with the test to predetermine the outcome and then justifying it with the word "sense" after the fact!  No no, for the test to be valid, the only variable that can be allowed is the method of grinding, all other things must be fixed (at any angle incidentally....the results would always be the same).

So, I have now exorcised the demon.  I understand both points of view entirely but I believe the fixed chord model is a more appropriate fit to the original question than the fixed angle model.  The fixed chord model is the only truly objective test and ironically (you're going to love this) Mr Euclid would HAVE to agree because being an empiricist, he would have no other rational choice :-)

[Herman Trivilino]

To be fair to the original question, there is no way we can mess with those chisels in any way.  It's almost like they need to be just fixed in space and the grinding media brought to them with no movement of the chisels at all.  Only then are the test conditions met only then is the test a fair comparison.  Now does anybody dispute that those conditions are reasonable and fair?  So when we do that, inevitably, of course, perfect sense....the chords are what remain the same because the chisels haven't moved at all. 

I dispute that. To be fair to the original question it makes more sense to me that, instead of the angle of the chords being the same, the angle of the tangent lines at the tips are the same. And the reason is simple, the Angle Master jig is the way we measure the angle, and that measures the latter not the former.

[Rob]

To be fair to the original question, there is no way we can mess with those chisels in any way.  It's almost like they need to be just fixed in space and the grinding media brought to them with no movement of the chisels at all.  Only then are the test conditions met only then is the test a fair comparison.  Now does anybody dispute that those conditions are reasonable and fair?  So when we do that, inevitably, of course, perfect sense....the chords are what remain the same because the chisels haven't moved at all. 

I dispute that. To be fair to the original question it makes more sense to me that, instead of the angle of the chords being the same, the angle of the tangent lines at the tips are the same. And the reason is simple, the Angle Master jig is the way we measure the angle, and that measures the latter not the former.

Herman you're completely missing the point.  This question was never specific to the Tormek universe. In fact, as I mentioned, it came up on a completely different forum and was an attempt to make some sense out of the urban myth.  The anglemaster is a red herring, it has nothing whatever to do with the wider assertion that's been out there for donkeys years.

If you add in the caveat of how would you use it in a real world, practical sense then I would entirely agree with you and will myself in future use the exact method discussed here ie have the bevel tip ground to 25 degrees, leaving the chord at a blunter angle.  But for the purposes of answering the original question, I maintain that for the test to be reasonable no other conditions can be varied other than the grinding media.  Only then is the test genuinely objective.

[SharpenADullWitt]

For as small the difference you all are talking, I think the real world factor would be the heat (during sharpening) that affected the blades temper at its edge and causing it not to hold the edge as long.