Traditional knife homework - Blade length to handle length ratio.

[jc57]

How do you measure the height of a tree you have to fell and which leans toward a house ? As you know trees always lean in the wrong direction.

This is a fairly straightforward calculation. You just need to assume a spherical tree in a perfect vacuum at sea level at the equator.

Then, the answer is, the tree just rolls into the sea.

 

[dennishy]

The one time digital calipers would've been helpful and its out of batteries. the robot uprising is happening before our eyes.

 

[Just Tom.]

This is a fairly straightforward calculation. You just need to assume a spherical tree in a perfect vacuum at sea level at the equator.

Then, the answer is, the tree just rolls into the sea.

You need to go sit in the corner facing the wall and think about how your behavior is affecting those who are here and WANT to learn

 

[5K Qs]

Math! What fun.

Don't you hate it when you hit the wrong emojicon and don't realize it? Glad you're enjoying the thread, Jer!

Gonna need some calculus for that.

But you want the right tool for the job. You wouldn't want to remove stems from fresh strawberries with a huge Bowie knife, even though you could; a smaller knife would be a much better choice for that job. Using calculus for Tom's homework is for losers.
Besides, differential calculus is so derivative. (Although folks who get excited about integral bolsters probably can't contain themselves when thinking of integral calculus.)

Calculus, maybe - or a piece of string

Piece of string, a ruler, and a calculator seem like the ideal technical tools for this job.

It's like he just gave his kids socks for Christmas.

Doing some thinking in a situation in which there's actual right and wrong instead of just expressing an opinion or claiming "But all truth is relative, and THIS is MY truth" is not valued very highly in our society.

Guys, it’s just 2 easy measurements, and simple division. It’s GOT to be a whole lot easier than remembering some obscure pattern number from a 1910 cutlery catalog or something…

Good point! Doing Tom's assignment is FAR easier for me than taking and posting knife photos!

I was expecting some calculus!

Speaking of expectations, I thought you'd be far more involved in the substance of this thread, Jack!
IIRC, you often write about how too little blade for the size of the handle can be a deal breaker for you. Maybe that's based mostly on your subjective perceptions rather than being data-driven, but wouldn't your argument be even more convincing if it included a quantitative component exactly like Tom's intent?

The golden whatchamacallit from the DaVinci Code?

I think the golden ratio is about 1.618, and its reciprocal, which is more relevant here, is 1/1.618 = .618.
Gotta ask again: Coincidence, ... or Grand Design???

Because the tang doesn't have an azimuth.

Or maybe because a tang has a choil (if we want to start a war of weird words)?

Is an azimuth like an annulus?

I've almost got it. I am constructing a Foucault pendulum at the moment. I have my slide rule out and properly oiled, just need to remember where I stored my sextant. Anyone have an astrolabe I can borrow?

I admit there are some unusual technical terms that can be bandied about, but I'll bet that's true in almost any area. (BTW, Vince, I just tell students an annulus is just a washer like you'd use with a bolt and nut, or its a cylinder with a hole bored through the centers of its circular bases.) But it's fun to start a sentence, "Kiss my ___" and fill in the blank with your favorite from azimuth, annulus, pendulum, astrolabe, clinometer, sextant, slide rule, tangent, choil, bolster, tang, liners, etc. and see the reaction of the "general public".

Still working on it....

View attachment 2045839

Remember when the Rolling Stones sang,
When I'm driving in my car
When a man come on the radio
He's telling me more and more
About some useless information
Supposed to fire my imagination
I can't get no, oh, no, no, no, hey, hey, hey
That's what I say
I can't get no satisfaction

They were obviously NOT singing about John, who has compiled a bunch of fascinating info in his post. Most of it is not relevant to Tom's homework from a content perspective, but almost everything John included has a fascinating history, whether it's Archimedes' methods for approximating the value of π, or for developing the formulas for volumes of cylinder and cone centuries before the "official" development of algebraic notation and calculus, or the story of the first discovery/invention (lots of philosophical fights over THAT distinction) of the "quadratic formula" by the Islamic mathematician Al-Khwarizmi (from whose name the word "algorithm" comes - another thing you could challenge someone to kiss) who did it all geometrically and actually had several "quadratic formulas", depending on the form of the quadratic equation he was trying to solve (and he published it all in a book whose title include the words "al jabr" from which our term algebra was formed when translated into Latin).

Eventually you would independently produce the same results that Benoit Mandelbrot came up with when thinking about the length of the coastline of Britain.

When I read the original post, I'd have never guessed that Mandelbrot's fractal geometry would appear!!


- GT

 

[5K Qs]

I tried this out of curiosity, inspired by the questions posed about true cutting edge length on blades with more 'belly' curvature. I'd never given it much thought until reading this thread.
...

I just tried this method with a Buck 110LT folder with its upswept tip on a clip blade. I measured 3.375" cutting edge length accounting for the belly, as compared to a straight-line measurement of 3.250" as measured directly from the plunge to the tip. Handle length is 4.875" (4-7/8"). So, accounting for true cutting edge length as I measured it, I'd have this result below, for the ratio of cutting edge length relative to handle length:

3.375" / 4.875" = 0.692

Or, if just relying on the straight-line measurement from the heel of the cutting edge to the tip, as per the OP, I'd get:

3.250" / 4.875" = 0.667 ( Note: I just noticed I typo'd the edge length incorrectly here at 3.125" - edited to 3.25" & recalculated, to fix that. )
...

Another example I looked at - an A.G. Russell 'Cowboy' model in black Rucarta. ...
Accounting for the belly:
Cutting edge length = 3-9/16" (~91mm)
Handle length = 4-5/16" (~110 mm)
Ratio, cutting edge length relative to handle = 0.827

Disregarding the belly:
Blade length, edged portion = 3-5/16" (~86mm)
Handle length = 4-5/16" (~110 mm)
Ratio, edged blade relative to handle = 0.781

Ok, since you HAD to go measuring the belly, here are my measurements of the length of the cutting edge of a knife with a lot of belly using 3 methods:

1. Using a ruler, a right angle, and trigonometry
2. Using a ruler and a piece of string
3. Using your cardboard method

I am somewhat surprised that the results of the 3 methods were practically identical, considering the uncertainty in the measurements made in dim light with a wooden ruler and my 1.25 magnification Easy-Readers.

...
I think my conclusion is that using a piece of string is good enough and a whole lot quicker…
...

Tom, that trig. method is pretty clever, and something I don't think I'd have thought of. It assumes that the belly of the knife is actually an arc of a circle, which seems kind of questionable to me; you can imagine drawing all kinds of curves that pass through the end points of the "chord" in your diagram. But I'm also surprised how close the results of various approaches are, and I'll bet they're almost always identical if we just expressed our ratios to the nearest tenth (i.e. the nearest 10 percent). I doubt if I could tell the difference in "real life" between a blade whose edge was 69% of the handle length and one whose edge was 67% of the handle length, as in David's first example of 2 methods of approximation.

Nevertheless, as all this can't fall lower, and as we have some mathematician aboard the train, i have a practical math related question. Here's.

How do you measure the height of a tree you have to fell and which leans toward a house ? As you know trees always lean in the wrong direction.

This, without endangering your life by climbing up the tree. Without jeopardizing your sanity by using an app on your phone. Without risking a calculation error knowing that there are only 3 categories of people, those who know how to count and those who don't.

Math teachers are allowed to answer though i don't always rely on their practical sense.

Dan.

You got it.
Measure with a rope the distance between the eye and your hand palm, arm stretched out in front of you.
Report the measure on a wood stick.
Cut the stick at the right length.
Hold the stick at its middle, arm stretched out, walk forward or backward until you aim with the butts both the top and the base of the tree.
The distance between the tree and you is equal to the height of the tree.
It's an application of the Thales theorem (proportionality of the triangles). Simple and reliable.

Dan.

Dan, I solved your problem on the back of an envelope this morning at breakfast, using proportionality of triangles and a stick of known height placed on the ground a known distance from the tree. But you'd be right not to rely on my "practical sense" since my method requires the ground between stick and tree to be perfectly level, and I ignore how far up the trunk the tree will be cut.
(I'm curious, does your (and Wild Willie's) method require that the tree be vertical? I tried to figure it out for a leaning tree, and that seemed to require another set of similar triangles.)

You need to go sit in the corner facing the wall and think about how your behavior is affecting those who are here and WANT to learn

Nice parody!
Reminds me of a Yogi Berra mythical quote that I often use: If the people don't want to come out to the ball park, how you gonna stop 'em?

- GT

 

[screened porch]

 

[Just Tom.]

Tom, that trig. method is pretty clever, and something I don't think I'd have thought of. It assumes that the belly of the knife is actually an arc of a circle, which seems kind of questionable to me; you can imagine drawing all kinds of curves that pass through the end points of the "chord" in your diagram.

EXTREMELY questionable, but given that the greatest precision I could achieve with my wooden ruler was 1/16 inch, and knowing from experience how small the difference would be if I were to calculate it as a series of curves with different radii, or as a spiral, and given the fact that I had to eyeball the point of curvature anyway, I figured it was good enough. Nobody is going to wind up with too much or too little dirt or concrete if I get it wrong….

There is also a little curvature to the “straight” part - the difference between the arc and chord distances on that part would be far less than the uncertainty in my measurements to begin with - notice even on the smaller radius arc, the chord distance is 1.75 inches vs. 1.82 along the arc - 1/16 inch difference about?

All the above is to say - using a piece of string is more than good enough (if not as much fun…).

 

[mrknife]

Don't you hate it when you hit the wrong emojicon and don't realize it? Glad you're enjoying the thread, Jer!


But you want the right tool for the job. You wouldn't want to remove stems from fresh strawberries with a huge Bowie knife, even though you could; a smaller knife would be a much better choice for that job. Using calculus for Tom's homework is for losers.
Besides, differential calculus is so derivative. (Although folks who get excited about integral bolsters probably can't contain themselves when thinking of integral calculus.)


Piece of string, a ruler, and a calculator seem like the ideal technical tools for this job.


Doing some thinking in a situation in which there's actual right and wrong instead of just expressing an opinion or claiming "But all truth is relative, and THIS is MY truth" is not valued very highly in our society.


Good point! Doing Tom's assignment is FAR easier for me than taking and posting knife photos!


Speaking of expectations, I thought you'd be far more involved in the substance of this thread, Jack!
IIRC, you often write about how too little blade for the size of the handle can be a deal breaker for you. Maybe that's based mostly on your subjective perceptions rather than being data-driven, but wouldn't your argument be even more convincing if it included a quantitative component exactly like Tom's intent?


I think the golden ratio is about 1.618, and its reciprocal, which is more relevant here, is 1/1.618 = .618.
Gotta ask again: Coincidence, ... or Grand Design???


Or maybe because a tang has a choil (if we want to start a war of weird words)?



I admit there are some unusual technical terms that can be bandied about, but I'll bet that's true in almost any area. (BTW, Vince, I just tell students an annulus is just a washer like you'd use with a bolt and nut, or its a cylinder with a hole bored through the centers of its circular bases.) But it's fun to start a sentence, "Kiss my ___" and fill in the blank with your favorite from azimuth, annulus, pendulum, astrolabe, clinometer, sextant, slide rule, tangent, choil, bolster, tang, liners, etc. and see the reaction of the "general public".
View attachment 2047688


Remember when the Rolling Stones sang,
When I'm driving in my car
When a man come on the radio
He's telling me more and more
About some useless information
Supposed to fire my imagination
I can't get no, oh, no, no, no, hey, hey, hey
That's what I say
I can't get no satisfaction

They were obviously NOT singing about John, who has compiled a bunch of fascinating info in his post. Most of it is not relevant to Tom's homework from a content perspective, but almost everything John included has a fascinating history, whether it's Archimedes' methods for approximating the value of π, or for developing the formulas for volumes of cylinder and cone centuries before the "official" development of algebraic notation and calculus, or the story of the first discovery/invention (lots of philosophical fights over THAT distinction) of the "quadratic formula" by the Islamic mathematician Al-Khwarizmi (from whose name the word "algorithm" comes - another thing you could challenge someone to kiss) who did it all geometrically and actually had several "quadratic formulas", depending on the form of the quadratic equation he was trying to solve (and he published it all in a book whose title include the words "al jabr" from which our term algebra was formed when translated into Latin).


When I read the original post, I'd have never guessed that Mandelbrot's fractal geometry would appear!!


- GT

I usually just eyeball it

 

[Just Tom.]

I usually just eyeball it

Honestly, that is the correct method in this case. Remember - there is uncertainty in every measurement. The most elegant solution to this problem is still only as good as whatever uncertainty was in the numbers you plugged into it to begin with.

Of course you could reduce those uncertainties by making repeat measurements…

 

[Wild Willie]

(I'm curious, does your (and Wild Willie's) method require that the tree be vertical? I tried to figure it out for a leaning tree, and that seemed to require another set of similar triangles.)

In my experience it doesn't really matter. It's not exactly a precise measurement, but can get you within a couple of feet. Which is usually enough to know whether what's being dropped (be it a pin or entire tree) will clear what it needs to. I haven't actually sighted one with a stick in some time, after a while you get an idea of where things are at. "When in doubt, top it out" are good words to live by in situations where it'll be close.

 

[Prester John]

Okay, teacher, I'm back from the principal's office.
Here's my favourite knife:

Opinel No. 8. (I probably carry a Victorinox Spartan more often because of the Batman Utility Belt array of accoutrements on it--especially the wine and beer bottle openers--but the Opinel is a way cooler knife.). The dimensions look and feel perfect to me. Measuring, I get approximately 3 and 5/16 inches for the blade, and 4 and 5/16 inches on the handle. Converting to decimals, that is 3.3 and 4.3. Dividing the first number by the second, I get this answer:

Which I will round up to 7.7. Did I do it right?

 

[Prester John]

Reminds me of a Yogi Berra mythical quote that I often use: If the people don't want to come out to the ball park, how you gonna stop 'em?

I think my favorite Yogi quote is:"I don't even suspect it!" when he was asked "Don't you know anything?"

 

[Peregrin]

I'll play!
First off, aesthetically I like the main blade to handle ratio as close to 1 as possible. Just looks right to me. I'm referring to total straight line blade length. That's what my eye sees. Secondary blades, it doesn't really matter to me.
That said, I'll follow the rules and use the straight line blade length, cutting edge only (BC). I'll also add the total blade length (BT), maybe for extra credit...

Lately it's been one of these 4 that makes it into my pocket everyday.

Here's the calculator I'm using. Overkill? Yes! It's the old engineer in me.

Here's my simple equations:
CH (cutting edge to handle ratio) = BC (blade cutting edge, straight line length) / BH (blade handle length)
TH (total edge to handle ratio) = BT (blade total length) / BH (blade handle length)

First up, Eureka Jack.
CH = BC ÷ BH
.736 = 2.625 ÷ 3.563
TH = BT ÷ BH
. 807 = 2.875 ÷ 3.563

Next, John Lloyd Shadow Trapper.
CH = BC ÷ BH
.677 = 2.625 ÷ 3.875
TH = BT ÷ BH
. 758 = 2.938 ÷ 3.875

Next, Boker Gent.
CH = BC ÷ BH
.600 = 2.50 ÷ 3.750
TH = BT ÷ BH
. 750 = 2.813 ÷ 3.75

Last, but not least. Fox Livri.
CH = BC ÷ BH
.714 = 2.50 ÷ 3.50
TH = BT ÷ BH
. 785 = 2.750 ÷ 3.5

OK, It's late and I'm not guaranteeing there isn't a mistake in my work. I'm my father's son. He attended the University of Pittsburgh, and graduated with a BSEE. He used to tell a story about turning in a final Calculus exam to his professor. Dad told him, Dr. C here's all I know. After looking at his exam, Dr. C responded, well you don't know enough, see you next semester...

Good night! I hope I don't have mathematical dreams!

 

[Just Tom.]

I'll play!
First off, aesthetically I like the main blade to handle ratio as close to 1 as possible. Just looks right to me. I'm referring to total straight line blade length. That's what my eye sees. Secondary blades, it doesn't really matter to me.
That said, I'll follow the rules and use the straight line blade length, cutting edge only (BC). I'll also add the total blade length (BT), maybe for extra credit...

Lately it's been one of these 4 that makes it into my pocket everyday.

Here's the calculator I'm using. Overkill? Yes! It's the old engineer in me.

Here's my simple equations:
CH (cutting edge to handle ratio) = BC (blade cutting edge, straight line length) / BH (blade handle length)
TH (total edge to handle ratio) = BT (blade total length) / BH (blade handle length)

First up, Eureka Jack.
CH = BC ÷ BH
.736 = 2.625 ÷ 3.563
TH = BT ÷ BH
. 807 = 2.875 ÷ 3.563

Next, John Lloyd Shadow Trapper.
CH = BC ÷ BH
.677 = 2.625 ÷ 3.875
TH = BT ÷ BH
. 758 = 2.938 ÷ 3.875

Next, Boker Gent.
CH = BC ÷ BH
.600 = 2.50 ÷ 3.750
TH = BT ÷ BH
. 750 = 2.813 ÷ 3.75

Last, but not least. Fox Livri.
CH = BC ÷ BH
.714 = 2.50 ÷ 3.50
TH = BT ÷ BH
. 785 = 2.750 ÷ 3.5

OK, It's late and I'm not guaranteeing there isn't a mistake in my work. I'm my father's son. He attended the University of Pittsburgh, and graduated with a BSEE. He used to tell a story about turning in a final Calculus exam to his professor. Dad told him, Dr. C here's all I know. After looking at his exam, Dr. C responded, well you don't know enough, see you next semester...

Good night! I hope I don't have mathematical dreams!

Very nice. The only thing that would have made it even better would have been an actual physical calculator - preferably an old HP

 

[dantzk8]

Dan, I solved your problem on the back of an envelope this morning at breakfast, using proportionality of triangles and a stick of known height placed on the ground a known distance from the tree. But you'd be right not to rely on my "practical sense" since my method requires the ground between stick and tree to be perfectly level, and I ignore how far up the trunk the tree will be cut.
(I'm curious, does your (and Wild Willie's) method require that the tree be vertical? I tried to figure it out for a leaning tree, and that seemed to require another set of similar triangles.)

If the tree is leaning, you can stand perpendicular to the direction in which the tree is leaning and operate in the same way by leaning the stick. Report the distance in the leaning direction.
It won't make a significant difference and it will anyway be less than the margin of error that you must consider necessary for safety.
If the tree has a heavy lean it's not safe to fell it in the leaning direction. In this case the weight of the tree is not exerted in its axis and the notch, then the cut, can weaken the tree to the point of splitting the trunk. It rarely happens but it is very violent and dangerous. The best in that case is to choose an other direction for the fall, make a deeper notch and make the hinge narrower on the leaning side.
It requires precision and precision when handling a chainsaw it's like whittling a ball in cage with boxing gloves.

Dan.

 

[Will Power]

Look, if we all wanted to be practical and use common sense, everybody would just carry an Opinel #8 and there would be nothing to talk about. I would then have to find a new, even LESS practical hobby.

That's what happens when using a chainsaw without paying attention.

Dan.

There's too much felling of trees near houses anyway, so the tree-kamma got them The birds & squirrels approve and they aint need no calculus/trig/logarithms to tell 'em the height of a tree A Beaver would never have made such an elementary error either....

 

[Will Power]

I'll play!
First off, aesthetically I like the main blade to handle ratio as close to 1 as possible. Just looks right to me. I'm referring to total straight line blade length. That's what my eye sees. Secondary blades, it doesn't really matter to me.
That said, I'll follow the rules and use the straight line blade length, cutting edge only (BC). I'll also add the total blade length (BT), maybe for extra credit...

Lately it's been one of these 4 that makes it into my pocket everyday.

Here's the calculator I'm using. Overkill? Yes! It's the old engineer in me.

Here's my simple equations:
CH (cutting edge to handle ratio) = BC (blade cutting edge, straight line length) / BH (blade handle length)
TH (total edge to handle ratio) = BT (blade total length) / BH (blade handle length)

First up, Eureka Jack.
CH = BC ÷ BH
.736 = 2.625 ÷ 3.563
TH = BT ÷ BH
. 807 = 2.875 ÷ 3.563

Next, John Lloyd Shadow Trapper.
CH = BC ÷ BH
.677 = 2.625 ÷ 3.875
TH = BT ÷ BH
. 758 = 2.938 ÷ 3.875

Next, Boker Gent.
CH = BC ÷ BH
.600 = 2.50 ÷ 3.750
TH = BT ÷ BH
. 750 = 2.813 ÷ 3.75

Last, but not least. Fox Livri.
CH = BC ÷ BH
.714 = 2.50 ÷ 3.50
TH = BT ÷ BH
. 785 = 2.750 ÷ 3.5

OK, It's late and I'm not guaranteeing there isn't a mistake in my work. I'm my father's son. He attended the University of Pittsburgh, and graduated with a BSEE. He used to tell a story about turning in a final Calculus exam to his professor. Dad told him, Dr. C here's all I know. After looking at his exam, Dr. C responded, well you don't know enough, see you next semester...

Good night! I hope I don't have mathematical dreams!

My father was a civil-engineer, we were very close but not about maths...

Great taste in KNIVES there is what I can see, the Lloyd in particular but I'm very glad you took the plunge-at last- and got a Fox Livri I've been on about it for some time. Kind of reminds me of Michael Caine , using that Cockney voice of his before he became more Americanized... pointing " There yew are. Eye told you din't eye eh? All along bin telling yew, yew knew wot eye would do dindt'cha eh?"

 

[dantzk8]

There's too much felling of trees near houses anyway, so the tree-kamma got them The birds & squirrels approve and they aint need no calculus/trig/logarithms to tell 'em the height of a tree A Beaver would never have made such an elementary error either....

Every spring i see birds holding twigs in their beak. To measure the trees heights of course.
Squirrels are very different animals, very good climbers, they count the number of steps from the roots to the top.
Beavers are stupid, every tree they fell ends up in the water.What a waste!

Dan.

 

[Peregrin]

Very nice. The only thing that would have made it even better would have been an actual physical calculator - preferably an old HP

Thanks, Tom! At the time I couldn't afford an the HP, Reverse Polish Notation and all! I had to settle for a TI SR-50, I believe that was the model.
I was going to use this Radio Shack I picked up much later, but it needs new button batteries, which I'll have to order.

My father was a civil-engineer, we were very close but not about maths...

Great taste in KNIVES there is what I can see, the Lloyd in particular but I'm very glad you took the plunge-at last- and got a Fox Livri I've been on about it for some time. Kind of reminds me of Michael Caine , using that Cockney voice of his before he became more Americanized... pointing " There yew are. Eye told you din't eye eh? All along bin telling yew, yew knew wot eye would do dindt'cha eh?"

View attachment 2047943

Thanks, Will! Eye yew told me! Eye shoulda listened to yew! The Livri is an excellent knife.

 

[Just Tom.]

Thanks, Tom! At the time I couldn't afford an the HP, Reverse Polish Notation and all! I had to settle for a TI SR-50, I believe that was the model.
I was going to use this Radio Shack I picked up much later, but it needs new button batteries, which I'll have to order.

Thanks, Will! Eye yew told me! Eye shoulda listened to yew! The Livri is an excellent knife.

Nice!
I’m not an RPN fan myself either, actually, though many in my industry are. Most of the formulas I need to implement are written out algebraically anyway:

Surprisingly, my math skills are actually very weak - I would probably fail any class 5K Qs taught. I can just figure out the stuff I need to get my job done, or implement formulas that others have come up with.