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November 28th, 2016, 09:40 AM   #1
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Measuring a triangle of sorts

First time here, hope someone can give me a shove in the right direction. I enjoy math, although it does mystify me, and I am rather poor at it, as you will see. Here is my question:

Recently I was looking at triangles with two identical sides and trying to figure out a way to measure angles. The triangles happened to be certain knife blades that come pretty close to forming a triangle if viewed from the front, by which I mean having the point of the knife pointing between your eyes. These are not perfect triangles in that the area close to the edge has been slightly altered through sharpening, two new angles have been added for about the final millimeter before you get to the edge, but for my purposes I decided to ignore that area around the edge and pretend the blade formed a triangle. Just pretend. The two longer sides of the triangle, the blade's sides, form nearly level planes from the spine down to the edge, where they converge in sharpness.

Anyway, I started measuring the width of the blade's spine (the one, very short side of the triangle), then measured the height of the blade, which with a knife blade comes pretty close to the length of the two identical sides of the triangle. Then I divided the width by the height, just for fun and to see what would happen. This was always measured at a point as close to midway along the spine that I could manage, and it was all done in millimeters.

Here are three examples:
1. The chef's knife that slices best in my kitchen has a 23 cm blade length, a spine thickness of 1.8 mm and a blade height of 42 mm, thus: 1.8 ÷ 42 = 0.043, when measured from a point halfway between handle and point.
2. Another very good (but not quite as good with vegetables) knife in my kitchen with a 23 cm blade length has a spine thickness of 2.4 mm and a blade height of 44 mm, thus it weighs in at 0.055 (2.4 ÷ 44 = 0.055).
3. A fantastic old hunting knife I own, one with a 10 cm blade length, has a spine thickness of 3.7 mm and a blade height of 24 mm, which gives us a ratio of 0.154 (3.7 ÷ 24 = 0.154).

This I did with dozens of knives, some very thin, some thicker, all with this shape of blade, which I hope makes sense to those reading this: for all practical purposes a triangle when viewed from the front, with one side (the spine) very short and two much longer sides identical in length. Knife people call this a flat grind or full flat grind. If you look here, there is a drawing on the right, and it's something approaching the number 2 that I am referring to: https://en.wikipedia.org/wiki/Grind

What I "discovered" is that the number I arrive at when dividing the width of the blade by its height consistently tells me a story about the blade's thickness. Knives meant for slicing vegetables, which have to be thin, have results in the range 0.025 to 0.075. More sturdy knife blades, like those used for hunting purposes, are in the range 0.08 to 0.18. Much sturdier blades will get up as high as 0.25. Anything above that starts to look like an axe, a real wedge, and anything below 0.025 is so thin that stability questions begin to arise.

These results are consistent, and by looking at a number arrived at this way, I can pretty much tell what the knife will probably be good at. At least I immediately have a pretty accurate mental image of how thick or thin the blade is.

There are thousands of other factors that can enter in here, but I would like to simplify things just for the purpose of my question:

What on earth am I measuring here? What might this kind of ratio be called? Is it used for other things? It possibly or probably has no other practical value than telling the person writing this how thick a blade is (which will indeed tell me something about how it might perform when -- if well sharpened and with sides that are highly polished -- it tries to work its way through a raw kohlrabi or turnip).

Can anyone out there help me with this? Or tell me where I might find an answer? I'd really appreciate it.

Thanks in advance

B.
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November 28th, 2016, 11:03 AM   #2
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any way you can condense this tome to the essential knowns and unknowns?
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November 28th, 2016, 11:41 AM   #3
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Thanks for replying, and sorry for the length. My area of expertise lies elsewhere, and if we ever meet there, I promise I will do what I can to help.

We have a triangle with two identical sides which are considerably longer (20-25x longer) than the third side. I measure the short side, then one of the longer sides, then divide the short side by the long side.

The results I get that way, now that I have measured a lot of triangles and know what to look for, consistently tell me something about the distance between the two longer sides. Specifically: how thick the blade is that I am measuring. I'm not referring to blade thickness in mm, but whether or not the blade is thin enough to slice vegetables comfortably. All this confuses the daylights out of me. How can anything this stupidly simple tell me anything this complex?

Am I really measuring anything here? If so, what's it called and what else, if anything, might it be used for?
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November 28th, 2016, 01:23 PM   #4
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ok so you've got a blade that is being modeled by an isosceles triangle and you want to be able to tell how sharp it is based on your measurements.

There are only two parameters controlling an isosceles triangle and you've mentioned and equivalent set, the short leg, and a long leg.

Basically the smaller the ratio of the thickness of the blade to it's width the sharper it will be... up to a point where mechanical weaknesses in the atomic structure make it unable to keep an edge.

If we had perfect materials we would want all of our blades to have essentially zero thickness, but we don't so sharpness is a trade-off of

1) expense of material, plain steel is cheaper than titanium steel alloy

2) expected use and maintenance

Chef's knives can be made very sharp even though they are regularly used because they are constantly being sharpened. Hunting knives need to be made durable and a less sharp edge is tolerable if it buys durability. Surgeon's scalpels can be made extremely sharp because they are only used for a handful of cuts.

In a pinch, the quickest way to determine the quality of a knife is the quality of its steel. Only high quality steel will take a good edge and keep it.

Last edited by skipjack; December 1st, 2016 at 05:26 AM.
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November 30th, 2016, 02:03 PM   #5
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Quote:
Originally Posted by romsek View Post
ok so you've got a blade that is being modeled by an isosceles triangle and you want to be able to tell how sharp it is based on your measurements.
Not really. There are numerous factors involved in blade performance; steel quality is most certainly of extreme importance, then blade geometry (including all of the factors surrounding blade thickness, the form of the edge and sharpening angles) and blade smoothness (drag possibly caused by its flanks), to mention just three. In order not to violate forum rules by getting too far off topic, I have been looking at the single factor of the triangle or wedge that “flat grind” blades exhibit when viewed from the front.

Thanks to ideas I picked up here, I spent some time studying triangles and have “discovered” that the calculation I attempted to summarize in my first post here (the short side of an isosceles triangle divided by one of the long sides) results in radian.

Three examples:

Knife 1: Spine 1.8mm, each side 42mm; 1.8 / 42 = 0,043
Sides: a = 42   b = 42   c = 1.8
Angle ∠ A = α = 88.772° = 88°46'20″ = 1.549 rad
Angle ∠ B = β = 88.772° = 88°46'20″ = 1.549 rad
Angle ∠ C = γ = 2.456° = 2°27'21″ = 0.043 rad

Knife 2: Spine 2.5mm, each side 22mm; 2.5 / 22 = 0.114
Sides: a = 22   b = 22   c = 2.5
Angle ∠ A = α = 86.743° = 86°44'34″ = 1.514 rad
Angle ∠ B = β = 86.743° = 86°44'34″ = 1.514 rad
Angle ∠ C = γ = 6.514° = 6°30'52″ = 0.114 rad
.
Knife 3: Spine 5mm, each side 20mm; 5 / 20 = 0.25
Seiten: a = 20   b = 20   c = 5
Angle ∠ A = α = 82.819° = 82°49'9″ = 1.445 rad
Angle ∠ B = β = 82.819° = 82°49'9″ = 1.445 rad
Angle ∠ C = γ = 14.362° = 14°21'41″ = 0.251 rad

As can be seen in all of these examples, the radian of all three C-angles is identical to the figure I arrived at by dividing the short side of the triangle with one of the long sides. If that figure is multiplied by 57.3 (1 radian), we then arrive at the total angle of the blade, or close enough to it to have a good idea of the kind of resistance its shape will create when we try to slice a turnip (assuming it is very sharp and polished).

I assume that the Greeks were doing things this way long ago, but I was completely unaware of what exactly I was calculating and no one was able to tell me anything about it. I divided this by that and arrived at numbers that seemed to be telling me a story. It is nice to know that the numbers I kept arriving at via the primitive calculation method mentioned above do indeed make sense in that they are directly related to the shape of the blade. This also allows us to calculate the blade’s total angle using very limited means.
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November 30th, 2016, 10:10 PM   #6
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Sorry, was in a hurry and screwed up:
The equation in the first line of each knife example should read "÷" rather than "/", thus 1.8÷42, 2.5÷22, 5÷20. My European keyboard is lacking an American division sign (÷) and I always have to paste and copy it.
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December 1st, 2016, 04:13 AM   #7
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Sorry, I have yet another correction:
The wider the angle of the isosceles triangle, the more rad begins to differ with relation to the simple calculation I suggested (length of the short side divided by one of the longer sides).

In the example of knife 3 above it already begins to differ slightly (0.25 vs. 0.251). A triangle with a=22.8 b=22.8 c=18.1 differs much more (0.794 vs 0.816).

It appears that this simple calculation does perfectly well with a triangle as narrow as a knife blade, but its accuracy drops off considerably with an axe.

Does anyone here have an explanation for this? And again, what on earth am I calculating here?
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December 1st, 2016, 06:02 AM   #8
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If the angle is x radians, what you calculate is 2sin(x/2) (if the height is the "slant height" measured along the knife's surface), which is less than 0.1% smaller than x if x is less than about 0.155 radians.

As the sin(x) function is a standard trigonometric function, I've moved this to the trigonometry subforum.
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December 1st, 2016, 09:00 AM   #9
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So if I want to know the angle of the blade and don't have the means to measure it directly, I can either measure all three sides and enter those figures into a triangle calculator, or I can divide the short side by a long side, then multiply it by 57.3. I find that both methods are quite accurate, but my method works only up to an angle of about 15°, then the results begin to "angle off." One day I may understand more about the trigonometric functions mentioned by skipjack. For now, this is fine because you will not run into many knives with a blade angle higher than 15°. That, or even half of that forms too much of a wedge to be of much use in a kitchen, except with meat or other very flexible items. A good vegetable slicer will be more in the range of 2.5-3°, like knife #1 in post 5 above.

For any knife nuts listening: No, this is just one of numerous things that contribute to blade performance, but it is not to be neglected.

Thanks.
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