My Math Forum Measuring a triangle of sorts

 Trigonometry Trigonometry Math Forum

 November 28th, 2016, 11:03 AM #2 Senior Member     Joined: Sep 2015 From: Southern California, USA Posts: 1,601 Thanks: 816 any way you can condense this tome to the essential knowns and unknowns?
 November 28th, 2016, 11:41 AM #3 Newbie   Joined: Nov 2016 From: Europe Posts: 6 Thanks: 0 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?
 November 28th, 2016, 01:23 PM #4 Senior Member     Joined: Sep 2015 From: Southern California, USA Posts: 1,601 Thanks: 816 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.
November 30th, 2016, 02:03 PM   #5
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Quote:
 Originally Posted by romsek 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.

 November 30th, 2016, 10:10 PM #6 Newbie   Joined: Nov 2016 From: Europe Posts: 6 Thanks: 0 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.
 December 1st, 2016, 04:13 AM #7 Newbie   Joined: Nov 2016 From: Europe Posts: 6 Thanks: 0 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?
 December 1st, 2016, 06:02 AM #8 Global Moderator   Joined: Dec 2006 Posts: 18,142 Thanks: 1417 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.
 December 1st, 2016, 09:00 AM #9 Newbie   Joined: Nov 2016 From: Europe Posts: 6 Thanks: 0 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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