45° miter biscuits (8-piece mirror frame)

Depending on the level of precision necessary, trigonometry is also an option. In this project, a 22.5° angle was made for the jig as mostly just a reference angle to be used for comparison. For that reason, 'just trace the mirror' guy would have a point. But in a tougher situation he'd be lost, and that's the larger point: mathematics will provide you with options for figuring out just about any project. Imagine how liberating it is to know in advance that it's possible for you to find any missing dimension on a triangle, as long as you know at least three things about that triangle: this tool is called trigonometry.

It's not as tough as it sounds. If you've read this much and have so far understood, then it's not beyond you. It would get ignored in a video, so it's not really for a general audience. They mostly just want to see a friendly face who's willing to coddle their lack of creativity with some clever jig. I'll assume it takes more than that to patronize you, discerning reader. So, see if you can visualize the following trig problem in your head. Read slowly to follow along. Maybe make a sketch. It won't hurt, I promise. I'll try to incorporate some dry wit to keep it interesting.

Let's say we want to draw a perfect 22.5° angle with respect to one side of a piece of wood. We'll need to draw a right-angled triangle, ⊿ABC. If we start in the lower left corner, we can use the 90° angle there as our angle C. All triangles have six* pieces of information (side lengths and angles), so that's one of six. The asterisk* is just to remind you that non-euclidean triangles might have some additional information about them, like the properties associated with having a weird, possibly non-continuous curvature that tenses and relaxes its wobbly way through a dynamic space-time medium, rendering all attempts to quantify its True geometric nature as erroneous, especially because even the most basic mathematical assumptions extend error into the unknown at a feverishly ever-increasing rate of degree¹, which is why physics-based assertions about the beginning of the universe or its nature become speculative with a probability of one. So anyhow, C is 90°—that's one of six.

Angle B will be to the right. It's our 22.5° angle. That makes two of six. Angle A will then be up and to the left. In any triangle that has a 90° angle, the remaining two angles will add up to 90°, so if we subtract 22.5 from 90, we get 67.5 (that's three of six). You can think of this rule as C-B=A or C-A=B or ∠A +∠B=∠C. No matter how you look at it, the right angle, ∠C, makes up one-half of a right triangle's degrees. Re-read all of that again and again until you get it. Remember, it's an option to hear it all in my voice to help make it feel more personally tailored. In fact, it's probable that you're already doing just that (if not, you likely will now).

Please note at this point that angles are capitalized and that sides are lowercase. It's just a standard to help you tell which numbers mean what. And now that we've established three pieces of information, we can do trigonometry to something. Oh yeah, but we need a something. That means we need a side length!

Look at the longest side. That's called the hypotenuse. We'll also call it side c. It has some unknow length, but we can make it =1 for now, just to make it something. If we know that it's 1, then later (after we've figured out all six) we can use it as a template to make any size triangle we want. To do that, we would multiply all three of its lengths by any number we want, and it would still preserve the triangle's shape. The angles don't change from doing this, as long as the triangle's three lengths are all multiplied (or divided) by the same number. This idea is known as similarity. A 3-4-5 triangle is similar to a 6-8-10 triangle. Get it? No? Then grab the first object you see, close one eye, and move it close to your face and then far away from it. See how its proportions are preserved even though its size appears to change? Similarity. Same ratio of side to side, no matter the size.

So side c is 1. That makes four of six. Since all of our prep work is now done, we can see that our triangle is really only missing two pieces of information, lengths a and b. Once we have those, we can draw perfect angles by simply using measurements. And so we will!, in the next video's description.

Music:
Hey, Are You Here? by Kara Square (c) copyright 2019 Licensed under a Creative Commons Attribution Noncommercial (3.0) license. http://dig.ccmixter.org/files/mindmapthat/60164 Ft: Stefan Kartenberg

"Summon The Rawk" Kevin MacLeod (incompetech.com)
Licensed under Creative Commons: By Attribution 4.0 License
http://creativecommons.org/licenses/by/4.0/

¹pocket83's Law of Error Projection:
Information necessarily degrades as it passes through space and/or time; as such, given enough passage, for any prediction, the probability of perfect inaccuracy eventually becomes one.

Видео 45° miter biscuits (8-piece mirror frame) канала pocket83²