Quadrature of the Circle with Compass and Straightedge and a Surprising Result for the Value of π in π R^2
It is general believe and deemd to be proven that the value of π in the formula for calculating the area of a circle; i.e. π r^2, is identical to the value of π in theformula for calculating the circumference of a circle; i.e. 2π r, which is irrational.Therefore, quadrature (or squaring) of the circle with compass and straightedge (or ruler) has been deemed to be impossible. We show that this was a prejudice and proof that quadrature is possible and clearly delivers π = 3 in the formula π r2 for calculating the area of the circle. We also show a physical experiment thatunambiguously proofs this result.
18 comments
@Skyknight #198835
No, it is not based on that bible passage.
PS:
Off topic, but πr² has always struck me as weirdly disconnected from how we usually handle areas mathematically, though I never really cared enough to explore why. Recently I figured it out during one of the very rare occasions I was using that formula: expressed more conventionally, it would be d²(π/4). (d is the diameter; π/4 is the ratio between the circumference of a circle and the perimeter of a perfectly overlapping square.) Pretty sure I’m not even remotely the first person to have that epiphany, but still. It’s just an easier-to-remember-and-work-with formula which does the exact same thing, though at the cost of making geometry a bit harder for some people to grasp.
@Skyknight #198852
Just iconoclastic fantasies, as far as I can tell.
@Zinnia #198853
The circle (two-dimensional sphere) is defined as the totality of points (within a plane) at a given distance known as the radius, which is also how you draw one. As such, it makes sense to use the radiuds. And I do not see what your objection is.
@Bastethotep #198875
Area is given in square feet. We don’t measure squares, or for that matter, just about any other flat shape, from the center. If we did, the area of a square would be defined as 4r² instead of x². Which you can do, it’s just very nonstandard. One might object to the idea that squares have a radius, but they technically have two, and this does come up in unusual situations; r is the distance from the center to the middle of an outer edge, while R is the distance from the center to one of the corners.
It’s not so much an objection, but an observation: circles (and ellipses) are treated as a special exception in geometry, and the main reason why seems to be that it makes the math a little easier, as there appears to be no actual reason to otherwise.
@Zinnia #198877
@Bastethotep #198879
If you’re going to consider squares/rectangles to be “special cases” rather than the baseline for which all other shapes are compared, then are you advocating to use circles as the baseline for area measure instead? Then we’d be describing areas in terms of “circle-feet” (or circle-meters, or circle-miles, or whatever distance unit) then the area of a circle would be defined as 4r² and the area of a square would be defined as r²(16/π). Or d² and d²(4/π) respectively, either way. Which might be a bit confusing considering how ² is usually pronounced “squared” in English rather than “to the power of two”.
@Zinnia #198882 precisely because it’s a special case.
the circle in mathematics is the one with r=1 rather than D=1 or A=1, the unit circle, besides being the totality of points at 1 away from a point (and the unit disk all points 1 or less), making it of special mathematical interest, is (or mmore specifically, its [1,1] quadrant is) fundamental to trigonometry, and there are certainly many more features of interest an actual mathematician could describe.
because it is both the direct application of the concept of extending length towards higher dimensions and the easiest to construct, but all regular shapes are special cases. In real life, most things are irregular in shape. And furthermore, the irrational nature of π makes it ill-suited for defining a basic derived unit.
Couldn’t you also define circles as a shape where every point on the perimeter is the exact same distance from the point opposite to it? This definition doesn’t mention a center and would strongly suggest one should use d=1 for the unit circle rather than r=1.
I will also point out that some aspects of trigonometry are rather unintuitive to most people because the convention is to use radius and π, despite that π is associated with diameter, not radius. In the past few decades a minority of mathematicians have been advocating that things should be changed to use τ instead of π (τ=2π) because it makes it easier for beginners to see/understand exactly how trig works. While I haven’t seen anyone suggest it, you could also get the exact same results by replacing r with d.
…While I was originally making an observation, coming from a place where I rarely use circles or ellipses and found them inconsistent with the rest of the geometry math that I use, I’m now starting to wonder if I *should* be objecting to certain mathematical conventions.
@Bastethotep #198891
Yes, well, my fault for jumping all over the place reframing the math in various different ways without fully explaining any of it. It also probably doesn’t help that I didn’t demonstrate how any of it works with pictures. Let’s just leave it with that there are always alternative methods in math which do the exact same thing, meaning most conventions are arbitrary.
This conversation has resulted in me realizing just how arbitrary some of it really is, and a better idea how many (at least potentially useful, maybe even situationally better) alternatives there are. I mean, the math we use was basically cobbled together over millennia by many different people who expanded on it for different unrelated purposes, and our conventions are based on those original purposes. There’s no reason why those conventions would end up being perfectly consistent with each other so long as it all works without getting too messy.
…meanwhile - at, say… the Ryogoku Kokugikan in Tokyo - the circle squared:
image
Siggie loses by Yori-Kiri .
Ah, reminds me of the Sumo shown by Channel 4 of yore: the late great Chiyonofuji , a.k.a. ‘The Wolf’.
Confused?
So were we! You can find all of this, and more, on Fundies Say the Darndest Things!
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