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Simple math question?
A regular 4-gon has a perimeter of 100 meters?
A similar "physical" 4-gon has a perimeter 97.0882063242942 m
The formula for the physical 4-gon is?
Each "conner" has a different radius of
curvature based on a mathematical constant.
ref:
Fun if you like a simple mystery.
In fairness:
Both 4-gons are identical except that
the physical-4-gon has four θ=π/2 arc
segments instead of four vertices, and
each arc has a different radius of curvature.
The constant is well known. The trick is: What
are the four different radii? Thanks for answering Shythian.
Ok, the drawing is to scale. It takes
about 9 radii of the larger circle to
complete the side. Ergo, find the constant
(not π). The other 3 radii are constant^(1/n)
where n is an integer. I'm surprised, you came so close with your equation.
--------------------------------------
Yes. Your number is correct.
I get: 6.78417697
------------------------------------
@Shythian
Your yanking my chain here!
You've had bigger problems than
this fall out of your nose! ;)
Brief comment regarding the other question:"Conner": Mathematically, a point where any two straight lines (example: n-gon) meet has no size except in the context of reality (physics). Hence, it is not a physical entity, but mathematical place where one line ends and another begins. Consequently, it has no effect on perimeter calculation. The probability of perimeter measurement = perimeter calculation is basically zero. I believe Schythian's comments regarding "infinite radius of curvature" and the like are correct. Ergo, my trivial question regarding "physical n-gon".
Common sense dictates that just about any physical entity has an extremely complicated geometry that is changing in time (4D) as we speak. Hence trying to define said geometry mathematically is like trying to catch a leaf in a hurricane, but we try using averages and
the like.
2 Answers
- falzoonLv 71 decade agoFavorite Answer
r1 + r2 + r3 + r4 = 2(100 - 97.0882063242942)/(4 - π) = 6.78417697118436
My guess is that the well-known constant is e = 2.71828182845904...
and my other guesses are that r1 = e, r2 = e^(1/2), r3 = e^(1/4), r4 = e^(1/8),
so that r1 + r2 + r3 + r4 = 6.78417696891374,
which is accurate to 8 decimal places. Hope that's close enough.
- Scythian1950Lv 71 decade ago
What comes close is x^18 + y^18 = (2/25)^2, which has a perimeter of 97.00370153382046. This figure has a maximum diameter of 25 meters, so that it fits inside a square with a perimeter of 100 meters.
Edit: I understand your problem now, I'll see what I can do later today.
Edit 2: This is the same as asking what 4 numbers, "based on a well known constant", adds up to 6.784176971184346. No luck so far.
Edit 3: The whole problem, like falzoon has pointed out, is that â(n=0 to 3) e^(1/2^n) = 6.784176968913741, which doesn't quite match 6.784176971184346. Plus, when I use your diagram and measure the radii, nothing fits right. So I just gave up.
I even ran the number 6.784176971184346 through the Inverse Symbolic Calculator, it can't find anything either.
Give the 10 points to falzoon for having posted the closest answer first.
