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| The sometimes 'Beauty' of mathematics??? |
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| rstofer:
I like "Gabriel's Horn" better. Consider the curve 1/x for x between 1 and infinity - it's a really long horn... Compute the volume of revolution and you will find that it takes PI units of paint to fill it completely. Then compute the surface area and you will find that it takes an infinite amount of paint to cover the interior surface. That's just weird! Hints: Volume = integral from 1 to infinity of pi * radius squared dx but r = 1/x so integral of pi/x2 dx = pi Surface Area = integral from 1 to infinity of 2 * pi * r which is the integral of 2 * pi / x dx. Remember, integral of 1/x = ln(x) and the result diverges (runs off to infinity) |
| magic:
You may run out of paint on the exterior, but the interior cannot be painted with a layer of any thickness because most of it is too narrow ;) If you want to see some seriously fucked up consequences of playing with all that "infinity" stuff, look up the Banach-Tarski paradox :wtf: (not going to pretend I know the proof of that) |
| TimFox:
Another similar paradox applied to natural objects is to compute the coastline of Norway. Obviously, the area of Norway (with mountains projected back onto the globe) is finite, since you can draw a closed curve around it, of finite area, that encloses it. If you do the actual experiment of computing the physical coastline as you go to finer and finer map scales, the value (in meters) increases without bound. (Please don't quibble about atomic dimensions. This is math.) An ideal geometric figure that does the same: the Koch snowflake, with finite area but infinite perimeter. Start with an equilateral triangle, of side 1. Total perimeter = 3. Iteration 1, on each of the three sides of that triangle erect another equilateral triangle whose side is 1/3 of the first. This gives you a Star of David hexagram. Total perimeter = 4 Iteration 2, on each line segment (12 for this step) of that figure erect another triangle of length 1/3. etc. There exists a circle that encloses all of the resulting figures, but the perimeter increases without bound. The length after N iterations is 3 x (4/3)N The asymptotic area (after an infinite number of iterations) is (8/5) x the area of the original triangle. (I read this discussion many years ago in Mandelbrot's popular introduction to fractal geometry.) |
| rstofer:
--- Quote from: magic on August 05, 2021, 09:06:20 pm ---You may run out of paint on the exterior, but the interior cannot be painted with a layer of any thickness because most of it is too narrow ;) --- End quote --- If I can have quantifiably small internal diameters, I should be able to have really tiny paint molecules. My math skills are too weak for either of the last two paradoxes. I just try to balance my checkbook and that isn't easy as it sometimes requires imaginary numbers. |
| TimFox:
The interior surface area might be infinite, but the volume of paint required to paint the interior is finite due to decreasing clearance, even if the thickness goes to subatomic dimensions. The volume of paint must be less than the volume of the interior (equal to pi according to the post, I haven't checked), which must be less than an external volume that contains the horn. Paint at Home Depot is sold by volume. |
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