On a recent morning walk, as I rounded the curve on Peartree Crescent, it occurred to me that if I moved from the outside of the curve to the inside of the curve on the lefthand side of the street I could shorten my walk a tad. That got me wondering how big the difference was in the length of a circular road running all the way around between the outside of the road and the inside of the road. Well, I reflected as I trudged on, the circumference of a circle is equal to 2πr. So, if the road is 20 feet wide and the circumference of the circle measured from the inside of the road is 2πr, then the circumference of the circle measured from the outside of the road will be 2π(r + 20). If we subtract the circumference of the inside of the road from the circumference of the outside of the road we get 40π. Now π, I recall from high school geometry, is 3.14 and a little bit (which is why nerds and geeks call March 14 “pi day.”) So the difference between the two is roughly 62.8 feet.
By this point I had turned right onto Magnolia and was about
to turn right again onto Hawthorne for the long schlep to the end of that road and
back. And then it struck me: the increase in the length of the outside
circumference of the circular road over the inside had been calculated without
any consideration of the length of the radius of the inside circular edge of
the road. This meant, unless I had badly misunderstood something, that a circle
drawn around the earth just 20 feet off the ground would be the same 62.8 feet
longer than the circle drawn around the earth hugging the ground.
This seems so profoundly counterintuitive that I spent the
entire walk to the end of Hawthorne and back checking and double checking my
arithmetic in my head to make sure that I was right.
Sometimes one’s intuitions are wrong.
22 comments:
If one looks at a half dollar and at a dime, one would conclude that the circumference of the half dollar is longer than that of the dime, correct?
However, here is an experiment which proves that this is not the case. If one were to place the dime in the center of the half dollar, drill a hole through both and insert a dowel through the hole to keep the dime affixed to the half dollar, then position them on a flat surface and place a mark on the surface perpendicular to the edge of both the half dollar and the dime, then roll the two until the half dollar makes a complete rotation, both the half dollar and the dime will have traversed the same distance, proving that their circumferences are identical. I am wrong, you say? How so?
This is a well-known puzzle - standard high school math fare! (See, e.g.,https://mathimages.swarthmore.edu/index.php/Rope_around_the_Earth .)
Sorry, Anonymous, your rope around the Earth citation does not address the paradox I have offered.
Prof Wolff -
First an arithmetic point - I make 40 π about equal to 125 feet (40 times a bit more than 3), which is twice your figure.
That said, I found it intuitively surprising as well. But such is the joy of maths!
Prof. Wolff’s post set me to thinking about the staggered lanes in the long distance track events at the Olympics. Let me preface this by saying that I know next to nothing about the mechanics of track events and have never particularly enjoyed running long distances. I have run a one mile distance only once in my life, during basic training at Ft. Campbell, Ky., in 1970. I broke no time records and have not done it again since.
In long distance track events, the lanes are staggered, with the lanes further from the inside track being staggered ahead of the others to compensate for the shorter circumferences of the tracks as they get closer to the first lane. Now, if the runners were required to stay in their lanes, no runner would have a distance advantage and they would all run the same distance. However, the runners are not required to stay in their lanes. My question for the mathematics and track aficionados is the following: Doesn’t this actually give the runner who is running in the last lane from the first lane an advantage, so that if s/he immediately after the gun sounds, runs diagonally towards the inside tracks actually winds up running a shorter distance? If this is done immediately, so as to shorten the diagonal, does not the staggering of the outside lane thereby give this runner an advantage? Since this runner is further ahead of all of the other runners at the outset, this runner can avoid colliding with other runners by cutting across immediately, and can wind up running the shortest race of all the runners. If this is a stupid and idiotic question, attribute it to my lack of sports knowledge.
Is there much that's more obnoxious than someone offering a "paradox" that isn't really a paradox and, moreover, isn't much of a mystery at all, and then insisting that all subsequent comments on the thread be a response to their "paradox" rather than a response to the original post? Anonymous at 7:56 was clearly responding to Prof. Wolff and not to anonymous at 7:36. That should be expected considering this is a comment thread attached to Prof. Wolff's post and not a comment thread attached to anonymous at 7:36's "paradox."
Moreover, it's distasteful for someone in a comment thread to adopt for themselves the role of teacher and try to elicit a response or instruct the other commenters. People who choose to go on Prof. Wolff's blog have consented to being lectured to to some extent by Prof. Wolff, but no one signed up for anonymous at 7:36's geometry course, so adopting the instructor's mantle by posing questions to which one already knows the answer is presumptuous. This is made worse by responding at 8:03 as if every subsequent comment in the thread is going to be a reply to the comment at 7:36. Anonymous at 7:36, maybe no one cares about your not at all puzzling "paradox." If you want to make a comment, make a comment, but don't try and hijack the thread so that you can play teacher. Add to this the insistence on commenting anonymously even after Prof. Wolff has declared his preference for commenters to comment using their names and it's all a bit too much to stand.
I guess none of this is new, I'm not the first person to notice that Prof. Wolff's blog would be a more pleasant place to spend time if commenters were better behaved.
T.J.,
The paradox I posed was in the spirit of Prof. Wolff’s observation regarding the different circumferences and their geometric and geographical implications. Moreover, the half dollar/dime paradox is not nearly as simple as you claim. If it’s so simple, what’s your answer – which you can decline answering at your discretion. The paradox was first noted by Aristotle in his Mechanica and in fact puzzled mathematicians for centuries. The answer is not as simple as you pretend.
Moreover, many of the commenters on this blog have offered commenters which diverge from Prof. Wolff’s original post in ways that add different perspectives to the original post, and many do so anonymously.
In light of T.J.’s dismissive comment, I will provide the answer to the paradox, which will demonstrate two things: that my comment was in keeping with Prof. Wolff’s post aimed at showing that one’s intuition is not always correct; and further, that T.J. does not know what he is talking about.
First, regarding T.J. T.J. thinks that the answer to the paradox is simple, and only a dumb-ass would think that the paradox is really a paradox. His likely explanation goes something like this: The paradox is a fallacy because it is confusing revolutions with circumference. The fact that the two coins each complete one revolution does not mean that they travel the same distance. We know this, because we know that their circumferences are in fact different. This is a circular argument, since it presumes what is intended to be proved – that the circumferences are different.
The fact of the matter is that neither coin as it travels through space is traveling the distance of its circumference. It took many mathematicians many centuries to figure this out. If you consider the point on the half dollar which is touching the flat surface, as the half dollar rotates, it does not trace in space the distance of its circumference. It traces a curve, known as a cycloid. The corresponding point on the dime also does not trace through space its circumference. It traces a different cycloid, which is shorter than that of the half dollar. This is related to Prof. Wolff’s observation that one’s intuition, that each coin traces its circumference, is not necessarily correct.
Did you know that, T.J.? And, by the way, for a person who uses his initials to post, rather than his/her full name, to chastise others for posting anonymously is the height of chutzpah.
Any responses to my track question?
It does seem counter intuitive to me too. I am still scratching my head.
But my grandson, who majors in mathematics, says it is true.
Tom,
Thank you for that confirmation.
Do you, or your grandson, have a response to my track question?
Susselman, Tom was not responding to you. He was responding to the blog post. You have a comedic sense of your own importance.
Anonymous,
How can you be positive of that?? I guess it's up to Tom to clear this up, if he wishes. I drew my conclusion from the location of his comment in the sequent of comments, which of course would not preclude that he was responding to the original post, rather than my comment. I also inferred that he was referring to my comment since of the two examples of counter-intuitiveness, the rope circling the Earth, versus the path formed by a rotating coin, the latter seems more counter-intuitive than the former. This is supported by two observations: (1) Anonymous at 7:56 A.M. was not impressed by Prof. Wolff's post and thought that it was rather common knowledge, "standard high-school math fare"; and (2) Tom was still scratching his head about the mathematical conclusion he had consulted his grandson about. If it was as obvious as Anonymous claimed it was, I doubt that Tom would be scratching his head. No big deal, either way. And thank you Anonymous for pointing out to me that I am rather full of myself. Good to know.
Nerds and geeks in the US, of course, with its weird date format.
I was responding to Prof Wolff's original post. I had to talk it through with my grandson and do the math. The math is straightforward which makes it so curios and interesting. (and I think it is 125.6).
I kept trying different Values in my head, like .1 feet for the inner radius and 1 million feet for the distance between circles. I could not break it
More on track example later
Mea culpa. I will proceed to mortify my flesh.
As for track and field long distance races are not staggered, only shorter sprints where runners remain in lane the entire race.
They use a curved starting line for long distance races, so it is almost impossible to run diagonally without running into someone. Most runners consider the outside lane a poor starting position.
Professor Wolff --
For me, what I was most struck about this post was the fact that you were trying to figure out a way to shave distance from your walk. On my walks, I am always trying to figure out how to add distance -- how to increase my movements as much as possible. I am so pressed for time that strategies to incorporate exercise is the key challenge.
-- Jim
Actually, when it comes to circles, "the circumference divided by the
diameter" produces the same value regardless
of their radius,
You're talking about increasing the Earth radius by 20 feet.
Normally, that radius is 20.902m feet. So that becomes
20,902,020, shall we say, instead of 20,902,000.
The circumference is 131,480,000 feet.
So, 131,480,000/20,902,000*2 = ?/209,902,020*2
And then, 131,480,000*209,902,020*2 = ?*20,902,000*2
So, 131,480,000*209,902,020/20,902,000 = ? new circumference
= 131,480,012.5
The difference between that and 131,480,000 is 12.5.
I muse that I have found myself correcting your math before -- I had fun mocking your statement that you taught yourself linear algebra because you needed to debunk mainstream economics. That was, like, a busy morning! But I dunno, maybe I'm sticking my neck out -- I haven't double-checked my own math, here..
I did garble this.
'So, 131,480,000/20,902,000*2 = ?/209,902,020*2'
No, it's 131,480,000/20,902,000*2 = ?/20,902,020*2
'And then, 131,480,000*209,902,020*2 = ?*20,902,000*2'
No, it's 131,480,000*20,902,020*2 = ?*20,902,000*2
'So, 131,480,000*209,902,020/20,902,000 = ? new circumference'
No, 131,480,000*20,902,020/20,902,000 = ? new circumference
'= 131,480,012.5'
No, 131480125.8
'The difference between that and 131,480,000 is 12.5.'
No, it's 125.8. I take this for the same answer that you got!! Point being, maybe, that although finding the answer requires only basic geometry, but the important thing here is to get a sense of proportion such that the really odd thing is that an extra metere of string will lift it 15cm off the surface of the Moon, or Jupiter, or a ping-pong ball.
I'm still just jerking you around, we're not there yet:
'If we subtract the circumference of the inside of the road from the circumference of the outside of the road we get 40π. Now π, I recall from high school geometry, is 3.14 and a little bit (which is why nerds and geeks call March 14 “pi day.”) So the difference between the two is roughly 62.8 feet.'
40π, and this is 'the difference between the two', which is what, now? 3.14*40 is 125.6.
Someone told me this paradox a couple of years ago and I didn't believe it. I had to keep checking the math. An extra 6 feet (or so) of rope's length and you can raise the rope by a foot all around the earth?
Maybe an issue here is that a foot is a decent-sized measurement by human standards, so it can feel significant to raise the rope that high off the earth's surface. But if the radius of the earth is (at the Equator) around 3963 miles, or 20,924,640 feet; and if its circumference is therefore nearly 25,000 miles, or 131,473,390 feet; then you're just saying that if you increase the length of the rope by 1/131,473,390, you extend the radius of the circle by around one-twenty-one-millionth. Does this dissipate some of the air of paradox?
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