canuckInOR
carpal tunnel
Registered: 13/02/2002
Posts: 3212
Loc: Portland, OR
So, I got an email from my mom, with the subject of Beauty of Mathematics, containing a youtube link. Since I have a degree in math, it seemed appropriately nice.
I was (perhaps naively) expecting something like this ViHart video:
*sigh*
I got this "inspirational" video, instead:
Okay, mom, the first part is neat. But the rest of that? It's not "mathematics", it's trite numerology, and it's idiotic garbage. (If you were smart, and skipped the second vid, they assign numbers to letters of the alphabet based on position, and then add up the numerical value of various phrases.)
I'm on the fence trying to decide if I ought to tell my conservative mother that if you continue with that theory, internet porn is even better than love of god.
I had never heard of Vi Hart before. She has a lot of videos out there and is a fantastic talent.
Has she been able to leverage these videos into a money-making operation? If so, whatever she earns isn't enough!
Thank you for broadening my horizons. I'm going to watch more of her videos over the next few days.
On the topic of mathematics...
Many years ago (like about 40 or so) I came up with a math problem that intrigued me. I was never able to solve it, so on a whim I gave it to my office manager, who did. I don't remember her solution, only that it involved some outside-the-box thinking. Here's the problem:
Visualize an analog 12-hour clock with hour hand, minute hand, second hand. We know that at 12:00:00 all three hands are in perfect mathematically defined alignment. At what other times if any (such as five minutes and a few seconds after one, or maybe 21 or 22 minutes and a few seconds after four) are all three hands in perfect alignment. Prove (or disprove) it.
Fun, eh?
tanstaafl.
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"There Ain't No Such Thing As A Free Lunch"
I think that second video converted me to some form of polytheism. After all, love of gods is 120%. I've heard about those 100, 101 and 110% people before. 120% sounds even better.
Registered: 29/05/2002
Posts: 798
Loc: near Toronto, Ontario, Canada
Originally Posted By: tanstaafl.
... Here's the problem:
Visualize an analog 12-hour clock with hour hand, minute hand, second hand. We know that at 12:00:00 all three hands are in perfect mathematically defined alignment. At what other times if any (such as five minutes and a few seconds after one, or maybe 21 or 22 minutes and a few seconds after four) are all three hands in perfect alignment. Prove (or disprove) it...
Prove what exactly?
Off the top of my head, the clock hands would align eleven times as the hour hand travels around the clock face. The minute hand must run around 60 minutes plus five minutes (1/12th of the next hour) to catch up with the hour hand.
The second hand only matters when the other two are aligned, as it next catches up with the minute hand every 1+1/12th hour.
It is way past bedtime here, so if I messed this up, that is my excuse...
Registered: 13/07/2000
Posts: 4180
Loc: Cambridge, England
If numerology is the last refuge of the scoundrel, the last-but-one is that thing that I don't have a name for, where people do essentially the same thing with bogus etymology. It was brilliantly sent up by, who else, Jon Stewart, satirising, who else, Glenn Beck, by doing a whole presentation on "LIBERTARIANS": roaring "They lie! 'Lie' is right there in the name! [underlines it] But *who* is lying to us? Bert! [picture of Bert from Sesame Street] And who else is lying to us? Arians!" [flails arms, becomes too overcome to speak].
(In retrospect, finding the clip on Youtube would probably have been easier than typing that out...)
#361616 - 09/05/201406:53Re: Beauty of Mathematics...
[Re: tanstaafl.]
andy carpal tunnel
Registered: 10/06/1999
Posts: 5916
Loc: Wivenhoe, Essex, UK
Originally Posted By: tanstaafl.
I had never heard of Vi Hart before. She has a lot of videos out there and is a fantastic talent.
Has she been able to leverage these videos into a money-making operation? If so, whatever she earns isn't enough!
"Q: Can I give you money? You should really be able to make videos full time! A: I already make videos full time! Making my videos takes a lot longer than most people’s craziest estimates. Luckily, you’re not the only one who thinks my work is worth supporting. I am currently very generously supported by SAP, and before that, I was supported by Khan Academy. I am doing pretty good!"
_________________________
Remind me to change my signature to something more interesting someday
#361621 - 09/05/201416:28Re: Beauty of Mathematics...
[Re: tanstaafl.]
canuckInOR
carpal tunnel
Registered: 13/02/2002
Posts: 3212
Loc: Portland, OR
Originally Posted By: tanstaafl.
I had never heard of Vi Hart before. [...] Thank you for broadening my horizons.
My pleasure! I love her videos, along with SmarterEveryDay.
Originally Posted By: tanstaafl.
Visualize an analog 12-hour clock with hour hand, minute hand, second hand. We know that at 12:00:00 all three hands are in perfect mathematically defined alignment. At what other times if any (such as five minutes and a few seconds after one, or maybe 21 or 22 minutes and a few seconds after four) are all three hands in perfect alignment. Prove (or disprove) it.
The simple way to do it is to get an analog clock, and wind the hour hand around the entire face, taking note of the times when they overlap.
#361666 - 12/05/201420:07Re: Beauty of Mathematics...
[Re: K447]
canuckInOR
carpal tunnel
Registered: 13/02/2002
Posts: 3212
Loc: Portland, OR
Originally Posted By: K447
Originally Posted By: tanstaafl.
... Here's the problem:
Visualize an analog 12-hour clock with hour hand, minute hand, second hand. We know that at 12:00:00 all three hands are in perfect mathematically defined alignment. At what other times if any (such as five minutes and a few seconds after one, or maybe 21 or 22 minutes and a few seconds after four) are all three hands in perfect alignment. Prove (or disprove) it...
Prove what exactly?
Off the top of my head, the clock hands would align eleven times as the hour hand travels around the clock face. The minute hand must run around 60 minutes plus five minutes (1/12th of the next hour) to catch up with the hour hand.
That's actually quite incorrect. It all comes down to a question of precision -- how precise do you want to be?
The problem with your logic is that, as the minute hand moves the 1/12th of the next hour to catch up to the hour hand, the the hour hand is still moving. So in truth, the minute hand must move 1/12th of an hour plus the delta that the hour hand has moved in those 5 + delta minutes. Similarly for the second hand.
The simple way to look at it is as a function of percentage. The hands will align when they're the same percentage of the way around the clock face. You can describe that as the following set of functions (in terms of seconds -- we could arbitrarily pick any other unit, like tenths of a second, if we wanted to, but seconds is quite sufficient):
Code:
h(x) = x / 43200
m(x) = (x mod 3600) / 3600
s(x) = (x mod 60) / 60
for x in [0,43200)
Alignment occurs for the set
Code:
{ x | h(x) = m(x) = s(x), 0 <= x < 43200 }
At this point, it's easier to break out a bit of C -- I lack the knowledge to continue representing this in pure mathematical notation.
But none of those percentage numbers are actually equal, and since Doug did say "perfect mathematical alignment", I'd say this level of precision isn't sufficient. As a consequence the only acceptable answer is...
Never (as Doug hinted, with his underlined "if any").
Thank you, irrational numbers.edit: Thank you Peter, for reminding me that these are not irrational, as they're just a decimal representation of fractions...
Registered: 13/07/2000
Posts: 4180
Loc: Cambridge, England
Another way of looking at it, is to say that the position repeats after 12 hours. In those 12 hours, the hour hand goes round once and the minute hand twelve times, so the minute hand overtakes the hour hand exactly eleven times. Because the speeds are constant, those moments must be exactly one-eleventh of a 12-hour period apart.
Similarly, the minute hand goes round 12 times and the second hand 720 times. So the second hand overtakes the minute hand 708 times, exactly 1/708th of a 12-hour period apart.
The three hands exactly align whenever the minute hand overtakes the hour hand at the exact moment that the second hand is overtaking the minute hand. That time would have to be an exact number of elevenths and an exact number of 708ths. But as 11 and 708 are coprime, there is no such fraction: the only answer is zero elevenths, which equals zero 708ths, which is 12:00:00.
Registered: 13/07/2000
Posts: 4180
Loc: Cambridge, England
Originally Posted By: Roger
Thank you, Erlang:
But that code also returns just the one match if you ask it whether the minute hand and hour hand ever coincide (ignoring the second hand). So it falls a bit short of proving that the three-way event never happens, because it only considers events which happen at an integer number of seconds. The numbers involved are not irrational, but nor are they integers (elevenths of a 12-hour period).
It occurs to me that if your clock doesn't have a sweep second hand, then that's a whole different calculation.
#361679 - 13/05/201414:45Re: Beauty of Mathematics...
[Re: peter]
canuckInOR
carpal tunnel
Registered: 13/02/2002
Posts: 3212
Loc: Portland, OR
Originally Posted By: peter
Thank you, coprime numbers.
Ah, wonderful. On the way home, I started thinking that there had to be an easier proof such that the hands didn't share any common factors, but I was stuck thinking with the numbers in my previous solution, which were clearly non-coprime.
Registered: 13/07/2000
Posts: 4180
Loc: Cambridge, England
And if the French Revolution's audacious idea of decimal time (1d=10h, 1h=100m, 1m=100s) had stood the test of, uh, time -- then every time the hour and minute hand coincided, the second hand would be there too. (The "11" and "708" become "9" and "990", and every exact ninth is also an exact 990th.)
The even more interesting question is, given X hours per day (or per half-day, if that's what your clock-face measures) and Y minutes per hour, how many triple alignments are there as a function of X and Y?
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