Wikipedia:Reference desk/Archives/Mathematics/2015 February 17

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February 17

Slicing a cube

Some probability problems are easy to solve geometrically, for example the probability that two people arrive within 10 minutes of each other if both come independently at random at any time within an hour. Drawing a 60 by 60 square, plotting the lines x = y ± 10 and considering the area between them readily gives the value 11/36. But what if there are three such people? Drawing a 60 by 60 by 60 cube and considering only the part between the pairs of parallel planes x = y ± 10, y = z ± 10 and z = x ± 10 should suffice, but I can't evaluate the remaining shape, nor indeed visualise it clearly. Is it a hexagonal prism with hexagonal pyramidal endcaps and what is its volume? More generally, what is the volume for everyone arriving within t minutes of each other?→31.54.246.96 (talk) 10:34, 17 February 2015 (UTC)

Instead of thinking of it as a cube, imagine looking down at one vertex of the cube so that the axis of sight is one of the four main diagonals of the cube. The projection onto two dimensions is a hexagon, subdivided into equilateral triangles by the edges of the cube (which are coincident with the diagonals of the hexagon). If we want the probability that the individuals arrive within t hours (so I'm using units with $0\leq t\leq 1$), then we mark points on the diagonals of the hexagon that are a ratio of t out from the center. Joining up these six points gives a smaller hexagon, and the desired probability is the volume of the portion of the cube that lies over that hexagon. Note that the six equilateral triangles are projections of six chambers of the desired volume. There is a scissor construction (I believe) that reduces an individual chamber to a triangular prism. I get, for the height of the prism, $(1-t/2){\sqrt {3}}$ (using similar triangles with the main diagonal of the cube and one of the six marked points). We can probably work out the area of a cross-section of the prism using geometry, but it's easier to observe that the area must be proportional to $t^{2}$, say $\alpha t^{2}$ (α is to be determined). So the probability is $6\alpha t^{2}(1-t/2){\sqrt {3}}$. When $t=1$, we get the whole cube, so $6\alpha {\sqrt {3}}/2=1$. Hence $\alpha =1/(3{\sqrt {3}})$ (which feels sort of right). Hence, according to this rough reckoning, the probability is $2t^{2}(1-t/2)$. But I could be wrong, and I haven't really checked in detail, so YMMV. Sławomir Biały (talk) 14:44, 17 February 2015 (UTC)

$2t^{2}(1-t/2)$ gives 11/216 if t=1/6 for 10 minutes, approx. 0.051, but simulation gave 0.074, the simplest fraction nearby being 2/27. So it seems that there is a flaw somewhere.→31.54.246.96 (talk) 17:28, 17 February 2015 (UTC)

Would need to check the similar triangles argument. This is where the $t/2$ comes from. It's conceivable that I've got the wrong proportions, and it should be $2t/3$ instead, which would give $\alpha =1/(2{\sqrt {3}})$, and so probability for ten minutes $2/27$ on the nose. But I don't have time to look into it. Sławomir Biały (talk) 18:11, 17 February 2015 (UTC)

An approach that seems to generalize to n dimensions is this: You're trying to find the volume of the polytope 0≤x_i≤l, for 1≤i≤n, and |x_i-x_j|≤a, for 1≤i,j≤n. Divide up the region according to the order of the x_i when sorted. There are n! possible orders and each has the same volume by symmetry, so the desired volum is n! times the volume of the polytope 0≤x₁≤...≤x_n≤l, x_j≤x_i+a. But the last condition, given that the x's are sorted, is the same as saying x_n≤x₁+a. So the volume of the smaller region is represented by the integral

$\int _{0\leq x\leq y\leq l,y\leq x+a}(volume\ of\ simplex\ \{x\leq x_{2}\dots \leq x_{n-1}\leq y)\}dydx$

or

$\int _{0\leq x\leq y\leq l,y\leq x+a}{\frac {1}{(n-2)!}}(y-x)^{n-2}dydx.$

So the volume of the original solid is

$n(n-1)\int _{0\leq x\leq y\leq l,y\leq x+a}(y-x)^{n-2}dydx.$

which reduces the problem to two dimensions again. --RDBury (talk) 22:11, 17 February 2015 (UTC)

PS. I made a small correction in the integrand. The final result is a^n-1(nl-(n-1)a).--RDBury (talk) 22:40, 17 February 2015 (UTC)

That final result should be a^n-1(n-(n-1)a), removing a rogue "1", which would make Slawomir's one $3t^{2}-2t^{3}$, where a and t are the same thing. The volume for 1/6 would then be 2/27, as suspected. Thanks to both. Would anyone be able to show an image of the polytope embedded in the cube?→31.54.246.96 (talk) 09:48, 18 February 2015 (UTC)

Actually the "rogue 1" was an l. Maybe I should have used $a^{n-1}(nl-(n-1)a)$. Of course if you're taking l=1 then it the same. --RDBury (talk) 14:29, 18 February 2015 (UTC)

Set Theory

How is the following statement expressed by a well-formed formula (having only x,y as free variables): "x belongs to...that belongs to...that belongs to y". Please notice, that we have no previous information about the length of the chain, and that a well-formed formula must not include the sign of three-dots... HOOTmag (talk) 13:28, 17 February 2015 (UTC)

Something like $\exists a\exists b.x\in a\land a\in b\land b\in y$? AndrewWTaylor (talk) 13:33, 17 February 2015 (UTC)

Your suggestion could have been a solution if we had been informed about the length of the chain. However, unfortunately, we have no previous information about the length of the chain... HOOTmag (talk) 13:38, 17 February 2015 (UTC)

Ah, sorry, I misunderstood. Can the chain be infinite? I guess you need to have a set S of the intermediates, and say something like

$\exists S.\forall a\in S:x\in a\lor a\in y\lor \exists b\in S,c\in S.b\in a\land a\in c,$

(i.e. every element of S either contains x, belongs to y, or is "between" two other elements of S) but that probably doesn't completely cover it. AndrewWTaylor (talk) 14:05, 17 February 2015 (UTC)

Addendum to that - you have to add that there exists an element of S that belongs to y, otherwise you could just have an infinite chain starting with x. By the Axiom of regularity you can't have an infinite "descending" chain from y, so maybe that's enough.. AndrewWTaylor (talk) 14:24, 17 February 2015 (UTC)

Wrong. Check: x = y = { 0 }, in which case - your well-formed formula is true (e.g. S = y) - whereas my original English statement (with the three dots) is false.

As for your question: For practical purposes, let's assume the chain is finite. HOOTmag (talk) 15:36, 17 February 2015 (UTC)

I think it suffices to add the requirement that $\exists a:x\in a\land a\in S$. If S has at least two elements, this follows automatically from regularity; but you have to account for the case that S has just one element.

If you want to include the case that the chain is of length 0, disjunct the whole formula with $x=y$. -- Meni Rosenfeld (talk) 18:54, 17 February 2015 (UTC)

Wrong. Check: x = { { 0 } } , y = { 0 } , in which case: the well-formed formula is true - e.g. S = { 0 , { x } }, whereas my original English statement (with the three dots) is false. HOOTmag (talk) 19:44, 17 February 2015 (UTC)

You can do this in a bunch of ways. The set of all x that satisfies your condition is called the transitive closure of y. You might be able to find refs in that article.

One way, probably more complicated than necessary if you want to get all the way down to the language with just epsilons, but quite faithful to your idea, is to say something like: "There exists a natural number n and a mapping f whose domain is n, such that f(0)=x, f(n−1)=y, and for every m less than n−1, f(m)∈f(m+1)". This references various other concepts (natural number, mapping, plus) but all of these have routine definitions that you can also look up. --Trovatore (talk) 19:52, 17 February 2015 (UTC)

To summerize, the simplest solution is: "x belongs to the transitive closure of y". HOOTmag (talk) 20:21, 17 February 2015 (UTC)

Hm, I've realized there are some additional issues with Andrew's original suggestion which are not easily removable. You can have an infinite ascending chain starting at x and an additional element of y. Or you can have a descending chain from y which terminates not on x, but on another element of y.

So I can't really think of anything better than bringing in some heavier machinery, as in Trovatore's suggestion. -- Meni Rosenfeld (talk) 19:58, 17 February 2015 (UTC)

"Heavier machinery"? x belongs to every S that contains both every element of y and every element of any element of S. HOOTmag (talk) 20:27, 17 February 2015 (UTC)

I was thinking about the one which defines a sequence of sets with each entry an element of the next, but yeah, this seems to work. -- Meni Rosenfeld (talk) 20:59, 17 February 2015 (UTC)

I think it's a fine definition, if all you want is to find a formula that characterizes the relation extensionally. However, if you want to prove that it's non-vacuous, you'll have to do a little bit of work. To start with, you have to show that there is some set S containing etc etc etc, since otherwise every set x whatsoever would satisfy the formula. The simplest way to do that, unless I'm missing something, is to go through something like the definition I offered. --Trovatore (talk) 21:07, 17 February 2015 (UTC)

Is it an anomaly that the month of February 2015 contains four occurrences of each day of the week? (February 10)

Back in 1972, a contributor to the Daily Telegraph letters page asked when the full moon last fell on Tuesday, February 29. A respondent said he was sure it hadn't happened since the calendar was reformed in 1752. He was wrong - it last happened in 1820. However, the interesting thing is that with 29.5 days in a month and 28 years before leap day cycles back to Tuesday this will happen on average every 826 years. I am sure this is where the 823 - year cycle you have been discussing this week originates.

In the Julian calendar, if March 1 falls on a Wednesday and it is not a leap year it will also fall on a Wednesday 823 years later. In the Gregorian calendar this is reversed - if March 1 falls on a Wednesday in a year which is within the first 76 years of the century and is a leap year then it will also fall on a Wednesday 823 years later. 156.61.250.250 (talk) 18:48, 17 February 2015 (UTC)

Very good! Here is a link to Joseph A. Spadaro's archived question: WP:Reference desk/Archives/Mathematics/2015 February 10#Is it an anomaly that the month of February 2015 contains four occurrences of each day of the week? -- ToE 19:59, 17 February 2015 (UTC)

156.61.250.250, did you just think this through or possibly locate an old article you half remembered, or did you just coincidentally run across the Daily Telegraph article and remember last weeks quesion? -- ToE 20:15, 17 February 2015 (UTC)

As a Telegraph reader, I peruse the letters page over breakfast. When the 823 - year cycle came up on the reference desk I put two and two together. If you're interested, there's more on these long cycles here: [1]. 156.61.250.250 (talk) 09:46, 18 February 2015 (UTC)

Note that the date of the full moon depends on your time zone. The relevant full moon in 1972 was on February 29 at 03:12 UT, which means it was still February 28 in Newfoundland (time zone −3:30) and all time zones west of there. The one in 1820 was on February 29 at 01:06 UT; since time zones had not yet been invented, this means it was still February 28 at all longitudes west of 16°30'W. (Though since that's a larger section of the world, it is true, at least, that all places that had that February 29 full moon in 1972 also had the one in 1820. And the Telegraph is a British paper, so the 1972 question may have been intended to implicitly refer to their time zone.) (I should note that daylight saving time and the International Date Line further complicate these statements.)

Also note that the synodic month is actually a bit over 29½ days; it's more like 29.53. It is true that 826 years in the Julian calendar works out an an integer multiple of 29½ days, though (365¼×826 = 301,696½ = 10,227×29½). But neither 823 nor 826 years is close to multiple of 29.53, so the phases of the moon won't repeat on the same dates after either of those intervals. For example, to stay clear of the Gregorian calendar I'll go back to the year 600. The first full moon of 600 was on January 6 at 08:44 UT. 823 years later would be 1423. In fact there was a January 6 full moon (time zone UT) in 1414 and not another one until 1433.

The fact that the numbers don't work doesn't mean that they weren't the origin of a false claim about the calendar, though.

Source for the moon dates is Fred Espenak's 6,000-year catalog of moon phases, which is currently missing from its normal home at http://eclipse.gsfc.nasa.gov/phase/phasecat.html but can be found here via the Wayback Machine. --70.49.169.244 (talk) 23:54, 18 February 2015 (UTC)

Even if 823 or 826 were a multiple of 29.53 that wouldn't mean anything. You would first have to convert the 1 March of the first year into its Julian Day number, do the same with the second, take the difference and divide by 29.53058872. (This is simplistic, because the length of the synodic month is changing with delta T). Then take the decimal part of your answer, multiply by 29.53058872, add or subtract 14.76529436 and you have (roughly) the age of the moon on 1 March of the later year. 156.61.250.250 (talk) 11:54, 19 February 2015 (UTC)

BTW that AD 600 time you quoted is in yet - to - be - invented British Summer Time. 156.61.250.250 (talk) 12:01, 19 February 2015 (UTC)

Taking another look at the figures, 29.53058872 x 28 = 826.8564842, = 826 years 313 days. In AD 600 the paschal full moon fell on April 4, so on March 1 the moon was ten days old. 313 days after March 1 brings us to January 8, so using this system the moon should have been ten days old on 8 January 1427. Now the paschal full moon that year fell on 13 April, so on 8 January the moon was eight days old. But we have to allow for the fact that the Julian calendar gains on the moon at the rate of one day every 307 years. Factoring that in, the figures agree. This is one example of the self - evident truth that if you multiply any number by 4 and call it Julian years, the number of days in that period of Julian years will be a multiple of that number. 156.61.250.250 (talk) 09:47, 20 February 2015 (UTC)