Wikipedia:Reference desk/Archives/Mathematics/2022 June 18

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June 18

What if we defined 0! as 0 rather than 1??

I understand the logic of defining 0! as 1, but I still find it interesting to know what would happen if we defined 0! as 0. If I'm right, it would force division by 0 to occur in many formulas that depend on factorials. Georgia guy (talk) 00:20, 18 June 2022 (UTC)

The recurrence relation n!=n*(n-1)! would break, and the relation between the factorial and the gamma function would also break. Also, power series like $\exp(x)=\sum _{k=0}^{\infty }{x^{k} \over k!}$ would have to be patched up. It doesn't seem so good overall. 2601:648:8202:350:0:0:0:90B2 (talk) 01:30, 18 June 2022 (UTC)

It wouldn't force division by 0, it would force us to define those formulas by cases. The binomial coefficient would be n!/(k!(n-k)!) unless k = 0 or k = n, for example.--2406:E003:812:5001:F4DC:2FE2:9FCD:9FE6 (talk) 03:28, 18 June 2022 (UTC)

The question is a bit like, "What if we redefine the + operation so that n + 0 = 0?" It would mean that the operation becomes practically useless.  --Lambiam 09:32, 18 June 2022 (UTC)

But WHY...?? Arithmetically, the factorial function can be defined as a product of natural numbers: $n!=1\times 2\times \dots \times n.$ This implies a recursion $n!=(n-1)!\times n,$ hence $(n-1)!={\frac {n!}{n}}$. If we want if to work for $n=1,$ we need $0!=(1-1)!={\frac {1!}{1}}=1.$ Otherwise the previous equation would work for all natural numbers except $1$ – and we don't like exceptions in basic notions...
In combinatorics, the factorial $n!$ expresses the number of permutations of an $n$–element sequence. For $n=0$ there can be defined exactly one such sequence, namely the empty sequence, and it has exactly one permutation (because you cannot reorder its elements, since it has no elements). Redefining $0!$ as zero makes the redefined factorial function useless for zero-length sequence case, which appears an exception in a basic case as well. --CiaPan (talk) 21:53, 19 June 2022 (UTC)

What's the chance a random place is <x nautical miles from the nearest integer coordinate line?

True random points NOT "pick a random longitude then a random point on it and call that a random point" (is there a name for this paradox, where all longitudes are equally likely and all latitudes on that longitude are equally likely yet the northernmost acre would be picked about 200,000 times more often than another specific acre in the tropics when points per acre should be the same everywhere)

2. What's the chance if the latitude is y? Since the 360 integer meridians start 60 nautical miles apart and eventually become only a few miles apart then nanometers. Sagittarian Milky Way (talk) 02:10, 18 June 2022 (UTC)

I don't know a name for this, but it is an instance of the more general problem that one has to be careful when talking about a "random" element of a set, since different procedures for picking a random element will generally give rise to different probability distributions over the possibility space. A well known example of the phenomenon is known as the Bertrand paradox. To get a uniform distribution (by area) over the sphere, the latitude should be picked as the arcsine of a random value with a uniform distribution over the interval [−1, +1]. When instead the latitude is picked uniformly from [−90°, +90°], the relative density (per unit of area) at latitude $\phi$ is given by:

$\left({\frac {d}{d\phi }}\sin \phi \right)^{\!{-}1}=(\cos \phi )^{{-}1},$

which tends to infinity as we get closer and closer to the poles. That the density is infinite (or undefined) at the poles is not a paradox, since the area occupied by the poles is zero.  --Lambiam 09:09, 18 June 2022 (UTC)

I'm very stupid for I could've imagined a box cos(latitude)*60nmi wide and 60 long on a plane, 30x60=1800nmi² for the latitude band 60°N±0.5, 10x20x2=400nmi² will be ≥10nmi away, (30-5.72949*2)*(60-5.72949*2)=900 or half will be ≥5.72949nmi away and so on. A trapezoid and looking up degree sizes instead of using 60xcos(lat)*60 might be slightly more accurate. Maybe there's some formula for the whole Earth average? Shorter than calculating the areas inside 90 boxes and a box 2x miles smaller in each detection and summing the two categories and dividing the lower acreage one by the other? Sagittarian Milky Way (talk) 14:26, 18 June 2022 (UTC)

Let $R$ be the radius of a sphere (approximating the surface of Earth). The spherical area of what appears as a rectangle under the Mercator projection given by $[\phi _{1},\phi _{2}]\times [\lambda _{1},\lambda _{2}]$ (in radians), equals $(\sin \phi _{2}-\sin \phi _{1})(\lambda _{2}-\lambda _{1})R^{2}.$ The side lengths in reality along the meridians are $(\phi _{2}-\phi _{1})R.$ Measured along the parallels, the two sides have in general different lengths, $\cos \phi _{1}(\lambda _{2}-\lambda _{1})R$ and $\cos \phi _{2}(\lambda _{2}-\lambda _{1})R.$ I don’t know if this answers your question, because I'm not sure I understand it.  --Lambiam 15:19, 18 June 2022 (UTC)

If you're looking for a simple closed-form answer, it's very unlikely there is one.

a variant of (2) is relatively easy to answer: what's the probability that a random point of latitude y is further than x from every integer meridian?

Assuming a spherical Earth (we'll use the unit sphere) and the intrinsic metric being used to define "further away": the points of latitude y form a circle of radius $r=\cos y$. The chance that a random point on that circle is further away than x from every one of 360 evenly-spaced points on that circle is:

• 0, if $x\geq 720\cdot 2\pi \cos y$
• $\left(1-720\cdot {\frac {x}{2\pi \cos y}}\right)$ otherwise

(Note again that I assume x and y have been normalized to the interval [-1,1])

In general, you'll have to compute the areas I_n where at least n of the circles (around the integer coordinates) overlap for each n, and calculate

$\sum _{n=0}^{\infty }\left(-1\right)^{n}I_{n}$

This is possible, but tedious. Given that I_n = 0 for n > 360 * 180, the sum is finite, at least...

If you have access to a computer, your best bet is either Monte Carlo simulation (easy, but never quite accurate) or using a system that allows geometric data types and handles their intersections etc. for you. It's possible the GIS extension to an SQL database is sufficiently powerful to do the trick (in particular, I'm thinking about postgresql). IpseCustos (talk) 16:05, 18 June 2022 (UTC)

I wish I knew how to program, I could answer many personal curiosity questions with Monte Carlo simulations. Sagittarian Milky Way (talk) 17:01, 18 June 2022 (UTC)

I now finally understand the question. I initially read the title of the post as an identifying tag and the body, starting with "True random points ..." as being the actual question. The distance of a point from a given meridian is in general somewhat less than that distance measured along a parallel. (They are the same if the point is on the meridian or on the equator.) It is possible to give a closed expression for the fraction of points on the parallel of given latitude $y$ that is further away than a given distance $x$ from any of the meridians, but it is a very messy expression. If $f_{x}(y)$ denotes that fraction, the average over the whole sphere is given by

$F(x)={\frac {1}{2}}\int _{{-}{\frac {1}{2}}\pi }^{{+}{\frac {1}{2}}\pi }f_{x}(y)\cos y\,dy.$

For a given numerical value of $x,$ the value of $F(x)$ can be approximated by numerical integration, which gives very precise results much faster than Monte Carlo methods.  --Lambiam 09:45, 19 June 2022 (UTC)

I've written a small program and posted in on the OP's talk page, and the actual functions (I misread their definition at first) don't seem overly interesting.

You're right, I was confused about the point that the distance along the parallel is different from the actual distance to the meridian! Sorry about that. IpseCustos (talk) 10:14, 19 June 2022 (UTC)

The problem is less messy than I thought. For the sake of keeping the formulas neat, let us use units such that the radius of the sphere is $1$ unit. So a nautical mile is then ${\tfrac {2\pi }{360}}$ units. Also, let us consider the slightly more general case of $n$ equidistributed meridians, where in the original problem $n=360.$ In what follows, distances are always measured as the lenth of a great-circle segment; if $\mathrm {O}$ is the centre of the sphere, the distance from $\mathrm {A}$ to $\mathrm {B}$ is then equal to the angle $\angle \mathrm {AOB}$, measured in radians.

Imagine that the $n$ meridians are painted white with a very broad brush, of width $2x$ – that is, $x$ units on either side. We assume that $x\leq {\frac {\pi }{n}}.$ Projected on the equatorial plane, the width of the brush stroke is then $2\sin x.$ Consider how the parallel circle at latitude $\phi ,$ which has radius $\cos \phi ,$ interacts with a single fat white meridian. As an ant crawls along the parallel, it traverses the meridian once each full round. (Meridians are half circles, running from one pole to the other.) The angle subtended by the white segment of the parallel is equal to $2\arcsin {\frac {\sin x}{\cos \phi }}.$ We have $n$ such segments on a total angle of $2\pi ,$ so the fraction within distance $x$ of any of the $n$ meridians is given by

$f_{x}(\phi )={\frac {n}{\pi }}\arcsin {\frac {\sin x}{\cos \phi }}.$

However, this is not valid when we get close to the poles, where the brush strokes overlap and $f_{x}(\phi ),$ according to this formula, exceeds $1.$ The critical latitudes $\pm \phi _{*}$ where the strokes merge into solid white polar caps are given by the equation $f_{x}(\phi _{*})=1,$ solved by

$\phi _{*}=\arccos {\frac {\sin x}{\sin {\tfrac {\pi }{n}}}}.$

So, instead of integrating $f_{x}(\phi )\cos \phi ,$ we need to integrate $f_{x}^{*}(\phi )\cos \phi ,$ where $f_{x}^{*}(\phi )$ is given by

$f_{x}^{*}(\phi )={\begin{cases}f_{x}(\phi )&{\text{if }}|\phi |\leq \phi _{*}\\1&{\text{otherwise}}\end{cases}}.$

I don't think this can be done analytically.  --Lambiam 18:32, 19 June 2022 (UTC)

We can get pretty close. Consider the area of the sphere below the critical latitude, versus the area of the stripe below the same latitude. It's not trivial because the end of the stripe is not cut off by a great circle, but how much accuracy do you want? —Tamfang (talk) 02:39, 20 June 2022 (UTC)

That is correct. If $n=360,$ this approximation is off by less than 3.1×10⁻¹⁷, and as a relative error by less than 10⁻¹². That is less than the error introduced by using a spherical model instead of a more accurate ellipsoid.  --Lambiam 08:57, 20 June 2022 (UTC)

more official name for "Half tetrahedron"?

Is there a more official name for the solid with one square face, two triangular faces and two trapezoidal faces that if you match two on the square face you get a regular tetrahedron? I can't find it described with any other name. :(

There is no naming committee in mathematics (that I know of, at least) like IUPAC for chemistry or IAU for astronomy. Random mathematical objects are generally not given names at all unless they serve a useful purpose or are particularly interesting examples. So if you've done a thorough search and have been unable to find the object described in a published paper or textbook then the chances are it has no name. The solid you described is mentioned in Wedge (geometry), but only by description. --RDBury (talk) 22:46, 18 June 2022 (UTC) PS. @Naraht: Don't forget to sign your posts.

Oops. And yet, we have such lovely names as "triangular gyrobianticupola" and "Hebesphenomegacorona"Naraht (talk) 15:27, 21 June 2022 (UTC)

Note that those have regular faces. The solid in question is topologically equivalent to an uniform triangular prism. —Tamfang (talk) 01:58, 24 June 2022 (UTC)

Some things are named that don't really need to be, and some go unnamed but deserve to have a name. It can be unfair. Maybe a mathematical naming committee would help with this kind of thing, maybe not. Relevant clip from LotR. --RDBury (talk) 08:45, 24 June 2022 (UTC)

Or you can view this as an opportunity to name it yourself. Discover some wonderful non-obvious properties and have a paper on these published, in which you name it something like the tharan, the tetrahedrogen, or the hemitetrahedral amblysphene (see amblygon and sphene).  --Lambiam 09:44, 24 June 2022 (UTC)