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March 3

Expression for pi

originally mistakenly posted at WP:RD/M

The square of the Gaussian integral is pi, so my logic follows from expanding and integrating the integrand using a multivariate Maclaurin series, with use of the binomial theorem.

Am I then correct in claiming that, briefly, $\pi =\iint _{\mathbf {R} ^{2}}e^{-(x^{2}+y^{2})}dxdy=\lim _{(a,b)\to (\infty ,\infty )}4\sum _{n=0}^{\infty }(-1)^{n}(\sum _{j=0}^{n}{\frac {a^{(2j+1)}b^{2(n-j)+1}}{(n-j)!j!(2j+1)(2n-2j+1)}})$ ?--Jasper Deng (talk) 06:48, 3 March 2014 (UTC)[reply]

Is the question whether you may integrate a power series term by term and then take the limit of the integration limits? YohanN7 (talk) 08:13, 3 March 2014 (UTC)[reply]

It's a little more general than that. In addition, I'm expecting others to tell me that this is almost useless for computing pi, as I would expect from its lack of mention in the pi article, but another thing to establish is strictly that this limit exists (not helped by the fact that the series is conditionally convergent) and is equal to pi.--Jasper Deng (talk) 08:21, 3 March 2014 (UTC)[reply]

Are you sure that the series is conditionally convergent? It looks like the Cauchy product of two absolutely convergent series to me. —Kusma (t·c) 09:16, 3 March 2014 (UTC)[reply]

Actually, the inner summation sign is a result of the binomial theorem expansion (and subsequent integration using the power rule) of -(x²+y²)ⁿ, and thus is always a finite sum for finite n. The reason why I say it's conditionally convergent (yes, this is not quite correct in the usual sense, but...) is that, as (a,b)→(infinity,infinity), the series would diverge if not for the (-1)ⁿ coefficient. While I'm not that familiar with the concept of a Cauchy product, one could see that due to Fubini's theorem, I could've easily just have squared the univariate Taylor series for the error function.--Jasper Deng (talk) 09:32, 3 March 2014 (UTC)[reply]

Model railroad geometry.

I'm in the process of designing a system of interlocking track segments - kinda like a model railroad or a slot-car racing set.

I want to come up with a good set of track pieces that allow the maximum flexibility in track design with the minimum number of pieces (maybe just three: One straight, One curve and One 'turnout').

Obviously the curves should be some integer sub-multiple of 360 degrees - and I've noticed that some track systems use 45, 22.5 or 11.75 degree curved sections, while others use 60, 30 or 15 degrees.

There seems to be wide variability in the ratio of the length of the curved sections and the length of the straights.

Is there any logical reason why I should pick one particular set of dimensions to maximize flexibility of layout design?

(Sadly, I'm unable to come up with any firm criteria for this problem - but obviously a 1 degree or a 180 degree curved section would be impractical.)

I'm really just looking for the things I should look out for rather than a concrete answer.

SteveBaker (talk) 21:09, 3 March 2014 (UTC)[reply]

You may want to figure out what sort of track layouts/purposes you want before picking pieces. This shows some of the features that model railroad layouts have, there is quite a variety. Curvature can be just as important as degrees of arc[1] and some layouts use more than one curvature. If you don't want to pick just one curvature, use Flex Track. After reading Train track (mathematics), I now want a model railroad that runs on a triple torus. --Mark viking (talk) 22:32, 3 March 2014 (UTC)[reply]

You must also consider the radius of the curved pieces. A 45° curve based on a 30cm radius is longer than 45° curve on a 20cm radius. → Michael J Ⓣ Ⓒ Ⓜ 00:53, 4 March 2014 (UTC)[reply]

If your curves are (1/n) of a full circle, you will be severely limited in your track layout options if n is a prime. The more factors n has, the better. Once you have decided n, consider constructing a "figure of 8" track built from two circles meeting at a diamond crossover, and choose the length of your standard straight piece accordingly. You can either choose for symmetry, which will give you a single standard length, or choose an asymmetric design, which will lead to two different "standard lengths" in your system. I have seen various model railway systems that use both approaches. Note that if you decide to move from building just single track layouts to double track ones, you will need to add a second radius of curve, and the problem becomes more complicated. RomanSpa (talk) 01:08, 4 March 2014 (UTC)[reply]

The easiest way is to restrict angles to 90°, that way the lengths of straights will not depend on the circle radius. If you consider other angles things get trickier and you need irrational lengths. Consider the diagram based on 45°, if the radius is $r$ then the spacing of the grid lines is ${\frac {r}{\sqrt {2}}}$ , and the horizontal straight segments are all ${\frac {r}{\sqrt {2}}}$ , the diagonals are just $r$ . Hence you could probably get away with two lengths for the straights. If you go the 30° than you will need more lengths.--Salix alba (talk): 13:41, 4 March 2014 (UTC)[reply]

Hmmm - so the desire to be able to make a 'teardrop'-shaped loop (or a figure-8) is what drives the ratio of the lengths of the straight sections to the radius of the curves. That's an interesting observation. It also explains why some 'minimal' systems (such as the Duplo track that only has one straight, one curve and one turnout) uses a turnout design that's basically two of the curved sections merged into one part rather than one straight and one curve. That preserves the symmetry of a teardrop-shaped track - which eliminates any need for different lengths of straight section.

To be completely clear here - I'm making the track sections from scratch myself using a CAD system and a laser cutter, so this isn't a question of which commercial systems might work best. I have the luxury of a clean slate - I can make any curvature and length I want (although flex-track is not an option). Since we'll be selling the resulting parts, we want to keep the inventory of parts reasonably small - and I'm kinda hoping we can restrict the number of parts we use to just three (straight+curve+turnout) - but if there was a compelling reason to add one or two more designs to greatly enhance the flexibility of the system, then I'd consider it.

I don't think there will be much need for double tracks - so I'm not going to mess with different radii curves or get to grips with the mess that leaves you in with needing straight sections in more than one length.

Based on the thinking for the teardrop issue, it sounds like making two kinds of turnout would be the best way to improve flexibility.

SteveBaker (talk) 20:45, 4 March 2014 (UTC)[reply]

Play value is greatly enhanced if you make both left- and right-handed turnouts, as that makes it possible to add a passing loop at one end of a basic oval. (Depending on the connection method, you may also need to consider a variety of connection combinations.) RomanSpa (talk) 00:08, 5 March 2014 (UTC)[reply]

On a triangle grid, I think you can make a teardrop with just three kinds of track: straights of 2 grid units, 60° arcs of radius √3, and a switch combining the two (tangent at one end). —Tamfang (talk) 00:17, 5 March 2014 (UTC)[reply]

There does seem to be a need for different length straight lines to cope with the ways layouts connect. If one has curves that go 45° then one is stuck with people wanting a square root of two length as well as a unit length. The problem with 60° is that one can't make 90° and so get a right angle and people think in terms of rectangles, but otherwise it seems a good idea. If you use full 90° curves you could have a square for every element with track going from the centre of one side to the centre of another. Anyway A Boy and His Train (animation doesn't work on IE) shows a layout using 45° angles though the right hand one could be done easily using only 90°. [2] is one I think might look good with 60° curves. Dmcq (talk) 09:32, 5 March 2014 (UTC)[reply]

Right - but if I pick 30 degree curves, then I can still make 90 degree curves as well as making 60 degrees. (Also, from a practical perspective, 60 degree curves would be physically rather large for reasonable radii). SteveBaker (talk) 21:09, 5 March 2014 (UTC)[reply]

If you want right angles and diagonals you'll need straight lines with a √3 factor in them rather than the √2 for 45°. Dmcq (talk) 23:33, 5 March 2014 (UTC)[reply]

I would think 15° turn segments would be most practical, in that they can be assembled to make 30°, 45°, 60°, 90°, and many other angle turns. StuRat (talk) 17:25, 6 March 2014 (UTC)[reply]

One way of thinking about this would be to consider a Regular tiling of the plane. There are three Regular tilings made of either triangles, squares or hexagons. Consider the hexagon tiling on each piece draw a segment of track so that the ends always line up with the middle of the edges of the hexagons. This basically gives you three tiles, straight across, 60° curve and a point(turnout). As its based on a tiling the pieces are bound to fit together. The one thing you loose with this arrangement is that you can't make a 90° turn, but that might not be such a great restriction. You could do a similar thing with the square tiling giving straights 90° turns and points, or even triangles with 60° turns and Y-junctions.--Salix alba (talk): 09:57, 9 March 2014 (UTC)[reply]

The first of these three concepts is equivalent to mine above, except that a physical hex tile would take up extra space that could otherwise host another curve. —Tamfang (talk) 22:42, 9 March 2014 (UTC)[reply]