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November 27[edit]

Smallest area between three state capitals[edit]

Driving into work this morning I was contemplating which two US state capitals were closest together. Turns out it's Boston and Providence, but there are a few other close contenders, too.

Anyway, I got to thinking, what's the smallest area between 3 state capitals? I originally stipulated that they must be adjacent states, but frankly, if three capitals are on a perfectly straight line, that should be the winner. Generally, it looks like Boston-Concord-Montpelier is the winner so far, but I haven't been able to get accurate numbers for Springfield-Indianapolis-Columbus or Sacramento-Carson City-Salt Lake City, where being off by just a mile or so could be the difference between winning and losing. I may be leaving some others out too.

I had been using this to scout out potential winners, this to determine distances (not sure how accurate it is, a new source is probably necessary), and this to calculate areas once I got my distances. If anyone is up to the challenge, please post your findings!

Jared (t) 14:31, 27 November 2013 (UTC)[reply]

A quick look at a US map suggests that the answer is going to have to be in the northeast. It should be easy to find the distances between those cities, and then figure out which 3 points are closest. I see what you're getting at with the triplets of midwestern and western states, but they're far enough apart that the triangles formed by them might not be small enough to meet your criteria. In the western states they would need to be practically all on the same line (or the same Great Circle, technically). ←Baseball Bugs ^{What's up, Doc?} carrots→ 15:18, 27 November 2013 (UTC)[reply]

If the numbers are so close that a few miles makes the difference, I'd suggest that your real problem is figuring out where to measure from. Most cities are a bit wider than an infinitesimal point, so the question of how close they are is going to vary wildly depending on whether you're measuring from city center to city center (and how defined?), from closest border to closest border, or something else. Matt Deres (talk) 16:02, 27 November 2013 (UTC)[reply]

It's even worse than that -- to do it properly you'd have to use spherical trigonometry. I don't see any reasonable way to do this without writing a computer program to test all the combinations, and even then the answer is likely to depend delicately on the precise location assigned to each capital. (There are 19,600 possible triangles.) Looie496 (talk) 17:14, 27 November 2013 (UTC)[reply]

The straight line case of MA, NH, VT is boring. A much better question would be, by drawing a circle intersecting their state capitols, which three states define such a circle covering the least area. In this case Boston, Providence & Hartford versus Harrisburg, Dover and Annapolis seem the finalists. See this map. The New England trio form the smallest triangle, but the Midlantic trio are more equally spaced. μηδείς (talk) 17:46, 27 November 2013 (UTC)[reply]

Of course your statement that the question is boring/bad requires a special point to draw attention to it :-/ If you don't know the answer, or can't suggest references, it's ok to abstain. Happy thanksgiving to those who celebrate it this week! SemanticMantis (talk) 22:24, 27 November 2013 (UTC)[reply]

No, it's just boring. References aren't necessary, the candidates are obvious by inspection, and I am not bored enough to do junior high math. μηδείς (talk) 03:54, 28 November 2013 (UTC)[reply]

And your "better" version of the OP's question is what kind of math? High school at best. And neither of these questions has an obvious answer. Staecker (talk) 12:54, 28 November 2013 (UTC)[reply]

You seem to be looking for a fight, because the math I wasn't interested in doing was that answering my version of the question. μηδείς (talk) 01:30, 29 November 2013 (UTC)[reply]

OK- that's not clear at all from your comment. No picking here- Staecker (talk) 13:48, 29 November 2013 (UTC)[reply]

Some clarification from the OP, Jared (talk · contribs), would be nice. But given that that's his only edit in the last 6 months, I wouldn't hold my breath. ←Baseball Bugs ^{What's up, Doc?} carrots→ 13:39, 28 November 2013 (UTC) [reply]

This indicates that Boston-Concord-Montpelier is 155 mi, while Boston-Providence-Hartford is only 106 mi. ~Hue Sat Lum 17:42, 28 November 2013 (UTC)[reply]

That's a total straight-line distance. You want the areas of the triangles. MA-NH-VT is 545 sq mi, and MA-RI-CT is 1113 sq mi. (Couldn't figure out how to get Wolfram to do this with single queries. Staecker (talk) 13:56, 29 November 2013 (UTC)[reply]

According to Wolfram, the capitals for MD-VA-NC are exactly colinear (distances measured to nearest tenth of a mile). So that area would be zero- as said above the answer depends a lot on exactly where you locate the points on your triangle.

Thanks. I do see you have slogged through the math. Did you use Heron's Formula for the area of a triangle from its side's or something else? To find the are of the circle defined by the three points, a, b, & c, bisect the line segments a-b and b-c, and draw a line perpendicular to each from these points. Where the points meet will be on the opposite side of the circle, at a point d. The line segment b-d will be a diameter of this circle, from which its area can be calculated. See http://www.qc.edu.hk/math/Advanced%20Level/circle%20given%203%20points.htm. μηδείς (talk) 17:29, 29 November 2013 (UTC)[reply]

I haven't actually done any math- my links are to Wolfram Alpha which does all the computations for you. I didn't try to compute any of the circle areas, since the circle area seems to have very little to do with "how close together" the cities are, even less so than the triangle area. For instance consider three points right next next to each other, almost in a straight line but not quite. Then the circle given by those three points would be close to a great circle on the planet-- the area would be enormous. Something like this would be true in the case of MA-NH-VT. The circle for those three would be far larger than for example IN-OH-KY. In any case, it still seems to me that none of these questions have answers that are "obvious by inspection", and your question is no more or less "boring" than the OP's. Staecker (talk) 18:31, 29 November 2013 (UTC)[reply]

Where did it say those three points are on a single great-circle? ←Baseball Bugs ^{What's up, Doc?} carrots→ 14:33, 29 November 2013 (UTC)[reply]

Doesn't say it explicitly, but the length of two sides of the triangle add up exactly to the length of the third. That means (assuming flat geometry) the points are colinear. Staecker (talk) 15:11, 29 November 2013 (UTC)[reply]

That would certainly indicate that they are on the same great circle. In fact, any three points on any given great circle should equate to a "null" triangle, i.e. a triangle with area = 0. So it kind of comes back to what the OP really has in mind with this question. Is he looking for the 3 US capitals which are the closest to being co-linear with each other? Or is he looking for the 3 that are "closest together", as per Medeis' example? Either way, though, the answers seem to be here now. ←Baseball Bugs ^{What's up, Doc?} carrots→ 15:48, 29 November 2013 (UTC)[reply]

I think it's safe to assume the OP has in mind exactly what the OP said. "Smallest triangle area" is not the same as "closest to colinear". Medeis's question is something else entirely. Staecker (talk) 18:21, 29 November 2013 (UTC)[reply]

That's the danger of introducing something that has nothing to do with an OP's question, but is allegedly "a much better question". That may or not be serving the respondent's ego, but it has little or nothing to do with serving the OP. Sometimes OPs don't express themselves very well, and we can often do a better job of that than they did, in order to be clear about exactly what it is they're asking. But this diversion is not that. It's merely "a much better question". -- Jack of Oz ^{[pleasantries]} 20:37, 29 November 2013 (UTC)[reply]

Yes, the "danger". Lol. I don't think it's at all clear the OP even thought out his question, as different definitions of smallest give different answers, as shown. I also think the concept of convexity with its various senses is quite important in various aspects, such as defining a triangle that is likely to have a greater or lesser range of geophysical or weather variation. You who kvetch are quite free to complain my suggested three points that define the smallest circle is not what the OP said. (I think I pointed that out myself?) But unlike with the OP's question, you can't complain my criteria are undefined. I win again. μηδείς (talk) 22:03, 29 November 2013 (UTC)[reply]

You win? The OP's question is perfectly well-defined. Staecker (talk) 22:07, 29 November 2013 (UTC)[reply]

How about we have a talk about the shortest crow-fly route that visits the capitals of all the 48 contiguous states? Or whether there are an infinity of universes? Or how long a piece of string is? These have no less relevance to the OP's question than Medeis's circle question does, no matter how rigorously it may be defined. -- Jack of Oz ^{[pleasantries]} 00:26, 30 November 2013 (UTC)[reply]

Aren't three co-linear points effectively a "flattened" triangle with area = 0? ←Baseball Bugs ^{What's up, Doc?} carrots→ 20:49, 29 November 2013 (UTC)[reply]

Yes, but three points can form a very small triangle even if they're far from colinear (if you measure colinearity in terms of the angle formed by any two of the sides). Staecker (talk) 22:04, 29 November 2013 (UTC)[reply]

The plain truth is that the question is ambiguously stated, and we won't know what the OP really wants unless he responds... possibly 6 months from now. ←Baseball Bugs ^{What's up, Doc?} carrots→ 23:59, 29 November 2013 (UTC)[reply]

Thank you all for sharing your insight. I don't have any more detail to add because my original question was more of a thought experiment, leaving the solution open to the solver's interpretation. There are plenty of good ideas for how to determine the smallest area, some of which involve changing the original question slightly to make the answer more intriguing (e.g., the circle problem). I don't necessarily need a complete answer because as someone said, the math would be fairly trivial (unless of course you decide that using spherical trig is best for sake of completeness!). But, feel free to create your own restraints if you want to determine one possible solution. I may have quit editing long ago, but it's rarely six months between check-ups! Jared (t) 15:55, 2 December 2013 (UTC)[reply]