### The Volokh Conspiracy

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# Which Is Larger in Area, Colorado or Wyoming?

|The Volokh Conspiracy |

Hint: There's an interesting geometrical reason for the answer. Further hint: The first hint, coupled with your knowledge of how such puzzles are constructed, may suffice to give you the answer even if you didn't know, say, how many degrees of latitude or longitude each state's border traverses.

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1. Surface Area or 2 dimensional Map area?

1. Surface area; 2-dimensional map area would turn too much on what projection you use.

2. Is it Colorado? I think it’s Colorado. Is it Colorado?

1. Interestingly, if we eliminate the latitude effect by changing Kansas’ boundaries so that it becomes a latitudinal/longitudinal “twin” of Colorado, but one state eastward (with apologies to Nebraskans, Iowans & Missourians, or perhaps to Kansans), the answer would *still* arguably be Colorado.

Not because Colorado is “crinkly” with all its mountains and valleys (geographic areas aren’t – and really couldn’t be if you think about it – reckoned that way; rather, they are calculated as if all surfaces were smooth), but because Colorado is on average about a mile *higher* than Kansas, and as every:-) geometry student knows, a given number of square degrees on a 4001-mile-radius (ish) sphere has about 2/4000 more area than the same number of square degrees on a 4000-mile-radius sphere.

Sadly for the puzzle value of this particular flight of fancy*, the “arguably” was included advisedly; I doubt that regional average elevations are ever actually considered when calculating regional areas. Rather, I suspect that, for area-computation purposes, the earth is treated as a sphere, or at best as a “squashed” sphere (in which radius is a function only of latitude, being minimal at the poles and reaching its maximum at the equator).

*Thanks to Bruce Hayden 6.15.18 @ 8:43PM for the inspiration; I first thought he was making the above point, but I later realized he was talking about volumes, not areas.

3. Make it real challenge and add a legal spin to the question. Ask which has the greater volume. Who owns that which is under the surface, and for how deep. Likewise how much of the airspace is owned by the state/property owners, and how much is federalized.

1. My guess is Rost the average altitude of CO is higher, so combined with a slightly ghtl larger area, would make more volume, almost regardless of where the vertical cutoff was. What would be interesting, is if the average altitude of WY was greater, and calculating the vertical transition between each having a larger volume. It would be straight forward, if the borders were parallel vertically. But they aren’t quite, since they are closer to triangles, or maybe even pizza slice shaped.

4. Make it a real challenge, require an essay on who would care.

1. You call asking legal scholars to write an essay on a subject few care about a challenge?

2. The surveyors would care. Wyoming, being farther north, has to fit on a smaller area because it’s a rectangle projected onto the globe of the earth. Rectilinear surveying must make adjustment for lines of longitude getting closer together the farther north(or south) you measure.

5. The borders of both are defined as rectangles with the same degrees of latitude and longitude between their parallel sides, but Wyoming is further north, so its absolute area is less, as the lines of longitude converge the closer you get to the poles.

1. Interesting.

Here I thought they were defined by their previous existence as sovereign entities, each with its own specific culture, history, religious traditions, and so on.

Guess not.

1. That was a pretty stupid thing to think. That’s not even how they came up with the borders for the original 13 states.

1. I think bernard11’s comment was a somewhat misguided attempt at parodying a particular kind of state autonomy argument.

1. I don’t think it was misguided.

It was intended to mock a lot of federalist arguments about the sanctity of states’ rights and how we are a union of sovereign states who created the federal government instead of the other way around, etc.

By the way, Eugene, I’d be a lot more impressed with your zeal for free expression if you ever posted about the Trump Administration’s continuing attacks on the free press, instead of posting endlessly about kids wearing a t-shirt that someone doesn’t like.

Just a thought.

1. “I’d be a lot more impressed with your zeal for free expression if you ever posted about the Trump Administration’s continuing attacks on the free press, instead of posting endlessly about kids wearing a t-shirt that someone doesn’t like.”

IANAL, nor am I EV, but I’ll speculate:

1)Whether El Presidente can use the power of the government to punish the NYT for saying things he doesn’t like is not, I think, an interesting legal question: he can’t. In contrast, the law about what schools can do to kids is rather less settled, and thus more interesting to legal nerds.

2)El Presidente vs. the NYT is at least a big boy fight; the NYT can afford good lawyers and as many tanker trucks of ink as it likes. That’s rarely true of School Admin vs. Student. If one has a penchant for helping the underdog, speaking truth to power, etc, etc, the student needs the help more than the NYT does.

2. the Trump Administration’s continuing attacks on the free press

There are none. On the other hand, our heroic & noble PRESIDENT DONALD J. TRUMP does not suffer fake news and liars spreading hate, disinformation, and misinformation under the false premise that they are “press.” Nor should he.

3. Other than saying mean things like “failing” and “fake”, what attacks have there been?

Has he hacked into any PCs for running unapproved stories like they did to Cheryl Atkisson? Surveilled reporters as National security threats like James Rosen? tried to de-credentialize any orgs he didn’t like?

1. Just for starters, how about threatening Amazon because he doesn’t like what the WaPo writes about him?

Or Pruitt kicking reporters out of EPA presentations. Or Trump calling the media an enemy of the country.

Just for starters.

4. “I don’t think it was misguided.”

Me either. Like I said, I think it was stupid, not misguided.

“It was intended to mock a lot of federalist arguments”

Oh, I knew you thought you were being clever. A lot of people mistake stupidity for cleverness.

1. Oh really? in what sense was it wrong?

I’ve read countless comments – stupid ones, by the way – about how states are wondrous beings apart from their citizens, so that it is only just that they are entitled to equla representation in the Senate.

1. “Oh really? in what sense was it wrong?”

I already explained why your comment was so stupid.

“I’ve read countless comments – stupid ones, by the way – about how states are wondrous beings apart from their citizens, so that it is only just that they are entitled to equla representation in the Senate.”

It’s clear that you don’t understand the comments that you are trying to mock. Otherwise, you wouldn’t have made such a stupid comment trying to be clever. In order to be able to mock something effectively, you actually have to understand it. You don’t, so your juvenile attempt at gotcha just makes you look stupid.

That’s what we’re doing, talking about your comment.

2. That would be Texas!

2. TK-421: Correct.

3. Different latitudes and longitudes. Otherwise, Wyoming would be directly north of Colorado instead of being offset to the west. Rather, they each occupy seven degrees of longitude and four degrees of latitude. Since Wyoming is further north, it is area is less for the reason you mentioned.

4. That would be a correct analysis of the problem IF, and ONLY IF, one were to assume that the surface topology of both States were sufficiently similar so that the increased surface area of non-horizontal areas (i.e., mountains, which have a greater surface area than just the area of their bases) did not overcome any difference in the areas calculated merely based upon the distances from north to south and east to west of the border of these two quadrilateral States. But if one really wants to measure true surface area, one must account for topology – just as a simple example, the total summed surface areas of the four sides of a pyramid will be greater – perhaps much greater – than the area of the rectangular base of that pyramid. If you take a State like Colorado, over 50% of which is covered by mountains, its surface area is going to be materially greater than the area of the rectangular base of the state (and yes, I realize that neither Colorado or Wyoming is truly rectangular, since, as noted, the lines of longitude converge as they run north.)

1. Wouldn’t an attempt to measure surface area in that situation run into the 3D version of the coastline paradox?

1. That depends entirely upon the scale and level of acceptable error in measurement selected. Even measurements of coastlines yield finite results at scales as small as one foot, or even smaller. Measurements of the area of Colorado or Wyoming in a scale of acres (based upon on the ground surveys, for example) would yield finite results, but would show a much larger area than simply multiplying distances calculated based upon latitudes and longitudes. It’s just a matter of recognizing that there is ALWAYS an uncertainty of measurement, in whatever you attempt to measure, and that EVERY measurement has a limit on significant digits. Every measurement of the surface area of the Earth is just an approximation. We can estimate the total surface area of the Earth in square miles by using the formula for the surface area of a sphere (4piRsquared) by taking the mean radius of the Earth, but know that that estimate will be in errror to the extent that (1) the Earth is NOT truly spherical, in that the centrifugal force of Earth’s rotation forces Earth’s surface to bulge at the equator, and (2) surface deviations, like mountains, canyons, valleys, etc., make the surface deviate from that of a perfect sphere.

2. The coastline paradox is only a paradox if you think it’s possible to measure an infinitely small increment. Like Zeno’s paradox, the paradox lies in ignoring the physical world (for Zeno’s paradox, the solution is that it is a convergent infinite series). The current theoretically smallest possibly measurable distance is the Planck length (and even getting down to that point starts to get into quantum problems). So, in a distance future, there will be, almost certainly, a finite upper bound on the coastline measurement.

And the earth is still, at scale, flatter than a pancake (a spherical pancake, obviously. I’m not implying the Earth is flat…)

3. Well, of course one can measure surface area if one chooses a particular scale for the measurement The point of the coastline paradox, though, is that the results may vary sharply depending on whether one measures using a figurative yardstick or inch-measure (or anything else) — and, here, that therefore there may not be a well-defined answer for whether Wyoming has a larger area than Colorado, if one counts the bumpy parts, absent an arbitrary choice of scale for the bumps.

1. Exactly.

1. Go ahead, pick a scale. Even assuming a perfectly smooth sea surface, its contour is warped continuously by tides. That warping is in turn constrained by the shapes of ocean basins, which are, with continental drift figured in, not alike from day to day. It would be interesting to know, given the actual waves of the sea, and the tidal contour variation, how frequently there occur two moments when the earth’s surface area is the same, and by how much it varies.

Add to that sea level fluctuations, plus tectonic uplift and decline, affecting both the sea floor, and adjacent lands. Those result in water surface either extending over land, or withdrawing from land?with effects on the world’s surface area which would depend on whether the water in question was more or less rumpled than the land which was either submerged or exposed.

With change continuous, there is no point in picking any scale insufficiently large to deliver a repeatable result during the time frame contemplated. Bad news for climate change models, perhaps?or maybe for folks who demand too much accuracy from them.

1. A tremendous number of problems with climate science are driven by issues of scale and measurement error. A few examples: (1) the historical record of surface temperatures from 1880-2000, which shows warming trend of about 0.8 degrees C, is based, in very large part, on thermometer measurements taken by glass-mercury thermometers which, at BEST, had a measurement error of about 0.5 degrees C, without considering observation error; (2) satellite measurements of sea level changes are made with microwave frequencies with a wavelength of about 0.5 mm, while measured changes in sea level are running at about 2.5-3.0 mm per year, so the limit of actual measurement, plus or minus 0.5 mm, is about 17% of the annual rate of change; (3) the Earth has a total surface area (calculated by squaring the mean radius and multiplying by 4pi) of about 196,940,000 square miles; the most detailed Global Circulation Models in use divide that surfave area into a maximum of 3,600 cells – a 60×60 grid, meaning each individual cell has over 54,000 square miles, which is far too large to capture localized weather variations; imagine looking at a photograph at a resolution of only 6 pixels per square inch – you miss a WHOLE LOT of very important detail. Frankly, making policy judgments based on model predictions that sketchy is just insane, even assuming that both the data and the algorithms of multivariable interactions were complete and accurate (and neither of those assumptions is warranted).

6. Montana.

1. Sounding giraffes.

2. Are you a Dental Floss Tycoon? h/t to Frank Zappa.

7. colorado 104k square miles
wyoming 97k square miles

though wyoming being north would encompass greater long x lat

8. Seriously? Colorado is noticeably larger on every map I’ve examined.

9. Wyoming is further north, so its absolute area is less, as the lines of longitude converge the closer you get to the poles

Kyrie begs to differ.

10. It’s “farther”, not, “further”.
Guess that’s what happens when the Democrat Party runs the public schools.

1. Democratic Party, not Democrat Party if we’re correcting grammar.

11. If you mean surface area (e.g. the size of a sheet one would need to lay it flat over all the mountains and valleys in the state) I’m pretty sure it’s Colorado.

12. According to Google…wait, I wasn’t supposed to do that, was I?

13. From those hints, and without having seen any other comments, I will guess they cover the same span of latitude and longitude, which makes Colorado bigger as it is closer to the equator and its longitudinal lines are farther apart.

1. Right answer, and the right way to get information from the hints, I think.

1. Other than the clue that the question is, in fact, posed, I don’t see how to get the answer short of looking it up. The way it’s asked suggests that the latitude and longitude spans are the same, but is that the case? If the longitude spans are the same, but Wyoming has a larger latitude span, that ~could~ make up for longitudinal convergence.* It doesn’t, but I don’t see how I’d get that without more data.

Help me out here.

* For example, if Wyoming extended from Colorado to the Canadian border, it would then be a lot bigger than Colorado.

1. You’re certainly right that one can’t get this unless one knows the latitude and longitude spans are the same, or unless one knows something else (like, say, the area). But that’s why the hint, I think, is helpful — once you know that “There’s an interesting geometrical reason for the answer,” that strongly suggests that the latitude and longitude spans are the same, because then there’s a (possibly) interesting geometrical reason for the answer. If Wyoming spanned six degrees of longitude and Colorado spanned seven, then that wouldn’t be a very interesting geometrical reason.

1. Two days prior, on 6/13, I was browsing an atlas, and posed EV’s exact question?relative area of Colorado vs. Wyoming?to my son. In some ways, I think that’s more interesting, demonstrating as it does that unlikely-seeming coincidences are often seen anyway, not only because variations which generate them are so numerous, but also because the set of potential coincidences is practically infinite.

During that same atlas session, I came up with this question, “What is the western-most European nation which features within its contiguous boundary some point directly north of any point in Asia? I doubt any but citizens of that nation could answer off the top of their heads, and I wonder even about them.

1. Define “Asia.” Heck, define “Europe.” And what do you mean by “directly north of any point?” Do you mean some point north of “Asia’s” max latitude (say Greenland or Iceland, depending on how one wants to draw continental boundaries) or that for any point in Asia there is a point within this country which lies on the same line of longitude, eg Russia.

2. IIUC that answer has changed over time. 😉

14. Since this thread is on an off-topic subject, I hope you won’t mind if I use it as a place to ask about a topic the Conspiracy has yet to post about: the Inspector General’s report on the Clinton investigation.

The report seems to me to show pretty clearly that some FBI employees deliberately let Clinton “skate” on illegal activities because they favored her over Trump — in other words, because they were corrupt. And this included several grants of immunity that ought not to have been made.

So these questions arise: (1) How binding are those grants of immunity? (2) Is there some level of corruption that, if shown, would enable the prosecution of those FBI employees for granting the immunity, or do we have to let them all get away scot-free?

15. Colorado 104,094 sq miles
Wyoming 97,814 sq miles

This includes bodies of water.

16. Considering their borders are based on latitude and longitude, both having the same dimensions. They are not rectangular, but trapezoidal in shape with Colorado’s shorter northern boarder being the same length as Wyoming’s longer southern border. Therefore Colorado has a greater area.

17. According to Article III, doesn’t the Supreme Court have jurisdiction over, “. . . Controversies between two or more States. . . .”

1. ^^^ Priceless!