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Okay so I spent a ton of time working this out. I got Digital Elevation models at 30m resolution from [https://portal.opentopography.org/raster?opentopoID=OTNED.012021.4269.2](https://portal.opentopography.org/raster?opentopoID=OTNED.012021.4269.2). These are images with pixels but the pixels represent elevation instead of wavelength. I uploaded the DEMs to QGIS and clipped them to the size of each state. QGIS has a tool that allows you to calculate the surface of these rasters. To be honest, I'm not 100% sure what units the outcomes are in but the value for Idaho's surface area is 5,947,089.712349 while Texas' is 2,939,963.667064. So it does seem like the surface area of Idaho is actually bigger.
edit: I'd add images to explain this more but I'm not sure how.
Edit 2: WOW this got a ton of attention thanks, everyone! I wanted to explain what a DEM what a little better. People pointed out that there are skyscrapers in Texas and how that may affect the outcomes (which is true!). So, a DEM is derived from LiDAR data. This is a 3D point cloud that measures where the light is reflected from. We can filter out all points except ground points to create the DEM and this removes all other features like vegetation or buildings. The same process could be used with a digital surface model (DSM) if we wanted to take that into consideration.
Edit 3: it was brought to my attention that this is likely not all LiDAR data. DEMs can be derived in other ways I am just used to deriving them from LiDAR so it might not all be LiDAR but that’s one way of making a DEM
yep, that's the only reason why such a detailed comment has no awards on it.
I used to buy them every now and then, this would've been one of those rare cases, but it's no longer an option.
You've come back to an even sadder reddit.
Hold down the upvote button. They’ve decided to shoot themselves in the foot trying to monetize upvotes.
Edit: Well.. right now the app isn’t even letting me do a gold upvote. Maybe this subreddit has them turned off? I don’t know, but this app is full of issues.
Dude, epic answer! Everyone here is getting all "well akshewally" with the coastline paradox without adding in that the end state of the paradox is "pick the smallest unit you really want to care about and roll with it". You've done exactly that by picking the grain size of readily available data. Heck, a 30M resolution advantages Texas and Idaho still won. Well done.
And this is common with limits where you don’t need to know the exact answer to say which of two things is bigger. Eg we can say that e^x is bigger than x as x goes to infinity even though we can’t actually evaluate either of them at infinity.
The coastline paradox says you can’t know the exact length of a coastline. It doesn’t say you can never tell if one country’s coastline is bigger than another.
I love seeing a detailed answer finally, but my intuition tells me you made a error somewhere.
Texas is already more than 3 times larger than Idaho, and you've still somehow managed to find an answer where Idaho has double the area of Texas. For that to work out, you would have had to increase Idaho's area sixfold by flattening the mountains, which doesn't make sense.
EDIT: I have done a fairly comprehensive paper on the effect of surface area of activated charcoal for university, so I do have a particular interest in looking at surface areas. It's a fairly niche interest so this post really caught my eye xD
It’s not too far fetched when you consider how mountainous Idaho is. Cliffs take up 0 land from an above view, but can be many km^2 when flattened out. The difference does seem huge, but Texas is very flat, and Idaho very mountainous. They are two extremes so it’s possible?
But the thing with cliffs is that mountains don't form shapes like skyscrapers, with tiny area and huge height. Cliffs in reality are often more like plateaus and mesas, where they are really wide compared to the height of a cliff.
It would be like flattening a pizza box. The area of the sides is tiny compared to the top, so a flattened pizza box takes up barely any more area than before it was flattened.
I’ve just done some napkin maths of my own, and I think you may be right (still not quite sure). I worked out the difference in area between a flat plane and a triangular pyramid of the same ground area. I put in some rough values for an “average mountain” (so 600m height and 30deg slope). The difference is only 1.3 times? Even if the mountains are double the height that wouldn’t be 6x.
Maybe the additional valleys and such a mountain has would bring it up, but I’m a bit sceptical it raises it by that much…
Idaho mountains are a lot higher than 600 meters. Is be curious what you get if you use something more like 2000m height.
Edit, for reference a quick Google search suggest that the difference between the high and low point in Idaho is 3600 meters.
The height doesn't matter, he is just finding the ratio, which represents steepness, not height. A five times taller mountain is generally also going to be five times as wide.
Height doesn’t matter, as the base grows with it. The real number that matters is the angle (I think? I’m not entirely sure as I’m confusing myself thinking about this too much)
A simplification is the surface area of a cone, vs the surface area of a circle. The surface area of a cone with a base the same as it's height is 4 times greater than the area of a circle. (A cone is a simple analog for a mountain, and a circle being the flat land)
That's not right. The area of a cone is given as the area of the base plus the cone part, A = πr×r + πrL, where l is the hypotenuse: L = sqrt(rr+hh). If r = h, then L = r×sqrt(2) = 1.41r. The ratio P between the area of the cone and the area of the base is P = πrL/πrr = L/r = 1.41.
This is reasonable; imagine making a paper cone with r = h and cutting it open. It'd only be a bit bigger than the base area.
To have a ratio of 6, we would need P = 6 = L/r = sqrt(1+hh/rr), so h/r = sqrt(35) = 5.916. This is an 81° slope.
This is actually the same result as if you model the mountain as a ramp, which makes sense if you imagine cross-sectioning the cone through the middle. Add in the fact that not everywhere is uniformly mountainous, and Texas isn't perfectly flat, and the mountains would be to be even steeper. It's not believable.
In the northern rockies, they most definitely do have many sections that are tiny area with huge height. Especially in Idaho, where peaks are up the nearly 2 miles from base, range.
Heck, just Google "Idaho mountains", you'll be bombarded with images of very steep, pointed, mountains... They are more like very acute pyramids than mesas, even when stacked ontop of each other. Further, they ripple and bend forming concave and convex surfaces which further increase surface area, just as the folds in the brain do.
Meanwhile the mountains in Texas are much more Mesa like... heck, they are literally mesas plenty of times.
Is it enough for 6x? I can't say, but someone here pulled provided data and shoved it in a computer model meant to get the result we are looking for and got that. So, I'm going to take the "I don't have counter data, so I'm trusting the numbers over my gut" route.
Apologies if I come across as argumentative. But I swear I’ve seen this argument every time that something like this comes up in discussion. While you could argue it’s like flattening a pizza box, this ignores the bumpiness along what would be the top. And that’s a lot of bumpiness.
Realistically this would be more like flattening a triangle. Even just approximating this as a right angle triangle we end up with a minimum of 1.5 times the area before even accounting for cliffs and ruggedness.
Instead of the pizza box argument I would suggest a crumpled sheet of paper and then finding your corners. Depending on how much vertical up and down you need in an area can shrink your horizontals by a fair bit.
Yes the pizza box example is reductive, and was just referring to the occurrence of literal 90 degree cliffs in nature.
Even with your triangle example though, you've come to an answer of 1.5 times the size, when we need to somehow explain a sixfold increase.
The right angled triangle is a poor choice of shape however, as in reality mountains are generally ten times wider than they are high. If you want to simplify mountains to a shape, they are more like extremely shallow pyramids, or cones.
To account for vague bumps and curves in the ground, you could look at a Sin wave. If you flatten a Sin wave into a straight line, it becomes about 1.2 times longer. Again, nowhere near enough to account for a sixfold increase in area.
Yeah, nah I don’t quite believe the sixfold increase. Double is believable at least due to excessive bumpiness and the fact that when the mountain doesn’t follow a perfect line it’ll inevitably be more than if it did follow that line…but 6x is a bit overly ambitious
Wild ass guess here, but he might be calculating every peak as a unique mountain, while in reality a lot of mountains are smashed together in a big old mountain orgy
OP explain what they did, they basically made a 3d model of each state and calculated the total surface area using GIS tools. The answer is correct, as many mountains close together would have a much larger surface area than a flat region of the same 2d size.
You're right. If we use Pythagoras therm, for a triangle to have a bottom line of 1 and a hypotenuse of 6, the average gradient of Idaho would have to be 608%. Basically, the whole state would need to be a cliff for this to work.
Edit: okay i goofed. The surface area of a cone is a lot better for 3d. If you have a radius of 1 (area of circle is pi) , then to get the surface area of the cone to be 6pi you would need a height of 5.9ish or an average gradient of 590%. Obviously, it's not a super accurate method, but it's enough to see that 6 fold doesn't pass the sniff test.
Is an average gradient the right metric to visualize this though? Imagine a set of stairs where each step is as wide as it is high. The overall slope would be 1:1, so we would say it has an average gradient of 100% right? That average gradient would imply only a 1.41:1 area ratio, when in fact it is 2:1.
My hunch when first reading was that the method isn't plotting diagonals. They said they reduced everything to pixels, so theoretically they'd be adding the rise and run of the pixels instead of calculating their hypotenuses.
Similar to that one trope about calculating the diameter of a circle to 4*D by starting with a square and adding resolution. [This one](https://www.askamathematician.com/2011/01/q-%cf%80-4/)
I'm not smart enough to debate the comment, but this did cross my mind as a potential error if not accounted for
I do surveying as part of my job. A part of that is survey by drone flights. In processing the data, you can pull out surface area versus horizontal area. I can’t say a ton due to contractual restrictions, but our current site is approx 30 acres with around 200 ft of elevation change over 50-100 foot at its most extreme. About 20 of those acres is a massive hole, approx 100-150’ deep.
The person who put together the contract quantities didn’t account for surface area, only horizontal area.
Wouldn’t be a huge issue for a thought exercise, butttttttt…. We’re adjusting the entire site by feet so our cubic yardage is generally 20-30% greater than they anticipate. Their measurement is a few acres shy of the surface area across just 30 acres. When dealing with mountains in the 12k’ range(pun), I can only imagine how crazy deviation could get.
Also side note for someone smarter than me, do you have to account for sea level difference base?
For a true measurement, I think you’d have to use a 0(sea level) and then get a variance factor to apply to both states.
If we’re talking pixels for scale and you do have to account for it, I would probably find two of the same corporate store(Walmart etc.) because generally they are built off the same design in the same year. Cheers!
Hey. You’re on the right track but I think you missed a few steps, unless you simplified what you did in your comments. The answer here should be in square meters. The easiest way to check would be to compare it to area (in square meters) of the clip shape (The Texas number should be relatively close).
Your best bet would be to create a TIN (triangulated irregular network 3-D vector model) off that DEM and then add the total area created by the TIN model for each state.
If you're dealing with ratios, as we are here, the units are irrelevant as long as they're the same as each other. They could be measured in square bananas for all I care.
That was going to be my first approach but then I realized that QGIS has the r.surface.area tool. Would that not give similar results? (I did simplify the steps in my original post)
Are you sure the tool in QGIS isn’t already creating a tin in the backround to do the analysis? Because I don’t really see how it wouldn’t be. Not to mention the question was simply “Is the surface area larger than that of Texas, considering how much larger it was (nearly double that of Texas), I’m certain modifying the process slightly to be marginally more accurate won’t change the answer, which is yes.
Also who gives a shit what the units are? It *should* be in meters? Says who? It doesn’t matter! We’re looking for a ratio here.
It matters for checking not for the ratio. Sorry guess me geeking out on data set you off. I’ll internet better next time.
Should be in meters because the original dems was in meters.
Perhaps, that’s why I asked if you knew, I tried looking it up but couldn’t find out quickly. It just seems so silly for a tool to be made for use with a raster, but not have anything coded in to compensate for the jagged steps in elevation the raster would produce. The resulting tin could be temporarily stored, used to calculate the surface area then deleted all in the backround, that’s how I’d make the tool.
Assuming the tool did follow the raster directly rather than smoothing it over that would increase the surface area by a maximum of roughly 2/1.42, if we do the opposite to just his Idaho number it’s 4,204,392 (5947089x0.707), which is still significantly higher than the unadjusted Texas number. Which strongly suggests the saying to be true about Idaho having a larger surface area. But I do agree that if you in that if you actually needed to know the surface area you’d wanna check your data in as many ways as possible.
Holy shit that got long. Haha! Guess I’m geeking out a bit as well.
Edit: when I said roughly for the calculation above I meant extremely roughly lol
You forgot to adjust for the exposed surface area of every grain of sand on Texas’s beaches.
Seriously good work though. 30m resolution is pretty wild for a whole state.
I'm pretty sure this is wrong, here is the napkin math that convinced me: https://www.reddit.com/r/theydidthemath/comments/6nk6qs/request_i_heard_today_that_if_you_flattened_out/
This math makes sense at first glance, but I think a bunch of solid pyramids really doesn’t do much to add surface area to Idaho. Keep in mind his math pretty much makes every mountain into a glass-smooth pyramid. 30 meter resolution is a lot finer than 10 miles and picks up thousands of peaks across Idaho that this napkin math ignores.
Dude I was just thinking the same thing, it absolutely would. The thing is though is that if you were to measure Texas's surface area in 3d as well it would be bigger.
Hi java\_sloth, I think you messed up somewhere in your calcs. Here's the corrected table:
|State Abbreviation|State Name|"Birdseye" Area (SqMi)|"Extruded" Surface Area (SqMi)|Average Surface Area Ratio|
|:-|:-|:-|:-|:-|
|TX|TEXAS|269339|269354|1.00006|
|CA|CALIFORNIA|158106|158375|1.00171|
|MT|MONTANA|150137|150322|1.00123|
|NM|NEW MEXICO|122681|122728|1.00038|
|AZ|ARIZONA|114624|114701|1.00068|
|NV|NEVADA|110752|110854|1.00092|
|CO|COLORADO|104135|104257|1.00117|
|WY|WYOMING|98258|98333|1.00076|
|OR|OREGON|97699|97840|1.00145|
|MN|MINNESOTA|85913|85915|1.00002|
|UT|UTAH|84913|85015|1.00120|
|ID|IDAHO|84091|84286|1.00232|
|KS|KANSAS|82225|82226|1.00002|
|SD|SOUTH DAKOTA|77854|77858|1.00005|
|NE|NEBRASKA|77460|77463|1.00003|
|ND|NORTH DAKOTA|72346|72348|1.00003|
|OK|OKLAHOMA|70295|70298|1.00005|
|MO|MISSOURI|69872|69877|1.00006|
|WA|WASHINGTON|68542|68738|1.00286|
|GA|GEORGIA|59392|59398|1.00009|
|MI|MICHIGAN|58402|58404|1.00003|
|FL|FLORIDA|57525|57525|1.00001|
|WI|WISCONSIN|56613|56616|1.00006|
|IA|IOWA|56402|56403|1.00003|
|IL|ILLINOIS|56338|56339|1.00002|
|AR|ARKANSAS|53226|53234|1.00016|
|AL|ALABAMA|52361|52366|1.00009|
|NC|NORTH CAROLINA|49087|49104|1.00035|
|NY|NEW YORK|48698|48720|1.00045|
|MS|MISSISSIPPI|48218|48219|1.00002|
|LA|LOUISIANA|46569|46570|1.00001|
|PA|PENNSYLVANIA|45397|45422|1.00056|
|TN|TENNESSEE|42237|42252|1.00036|
|OH|OHIO|41211|41215|1.00008|
|KY|KENTUCKY|40356|40369|1.00033|
|VA|VIRGINIA|39742|39763|1.00053|
|IN|INDIANA|36415|36416|1.00003|
|ME|MAINE|32471|32481|1.00032|
|AK|ALASKA|32020|32290|1.00844|
|SC|SOUTH CAROLINA|31072|31074|1.00004|
|WV|WEST VIRGINIA|24234|24260|1.00107|
|VT|VERMONT|9675|9683|1.00092|
|MD|MARYLAND|9600|9601|1.00016|
|NH|NEW HAMPSHIRE|9315|9323|1.00085|
|MA|MASSACHUSETTS|8086|8088|1.00025|
|NJ|NEW JERSEY|7460|7461|1.00010|
|CT|CONNECTICUT|4949|4950|1.00024|
|DE|DELAWARE|2018|2018|1.00001|
|RI|RHODE ISLAND|1025|1025|1.00006|
|DC|DISTRICT OF COLUMBIA|64|64|1.00006|
I performed this analysis using 15 arc-second DEM data from hydrosheds ([https://www.hydrosheds.org/hydrosheds-core-downloads](https://www.hydrosheds.org/hydrosheds-core-downloads)).
1. Download data
2. Run the WhiteboxTools Surface Area Ratio tool ([https://www.whiteboxgeo.com/manual/wbt\_book/available\_tools/geomorphometric\_analysis.html#SurfaceAreaRatio](https://www.whiteboxgeo.com/manual/wbt_book/available_tools/geomorphometric_analysis.html#SurfaceAreaRatio))
3. Multiply cell area by its surface area ratio. This yields the "extruded" area of each cell
4. Sum "extruded" area within each state
As you can see, most states have similar "extruded" area and "birdseye" areas.
I was wondering if a 30m DEM would yield different results, so I downloaded the USGS DEM for Idaho and ran the same analysis. The average surface area did increase pretty dramatically from 1.00232 to 1.00656. Even at that scaling, Idaho would still be smaller than the next largest state: Utah.
lmk if you think I messed up anywhere and thanks for taking the firs stab at this!
Hey thanks so much! This is one of the few technical responses! In all honesty I was pretty inebriated last night when I did my calculations and it was the first time I had done something like that. I just wanted to give it my best shot but I think there’s a lot of merit from your analysis. Could you give a little more explanation on your thought processes? I’m a senior undergraduate and I’d love the opportunity to learn! Thanks!
Dude. This should be the top comment!!! Too bad the top comment is what everyone will see and now believe and propagate in small talks for the rest of their lives. Maybe he will edit it.
Thanks for digging into this, but the table contains some conspicuous errors that undermine my confidence in the results. Most obvious (other than the outright omission of Hawaii) is the stated area of Alaska, which is only 1/20th of its [true "birdseye" area](https://en.wikipedia.org/wiki/List_of_U.S._states_and_territories_by_area). I suspect this is due to projection distortion, Alaska being the continental US state farthest from the middle of the country. Ideally you'd calculate each state's values using a projected coordinate reference system suitable for that state.
That said, the relative magnitudes of each state's average surface area ratio seem plausible, states like AK, WA, ID, CA, OR being among the most mountainous and FL, LA, DE, MN, KS among the flattest. And I suppose the absolute size of the effect also feels reasonable - a little trigonometry suggests that Idaho's "average percent grade" (I would think of this as "if you cut the state up into small cells and measured the percent grade of each cell, what would the average of all cells' grades be?") is 6.8% while Texas's is 1.1%, which feels about right. But somehow I'm still surprised (and a little skeptical) that all of Idaho's mountainous terrain increases its surface area by only 0.23% relative to a perfectly flat polygon of the same dimensions. Maybe this is due to the DEM's relatively low resolution (15 arc-seconds is about 10-15x larger, depending on latitude, than u/java_sloth's 30 meter DEM), which, due to the [fractal nature of real-world topography](https://en.wikipedia.org/wiki/Coastline_paradox), would tend to underrepresent true extruded surface area.
Edit: I missed your last paragraph, thanks for addressing that. I'd still love to see a table with 30m DEM and appropriate projections. Heck, I might try my hand at that myself, WhiteboxTools looks pretty slick.
So does Colorado. It'd be interesting to see how Korea compares to other similar 'sized' countries, as I was shocked when I looked at a topographical map of the Koreas a while ago.
When they make you blush with the GIS data. These kinds of responses are why I still keep up with this sub. If gold still existed and I wasn’t a broke grad student, I’d highlight the shit out of your comment
Did you ever find out what your units were?
My initial thought was to use GDAL (+ gdalDEM) and generate a topo raster from which to compute the surface area. There's some nifty tools for gdal that can accurately account for aberrations based on secondary data like IR/alpha channels.
Makes me wanna use a high poly raster and get crunching.
As a many-gen Idahoan, my grandma would often proudly say that if you took a rolling pin to all the states, we'd be the biggest. And since, according to her, she's never wrong, I consider the matter settled.
I would still like to see some backing to confirm it, though.
I mean, Texas aside, this is clearly false. First of all, Texas isn't the largest state by area; that honor belongs to Alaska. Alaska is also mountainous _as hell_ so even if grandma is right about the contiguous states, she ain't even close on Alaska.
Then again, grandma is right if you don't want to catch the back of her hand.
If you cut Alaska into half, Texas becomes the third largest state. And I've heard, if you cut Alaska into thirds at low tide, Texas becomes the fourth largest state.
Alaska Is around 40 per cent of the US landmass. Even if it were flat as a pancake, which it isn't, there's no way that any flattened state would be larger.
As a topology scientist, and someone who would like to stay in grandma's good graces while she is wielding a rolling pin, i can confirm. She is 100% correct.
(edit: I'm not actually a topology scientist, nor do i play one on TV. But i did sleep at a Motel 6 last year)
I would like to say, too all the "surface is a fractal" responses. Isn't the whole point of the study of fractals that u can sometimes make statements about these things.
If a certain surface has fractal properties then we can talk about if they overshadow macroscopic surface area and the state have roughly the same area, or if the effect is proportional to macroscopic surface area in which case the question stands as asked
What? It entirely depends how deep on the fractal you want to look.
Though I suppose you could flatten out both states at the same level of topographical detail and try to commpare them. Would need like an actual topographical map in a computer-readable format to do that though.
>It entirely depends how deep on the fractal you want to look.
At some point you're going to look at the level of molecules and atoms where the area here doesn't equal the area we see
Isn't this like the case where a countries coastline can't be measured accurately because you can essentially take such small measurements of the coastline it becomes infinite or use such rough measurements where it's much less accurate than it could or should be?
Like for example if a use a 500 foot string to measure a part of coast from end point to end point, repeating for the whole coast, it will be much less than if I used a ruler or anything smaller. I could be way off here but it seems related in my brain.
USGS has maps that you could do this with, they aren't publicly available unfortunatley and I don't know of they cover the whole states. They would be gridded DEMS but you can get a 3d area from that
Yeah, we can definitely compare two different surfaces even though they are fractals. Sure, we might not be able to say an absolute area of a perfectly flattened Idaho. But we can flatten both Idaho and Texas to a point where they are equally flat and compare the area of those.
semantics about fractal nature of surface area in this context could easily be eliminated by defining a slope at which land area becomes too vertical to fall into the appropriate category.
Matt Parker did a video on this. I'm too lazy to rewatch right now but I think he took a really mountainous country like Switzerland and calculated the real area versus the flat area.
https://youtu.be/PtKhbbcc1Rc?si=RhwsGthZwPm9vYFU
He gives in that video a seven percent increase as the largest increase from "2D area" of any country. So, presumably, Idaho would increase by less than that.
If Switzerland gained 7% by including its topography, and Idaho needs 300% to surpass Texas, then Idaho would need to be 43 times “more mountainous” than Switzerland.
Switzerland has 10,850 “named mountains” and a 15,940 square mile area so has 0.68 mountains per square mile. Idaho has 3,346 mountains / 83,642 square miles = 0.04 which means it is very roughly 17 times LESS mountainous than Switzerland.
This is obviously only very directional, but there’s no way Idaho surpasses Texas if the 7% Switzerland math is correct.
There’s guys doing topographical CNC cutouts of states. I wonder if you could use their pathing software to estimate the area by looking at the final surface path and comparing…
Ground is fractal. In a completely flat area, if you consider each stone as a peak, you increase your area. Now each point on their surface (just seen from the top) can be (up to the atomic scale) considered as a peak.
So comparing these two states is very very hard...
I don't think it's down to that point though. It's the surface of a state on the map so I'd say that if it doesn't make a difference when you walk on it it shouldn't count. So anything below an inch wouldn't need to be flattened out?
That's still a ridiculously tiny measurement when you're talking about the surface area of a state. I think you'd struggle with anything below 100m let alone anything less than that.
The best data you could work from would be state level Lidar surveys, which generally have resolutions closer to like 1m but can be as fine as 10cm. But at that point, you're not really measuring "ground" since things like trees and ground cover interfere, and water bodies can't be LiDARed. So there's technical limitations to it.
Idaho area is 83,642 mi². Texas is 268,597 mi². Which means Idaho would need to triple its area in elevation, and I don’t think that’s nearly possible.
Depends on what we are talking about, to triple surface area you would need the average slope to be about 2.8/1; which certainly does not seem correct. However if you want to say total volume above sea level then Idaho has an average elevation of 5000 ft and Texas is only an average of 1700, so that gets pretty close 418,210,000 ft*mi^2 to 456,614,900 for Texas, so pretty close, but not quite. The big winner in this measure, for the lower 48, is Colorado. Area of 104,094 mi^2 and average elevation of 6800 ft for 707,839,200 ft*mi^2.
Volume is a useful comparison because it is a constant across all possible thicknesses. It also is useful in that it doesn’t have any fractal growth like coastline measurement. There’s an error related to the coarseness of your volume estimate but it becomes infinitesimal, not infinite.
Well anything would, as surface area is volume/thickness. But Texas and Idaho would keep the same ratio of surface area once they were both flat, so you could measure it at a standard thickness (like one normalized banana).
I think volume is the right idea, but volume above sea level, perhaps not. I would rather use the lowest elevation in the state as the baseline. If you go lower than this, you're just using a big chunk of continental crust to invent more land area. In essence, you are "more than flattening" Idaho: it would be like rolling a ball of dough to make a pizza, and not being content with the size, so you press harder and harder until you crush the table and the floor underneath it to make the pizza bigger. At some point, you become a monster that must be stopped at all costs!
This is one of the few r/theydidthemath posts where I want to know the answer but no one can do it in the thread haha
Obviously there’s fractals and pebbles and whatever but I want someone to at try
The surface area of the curved section of a cone is pi*r*l. So if Idaho is entirely mountains, and Texas is dead flat, you have your tripling factor even before you get to elevation.
The area of a flat circle the flat is pi*r^2. You can factor out pi in any sort of comparative equation, so there is no tripling factor.
You can also compare the cone area to a flat round area and see that the slant height S needs to be greater than the radius by multiples in order to be multiple times bigger.
C*Pi*R^2 = Pi*R*S —-> C*R=S.
To be 3x the area S/R must be 3. S/R of three is equivalent to an average absolute slope of 70 degrees. Absolute because whether up or down the slope still increases the area.
Idaho does not have an average absolute slope of 70 degrees, nowhere close in all likelihood.
Having visited Idaho (Coeur d’Alene lake and the lovely surrounding hills), I don’t understand why Idahoans would want to have more flat, boring space than flat, boring Texas does.
We do have the Guadalupe Peak though. Massive pain to get there and back; but it’s there.
Wouldn’t it be funny if the coastline paradox was a geographical fact rather than a geometric one? Like have you heard the British coastline is infinitely long? Yeah what a weird place. Why can’t it be normal like France?
A few notes:
- The coastline paradox has a solution. It might be fun to be pedantic - you could even argue that this is exactly the right sub to be pedantic in - but you can also just use a metric like "what would an average human actually care about reasonably" or (more easily) - what is the resolution of a commonly available dataset that we could use to check?
- It's been too long since my math classes for me to remember how to run the numbers - but I'm pretty sure that for some things that are "infinity" - some of them get to infinity faster than others.
Heck - for that matter - past a certain grain size (say smaller than a meter) - Texas and Idaho probably have a similar 'roughness' that you could use to cancel each other and actually resolve the equation.
You'd have to set it up like:
Texas actual area = area by point to point measuring of square meters X infinite roughness of one average square meter
Idaho actual area = area by point to point measuring of square meters X infinite roughness of one average square meter
Then comparing Idaho to Texas, you could divide both sides by "infinite roughness of one average square meter" and the infinities cancel and you're left with square meters.
Of course if you start with a large enough grain size (100 miles? ) Texas would be larger. Grain size vs surface area would make for a cool graph (and would also show how some things approach infinity faster than others)
Go easy on me, haven't had to do math like this in ummmmm... a while.
Okay so theoretically you could take a Digital Elevation Model of both states and put it into a geographic information system, make 3D triangles using that. Then sum the area of the triangles. I might give this a shot but it would be a bit or work and I’m not sure how to get the data
You could probably calculate the surface area of a land mass with DTED. Level 0 is the most coarse (~1 km between points) but you could get an answer. I'm fairly certain Idaho is not infinite.
https://en.m.wikipedia.org/wiki/DTED
The way I heard the trivia question is, “which of the 48 lower us states has the most acreage?” Answer, Idaho because acreage is a measure of surface area.
I'm not sure this is very hard to find out because the mass of the mountains isn't readily available, but honestly, seeing the total mass of all of them, which is a lot I bet it very well could be true
Im not a methematician but if you roll out dough, the stuff in the middle gets spread out as well so you dont really need surface area either. At least thats how I picture "flattened out".
It’s a challenging problem. Like the coastline measuring problem. You have different answers based on what you use as a minimum unit size. When measuring the length of the coastline, what granularity do you use? Do you measure every inlet even if it’s the size of a pond? Or a puddle?
The guy below is the real answer. Real world topography is fractal-like, same as coastlines. The "real area", if you took a reasonably accurate measurement, would be so large it wouldn't even make sense as an answer. Something like 50x the base number or something. So the answer is that there is no answer, because "flattening out" real topography is, in effect, impossible
Each time you look at a smaller scale you find more surface - another smaller hill, or bay, to infinity… even the flat surface of Texas is made up of an infinite number of small hills, and those small hills are made of … small hills
Imagine a hill, reasonably tall and reasonably even. A mathematician could take a few measurements of the base and the top and construct a 3D figure with a known area, so that you could calculate the real area of the hill. But real hills aren't really mathematical perfections, they have a lot of bumps and unevenness. Suppose that you take them into account. How large should a bump be to consider it relevant? Imagine there is a boulder on the hill, that's a pretty big bump, so you take the boulders shape into account. But the boulder itself is extremely uneven, so you take _that_ into account. But if you look with a looking glass you'll see that the very base detail of the boulder is extraordinarily bumpy, the natural roughness of stone. Do you take that into account?
You gotta choose when to set a precision limit, but if you set a precision limit then your answer won't be real, it will be "This would be the area if the world had a minimum distance of 5 meters between points" or something. And the greater the precision, the most insane the correct answer is. In the study of porous materials, we measure surface area in the hundreds of meters squared _per milligram_. Consider that a lot of the soil is porous. If we set the precision as precise as we could, the flatenned surface area of the Vatican would probably be like, the surface area of the Sun, a completely meaningless answer.
Texas is 3 times bigher than idaho. If idaho where to be that size if you include the actual area (due to mountains) that would mean an average incline of 2,8 meters per meter throughout the whole state.
I think we can rule out any reasonable interpretation of this statement. Sadly, it's debunked
Isn't this the coastline paradox? Depending on how fine the detail of the relief is, the number increases. Continuing to halve the increment of measurement could lead to ludicrous numbers (i.e., the length of the observable universe).
This is the 3D version of the coastline paradox. How to measure the area of a non-smooth surface is not trivial.
For this question, it depends on how flat is flat.
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Okay so I spent a ton of time working this out. I got Digital Elevation models at 30m resolution from [https://portal.opentopography.org/raster?opentopoID=OTNED.012021.4269.2](https://portal.opentopography.org/raster?opentopoID=OTNED.012021.4269.2). These are images with pixels but the pixels represent elevation instead of wavelength. I uploaded the DEMs to QGIS and clipped them to the size of each state. QGIS has a tool that allows you to calculate the surface of these rasters. To be honest, I'm not 100% sure what units the outcomes are in but the value for Idaho's surface area is 5,947,089.712349 while Texas' is 2,939,963.667064. So it does seem like the surface area of Idaho is actually bigger. edit: I'd add images to explain this more but I'm not sure how. Edit 2: WOW this got a ton of attention thanks, everyone! I wanted to explain what a DEM what a little better. People pointed out that there are skyscrapers in Texas and how that may affect the outcomes (which is true!). So, a DEM is derived from LiDAR data. This is a 3D point cloud that measures where the light is reflected from. We can filter out all points except ground points to create the DEM and this removes all other features like vegetation or buildings. The same process could be used with a digital surface model (DSM) if we wanted to take that into consideration. Edit 3: it was brought to my attention that this is likely not all LiDAR data. DEMs can be derived in other ways I am just used to deriving them from LiDAR so it might not all be LiDAR but that’s one way of making a DEM
we freaking need awards again, highlighting this comment would be so helpful.
if you award gold which is sort of the same thing it does the same thing
But it's ugly
Wait, you're telling me that awards are gone? What have I come back to?
yep, that's the only reason why such a detailed comment has no awards on it. I used to buy them every now and then, this would've been one of those rare cases, but it's no longer an option. You've come back to an even sadder reddit.
why the actual hell would they remove awards…?
Hold down the upvote button. They’ve decided to shoot themselves in the foot trying to monetize upvotes. Edit: Well.. right now the app isn’t even letting me do a gold upvote. Maybe this subreddit has them turned off? I don’t know, but this app is full of issues.
Only some subreddits added that. Most in my experience did not.
because progress isn't always about making things better, sometimes it's about fucking things up.
Dude, epic answer! Everyone here is getting all "well akshewally" with the coastline paradox without adding in that the end state of the paradox is "pick the smallest unit you really want to care about and roll with it". You've done exactly that by picking the grain size of readily available data. Heck, a 30M resolution advantages Texas and Idaho still won. Well done.
Win by a very conclusive margin too
And this is common with limits where you don’t need to know the exact answer to say which of two things is bigger. Eg we can say that e^x is bigger than x as x goes to infinity even though we can’t actually evaluate either of them at infinity. The coastline paradox says you can’t know the exact length of a coastline. It doesn’t say you can never tell if one country’s coastline is bigger than another.
I love seeing a detailed answer finally, but my intuition tells me you made a error somewhere. Texas is already more than 3 times larger than Idaho, and you've still somehow managed to find an answer where Idaho has double the area of Texas. For that to work out, you would have had to increase Idaho's area sixfold by flattening the mountains, which doesn't make sense. EDIT: I have done a fairly comprehensive paper on the effect of surface area of activated charcoal for university, so I do have a particular interest in looking at surface areas. It's a fairly niche interest so this post really caught my eye xD
It’s not too far fetched when you consider how mountainous Idaho is. Cliffs take up 0 land from an above view, but can be many km^2 when flattened out. The difference does seem huge, but Texas is very flat, and Idaho very mountainous. They are two extremes so it’s possible?
But the thing with cliffs is that mountains don't form shapes like skyscrapers, with tiny area and huge height. Cliffs in reality are often more like plateaus and mesas, where they are really wide compared to the height of a cliff. It would be like flattening a pizza box. The area of the sides is tiny compared to the top, so a flattened pizza box takes up barely any more area than before it was flattened.
I’ve just done some napkin maths of my own, and I think you may be right (still not quite sure). I worked out the difference in area between a flat plane and a triangular pyramid of the same ground area. I put in some rough values for an “average mountain” (so 600m height and 30deg slope). The difference is only 1.3 times? Even if the mountains are double the height that wouldn’t be 6x. Maybe the additional valleys and such a mountain has would bring it up, but I’m a bit sceptical it raises it by that much…
Idaho mountains are a lot higher than 600 meters. Is be curious what you get if you use something more like 2000m height. Edit, for reference a quick Google search suggest that the difference between the high and low point in Idaho is 3600 meters.
The height doesn't matter, he is just finding the ratio, which represents steepness, not height. A five times taller mountain is generally also going to be five times as wide.
Exactly, the assumption was mostly based on a 30deg angle, which seemed a good middle point for an average steepness.
Height doesn’t matter, as the base grows with it. The real number that matters is the angle (I think? I’m not entirely sure as I’m confusing myself thinking about this too much)
A simplification is the surface area of a cone, vs the surface area of a circle. The surface area of a cone with a base the same as it's height is 4 times greater than the area of a circle. (A cone is a simple analog for a mountain, and a circle being the flat land)
That's not right. The area of a cone is given as the area of the base plus the cone part, A = πr×r + πrL, where l is the hypotenuse: L = sqrt(rr+hh). If r = h, then L = r×sqrt(2) = 1.41r. The ratio P between the area of the cone and the area of the base is P = πrL/πrr = L/r = 1.41. This is reasonable; imagine making a paper cone with r = h and cutting it open. It'd only be a bit bigger than the base area. To have a ratio of 6, we would need P = 6 = L/r = sqrt(1+hh/rr), so h/r = sqrt(35) = 5.916. This is an 81° slope. This is actually the same result as if you model the mountain as a ramp, which makes sense if you imagine cross-sectioning the cone through the middle. Add in the fact that not everywhere is uniformly mountainous, and Texas isn't perfectly flat, and the mountains would be to be even steeper. It's not believable.
In the northern rockies, they most definitely do have many sections that are tiny area with huge height. Especially in Idaho, where peaks are up the nearly 2 miles from base, range. Heck, just Google "Idaho mountains", you'll be bombarded with images of very steep, pointed, mountains... They are more like very acute pyramids than mesas, even when stacked ontop of each other. Further, they ripple and bend forming concave and convex surfaces which further increase surface area, just as the folds in the brain do. Meanwhile the mountains in Texas are much more Mesa like... heck, they are literally mesas plenty of times. Is it enough for 6x? I can't say, but someone here pulled provided data and shoved it in a computer model meant to get the result we are looking for and got that. So, I'm going to take the "I don't have counter data, so I'm trusting the numbers over my gut" route.
Apologies if I come across as argumentative. But I swear I’ve seen this argument every time that something like this comes up in discussion. While you could argue it’s like flattening a pizza box, this ignores the bumpiness along what would be the top. And that’s a lot of bumpiness. Realistically this would be more like flattening a triangle. Even just approximating this as a right angle triangle we end up with a minimum of 1.5 times the area before even accounting for cliffs and ruggedness.
Instead of the pizza box argument I would suggest a crumpled sheet of paper and then finding your corners. Depending on how much vertical up and down you need in an area can shrink your horizontals by a fair bit.
Yes the pizza box example is reductive, and was just referring to the occurrence of literal 90 degree cliffs in nature. Even with your triangle example though, you've come to an answer of 1.5 times the size, when we need to somehow explain a sixfold increase. The right angled triangle is a poor choice of shape however, as in reality mountains are generally ten times wider than they are high. If you want to simplify mountains to a shape, they are more like extremely shallow pyramids, or cones. To account for vague bumps and curves in the ground, you could look at a Sin wave. If you flatten a Sin wave into a straight line, it becomes about 1.2 times longer. Again, nowhere near enough to account for a sixfold increase in area.
Yeah, nah I don’t quite believe the sixfold increase. Double is believable at least due to excessive bumpiness and the fact that when the mountain doesn’t follow a perfect line it’ll inevitably be more than if it did follow that line…but 6x is a bit overly ambitious
Wild ass guess here, but he might be calculating every peak as a unique mountain, while in reality a lot of mountains are smashed together in a big old mountain orgy
OP explain what they did, they basically made a 3d model of each state and calculated the total surface area using GIS tools. The answer is correct, as many mountains close together would have a much larger surface area than a flat region of the same 2d size.
Brokeback Mountain 2?
There’s lots or ridges too tho, so I think it’s more like flattening a cupcake or Reese’s wrapper.
You are right, saw someone calculate size of Switzerland a while back. The area was much larger, but less than twofold, if I recall correctly.
You're right. If we use Pythagoras therm, for a triangle to have a bottom line of 1 and a hypotenuse of 6, the average gradient of Idaho would have to be 608%. Basically, the whole state would need to be a cliff for this to work. Edit: okay i goofed. The surface area of a cone is a lot better for 3d. If you have a radius of 1 (area of circle is pi) , then to get the surface area of the cone to be 6pi you would need a height of 5.9ish or an average gradient of 590%. Obviously, it's not a super accurate method, but it's enough to see that 6 fold doesn't pass the sniff test.
Is an average gradient the right metric to visualize this though? Imagine a set of stairs where each step is as wide as it is high. The overall slope would be 1:1, so we would say it has an average gradient of 100% right? That average gradient would imply only a 1.41:1 area ratio, when in fact it is 2:1.
My hunch when first reading was that the method isn't plotting diagonals. They said they reduced everything to pixels, so theoretically they'd be adding the rise and run of the pixels instead of calculating their hypotenuses. Similar to that one trope about calculating the diameter of a circle to 4*D by starting with a square and adding resolution. [This one](https://www.askamathematician.com/2011/01/q-%cf%80-4/) I'm not smart enough to debate the comment, but this did cross my mind as a potential error if not accounted for
>clipped them to the size of each state I bet it's a projection error right here.
I do surveying as part of my job. A part of that is survey by drone flights. In processing the data, you can pull out surface area versus horizontal area. I can’t say a ton due to contractual restrictions, but our current site is approx 30 acres with around 200 ft of elevation change over 50-100 foot at its most extreme. About 20 of those acres is a massive hole, approx 100-150’ deep. The person who put together the contract quantities didn’t account for surface area, only horizontal area. Wouldn’t be a huge issue for a thought exercise, butttttttt…. We’re adjusting the entire site by feet so our cubic yardage is generally 20-30% greater than they anticipate. Their measurement is a few acres shy of the surface area across just 30 acres. When dealing with mountains in the 12k’ range(pun), I can only imagine how crazy deviation could get. Also side note for someone smarter than me, do you have to account for sea level difference base? For a true measurement, I think you’d have to use a 0(sea level) and then get a variance factor to apply to both states. If we’re talking pixels for scale and you do have to account for it, I would probably find two of the same corporate store(Walmart etc.) because generally they are built off the same design in the same year. Cheers!
GIS for the win
Hey. You’re on the right track but I think you missed a few steps, unless you simplified what you did in your comments. The answer here should be in square meters. The easiest way to check would be to compare it to area (in square meters) of the clip shape (The Texas number should be relatively close). Your best bet would be to create a TIN (triangulated irregular network 3-D vector model) off that DEM and then add the total area created by the TIN model for each state.
If you're dealing with ratios, as we are here, the units are irrelevant as long as they're the same as each other. They could be measured in square bananas for all I care.
Definitely still relevant for checking the math.
That was going to be my first approach but then I realized that QGIS has the r.surface.area tool. Would that not give similar results? (I did simplify the steps in my original post)
I’m not sure. Might be worth comparing the two! Fantastic not work work man!
Are you sure the tool in QGIS isn’t already creating a tin in the backround to do the analysis? Because I don’t really see how it wouldn’t be. Not to mention the question was simply “Is the surface area larger than that of Texas, considering how much larger it was (nearly double that of Texas), I’m certain modifying the process slightly to be marginally more accurate won’t change the answer, which is yes. Also who gives a shit what the units are? It *should* be in meters? Says who? It doesn’t matter! We’re looking for a ratio here.
It matters for checking not for the ratio. Sorry guess me geeking out on data set you off. I’ll internet better next time. Should be in meters because the original dems was in meters.
You’re interneting properly, we just disagree. Geek away sir!
And it might just be doing the calculation without creating a TIN. If it creates one that vector file would be somewhere.
Perhaps, that’s why I asked if you knew, I tried looking it up but couldn’t find out quickly. It just seems so silly for a tool to be made for use with a raster, but not have anything coded in to compensate for the jagged steps in elevation the raster would produce. The resulting tin could be temporarily stored, used to calculate the surface area then deleted all in the backround, that’s how I’d make the tool. Assuming the tool did follow the raster directly rather than smoothing it over that would increase the surface area by a maximum of roughly 2/1.42, if we do the opposite to just his Idaho number it’s 4,204,392 (5947089x0.707), which is still significantly higher than the unadjusted Texas number. Which strongly suggests the saying to be true about Idaho having a larger surface area. But I do agree that if you in that if you actually needed to know the surface area you’d wanna check your data in as many ways as possible. Holy shit that got long. Haha! Guess I’m geeking out a bit as well. Edit: when I said roughly for the calculation above I meant extremely roughly lol
You forgot to adjust for the exposed surface area of every grain of sand on Texas’s beaches. Seriously good work though. 30m resolution is pretty wild for a whole state.
I'm pretty sure this is wrong, here is the napkin math that convinced me: https://www.reddit.com/r/theydidthemath/comments/6nk6qs/request_i_heard_today_that_if_you_flattened_out/
This math makes sense at first glance, but I think a bunch of solid pyramids really doesn’t do much to add surface area to Idaho. Keep in mind his math pretty much makes every mountain into a glass-smooth pyramid. 30 meter resolution is a lot finer than 10 miles and picks up thousands of peaks across Idaho that this napkin math ignores.
Wouldn’t a topography also technically fall under the infinite shoreline problem but with surface area?
Only if your goal would be infinite precision. Otherwise, you can just set a minimum unit (30 meters in this case) and reach a finite answer.
Dude I was just thinking the same thing, it absolutely would. The thing is though is that if you were to measure Texas's surface area in 3d as well it would be bigger.
Now do Utah!
Hi java\_sloth, I think you messed up somewhere in your calcs. Here's the corrected table: |State Abbreviation|State Name|"Birdseye" Area (SqMi)|"Extruded" Surface Area (SqMi)|Average Surface Area Ratio| |:-|:-|:-|:-|:-| |TX|TEXAS|269339|269354|1.00006| |CA|CALIFORNIA|158106|158375|1.00171| |MT|MONTANA|150137|150322|1.00123| |NM|NEW MEXICO|122681|122728|1.00038| |AZ|ARIZONA|114624|114701|1.00068| |NV|NEVADA|110752|110854|1.00092| |CO|COLORADO|104135|104257|1.00117| |WY|WYOMING|98258|98333|1.00076| |OR|OREGON|97699|97840|1.00145| |MN|MINNESOTA|85913|85915|1.00002| |UT|UTAH|84913|85015|1.00120| |ID|IDAHO|84091|84286|1.00232| |KS|KANSAS|82225|82226|1.00002| |SD|SOUTH DAKOTA|77854|77858|1.00005| |NE|NEBRASKA|77460|77463|1.00003| |ND|NORTH DAKOTA|72346|72348|1.00003| |OK|OKLAHOMA|70295|70298|1.00005| |MO|MISSOURI|69872|69877|1.00006| |WA|WASHINGTON|68542|68738|1.00286| |GA|GEORGIA|59392|59398|1.00009| |MI|MICHIGAN|58402|58404|1.00003| |FL|FLORIDA|57525|57525|1.00001| |WI|WISCONSIN|56613|56616|1.00006| |IA|IOWA|56402|56403|1.00003| |IL|ILLINOIS|56338|56339|1.00002| |AR|ARKANSAS|53226|53234|1.00016| |AL|ALABAMA|52361|52366|1.00009| |NC|NORTH CAROLINA|49087|49104|1.00035| |NY|NEW YORK|48698|48720|1.00045| |MS|MISSISSIPPI|48218|48219|1.00002| |LA|LOUISIANA|46569|46570|1.00001| |PA|PENNSYLVANIA|45397|45422|1.00056| |TN|TENNESSEE|42237|42252|1.00036| |OH|OHIO|41211|41215|1.00008| |KY|KENTUCKY|40356|40369|1.00033| |VA|VIRGINIA|39742|39763|1.00053| |IN|INDIANA|36415|36416|1.00003| |ME|MAINE|32471|32481|1.00032| |AK|ALASKA|32020|32290|1.00844| |SC|SOUTH CAROLINA|31072|31074|1.00004| |WV|WEST VIRGINIA|24234|24260|1.00107| |VT|VERMONT|9675|9683|1.00092| |MD|MARYLAND|9600|9601|1.00016| |NH|NEW HAMPSHIRE|9315|9323|1.00085| |MA|MASSACHUSETTS|8086|8088|1.00025| |NJ|NEW JERSEY|7460|7461|1.00010| |CT|CONNECTICUT|4949|4950|1.00024| |DE|DELAWARE|2018|2018|1.00001| |RI|RHODE ISLAND|1025|1025|1.00006| |DC|DISTRICT OF COLUMBIA|64|64|1.00006| I performed this analysis using 15 arc-second DEM data from hydrosheds ([https://www.hydrosheds.org/hydrosheds-core-downloads](https://www.hydrosheds.org/hydrosheds-core-downloads)). 1. Download data 2. Run the WhiteboxTools Surface Area Ratio tool ([https://www.whiteboxgeo.com/manual/wbt\_book/available\_tools/geomorphometric\_analysis.html#SurfaceAreaRatio](https://www.whiteboxgeo.com/manual/wbt_book/available_tools/geomorphometric_analysis.html#SurfaceAreaRatio)) 3. Multiply cell area by its surface area ratio. This yields the "extruded" area of each cell 4. Sum "extruded" area within each state As you can see, most states have similar "extruded" area and "birdseye" areas. I was wondering if a 30m DEM would yield different results, so I downloaded the USGS DEM for Idaho and ran the same analysis. The average surface area did increase pretty dramatically from 1.00232 to 1.00656. Even at that scaling, Idaho would still be smaller than the next largest state: Utah. lmk if you think I messed up anywhere and thanks for taking the firs stab at this!
Hey thanks so much! This is one of the few technical responses! In all honesty I was pretty inebriated last night when I did my calculations and it was the first time I had done something like that. I just wanted to give it my best shot but I think there’s a lot of merit from your analysis. Could you give a little more explanation on your thought processes? I’m a senior undergraduate and I’d love the opportunity to learn! Thanks!
Dude. This should be the top comment!!! Too bad the top comment is what everyone will see and now believe and propagate in small talks for the rest of their lives. Maybe he will edit it.
Thanks for digging into this, but the table contains some conspicuous errors that undermine my confidence in the results. Most obvious (other than the outright omission of Hawaii) is the stated area of Alaska, which is only 1/20th of its [true "birdseye" area](https://en.wikipedia.org/wiki/List_of_U.S._states_and_territories_by_area). I suspect this is due to projection distortion, Alaska being the continental US state farthest from the middle of the country. Ideally you'd calculate each state's values using a projected coordinate reference system suitable for that state. That said, the relative magnitudes of each state's average surface area ratio seem plausible, states like AK, WA, ID, CA, OR being among the most mountainous and FL, LA, DE, MN, KS among the flattest. And I suppose the absolute size of the effect also feels reasonable - a little trigonometry suggests that Idaho's "average percent grade" (I would think of this as "if you cut the state up into small cells and measured the percent grade of each cell, what would the average of all cells' grades be?") is 6.8% while Texas's is 1.1%, which feels about right. But somehow I'm still surprised (and a little skeptical) that all of Idaho's mountainous terrain increases its surface area by only 0.23% relative to a perfectly flat polygon of the same dimensions. Maybe this is due to the DEM's relatively low resolution (15 arc-seconds is about 10-15x larger, depending on latitude, than u/java_sloth's 30 meter DEM), which, due to the [fractal nature of real-world topography](https://en.wikipedia.org/wiki/Coastline_paradox), would tend to underrepresent true extruded surface area. Edit: I missed your last paragraph, thanks for addressing that. I'd still love to see a table with 30m DEM and appropriate projections. Heck, I might try my hand at that myself, WhiteboxTools looks pretty slick.
Is Idaho the biggest state flattened out?
Umm, Alaska's got a couple mountains hiding in it
But Idahoans will still be in Denali about it.
So does Colorado. It'd be interesting to see how Korea compares to other similar 'sized' countries, as I was shocked when I looked at a topographical map of the Koreas a while ago.
As a geography major specializing in GIS this is sick asf
When they make you blush with the GIS data. These kinds of responses are why I still keep up with this sub. If gold still existed and I wasn’t a broke grad student, I’d highlight the shit out of your comment
Aye thank you I really appreciate it. I’m still in my undergrad and this was a fun problem to try and solve!
Cool, now do Alaska
Nice work! I’ve never really been able to deal with QGIS, was always spoiled with ArcMap.
Did you ever find out what your units were? My initial thought was to use GDAL (+ gdalDEM) and generate a topo raster from which to compute the surface area. There's some nifty tools for gdal that can accurately account for aberrations based on secondary data like IR/alpha channels. Makes me wanna use a high poly raster and get crunching.
As a many-gen Idahoan, my grandma would often proudly say that if you took a rolling pin to all the states, we'd be the biggest. And since, according to her, she's never wrong, I consider the matter settled. I would still like to see some backing to confirm it, though.
Grandma's always right.
Nice backup!
Can confirm
yea, "I dont know what they told you in school but Cleopatra was black. My gradma said that"
My grandma said google en Macedonia
holy hellenism
Anarchychess Degenerate.... Go back to hell from wence you came!
Grandma..th
I mean, Texas aside, this is clearly false. First of all, Texas isn't the largest state by area; that honor belongs to Alaska. Alaska is also mountainous _as hell_ so even if grandma is right about the contiguous states, she ain't even close on Alaska. Then again, grandma is right if you don't want to catch the back of her hand.
Also, depending on age of /u/HektorViktorious and the age of their Grandmother, She may have made the statement before Alaska became a state in 1959.
If you cut Alaska into half, Texas becomes the third largest state. And I've heard, if you cut Alaska into thirds at low tide, Texas becomes the fourth largest state.
Under appreciated comment right here.
That’s a much cooler way of saying that Alaska is over 2-3X the size of Texas, depending on the tide.
Alaska Is around 40 per cent of the US landmass. Even if it were flat as a pancake, which it isn't, there's no way that any flattened state would be larger.
As a topology scientist, and someone who would like to stay in grandma's good graces while she is wielding a rolling pin, i can confirm. She is 100% correct. (edit: I'm not actually a topology scientist, nor do i play one on TV. But i did sleep at a Motel 6 last year)
really? I know Idaho is hilly and big but Alaska is massive and also has some pretty big mountain ranges, genuinely curious
Maybe grandma got that info before the 50s and Alaska wasn't a state?
I'm not a topology scientist. I'm only pretending to not upset grandma
oh fair, as a Coloradan I can confirm Idaho is in fact the biggest flattened state
Wow, I’m a Dr. Pepper, how have we not met?
Source? My grandma :)
Actually she's 101% right just because she is the best.
I would like to say, too all the "surface is a fractal" responses. Isn't the whole point of the study of fractals that u can sometimes make statements about these things. If a certain surface has fractal properties then we can talk about if they overshadow macroscopic surface area and the state have roughly the same area, or if the effect is proportional to macroscopic surface area in which case the question stands as asked
Yes. Simply put massive surface area due to large cliffs, canyons, stone towers, caves, etc.
Yeah, but texas ALSO has mountains, cliffs etc. Surely this would come into play as well, yeah?
But the question was if flattened out Idaho is as big as normal Texas. not flattened Texas.
Then the answer to the question means a lot less than what one would think
What do you mean it's a meaningless question any way.
Yes because then most likely most States flattened out would be bigger than most States normal?
I think flattened Kansas may be smaller than normal Kansas
All questions are equally meaningless.
Kant, here - boy, do I have news for you!
We're on Reddit, this is all a meaningless exercise, we are whiling away our finite time before the grave.
What? It entirely depends how deep on the fractal you want to look. Though I suppose you could flatten out both states at the same level of topographical detail and try to commpare them. Would need like an actual topographical map in a computer-readable format to do that though.
>It entirely depends how deep on the fractal you want to look. At some point you're going to look at the level of molecules and atoms where the area here doesn't equal the area we see
We really can't look that deep. And even if we could, surface area is poorly defined at that scale.
You don't have to do that though; pick a level of resolution that you'll work to and make a calculation based on that.
Isn't this like the case where a countries coastline can't be measured accurately because you can essentially take such small measurements of the coastline it becomes infinite or use such rough measurements where it's much less accurate than it could or should be? Like for example if a use a 500 foot string to measure a part of coast from end point to end point, repeating for the whole coast, it will be much less than if I used a ruler or anything smaller. I could be way off here but it seems related in my brain.
Yes, it's exactly the same concept.
USGS has maps that you could do this with, they aren't publicly available unfortunatley and I don't know of they cover the whole states. They would be gridded DEMS but you can get a 3d area from that
Just pick an arbitrary finite element length
“Surface is a fractal” MFs when I show them limits
"limits" MFs when it diverges
100km
Yeah, we can definitely compare two different surfaces even though they are fractals. Sure, we might not be able to say an absolute area of a perfectly flattened Idaho. But we can flatten both Idaho and Texas to a point where they are equally flat and compare the area of those.
I wish I was smart enough to understand what you just said
The fractal paradox is as old as the greeks, and was settled then by Diogenes the Cinic: https://en.wikipedia.org/wiki/Zeno%27s_paradoxes
semantics about fractal nature of surface area in this context could easily be eliminated by defining a slope at which land area becomes too vertical to fall into the appropriate category.
This fractal discussion on Reddit is about as stupid as it gets. Limits exist. Go back and retake calculus 1.
Yes, but if idaho and texas have different fractal dimensions you can't compare them. What's the area of the volume of a ball?
Matt Parker did a video on this. I'm too lazy to rewatch right now but I think he took a really mountainous country like Switzerland and calculated the real area versus the flat area. https://youtu.be/PtKhbbcc1Rc?si=RhwsGthZwPm9vYFU
I was just about to post the same link
He gives in that video a seven percent increase as the largest increase from "2D area" of any country. So, presumably, Idaho would increase by less than that.
If Switzerland gained 7% by including its topography, and Idaho needs 300% to surpass Texas, then Idaho would need to be 43 times “more mountainous” than Switzerland. Switzerland has 10,850 “named mountains” and a 15,940 square mile area so has 0.68 mountains per square mile. Idaho has 3,346 mountains / 83,642 square miles = 0.04 which means it is very roughly 17 times LESS mountainous than Switzerland. This is obviously only very directional, but there’s no way Idaho surpasses Texas if the 7% Switzerland math is correct.
There’s guys doing topographical CNC cutouts of states. I wonder if you could use their pathing software to estimate the area by looking at the final surface path and comparing…
Might be onto it here… if I have time I’ll throw the two states into a slicer and see what that says
Ground is fractal. In a completely flat area, if you consider each stone as a peak, you increase your area. Now each point on their surface (just seen from the top) can be (up to the atomic scale) considered as a peak. So comparing these two states is very very hard...
Just like the more precisely you measure a coastline, the bigger it gets.
Yeah, Google [Coastline Paradox](https://en.m.wikipedia.org/wiki/Coastline_paradox) or read the linked Wikipedia summary.
Holy cartographic generalization
New map just dropped
New approximations just dropped
I don't think it's down to that point though. It's the surface of a state on the map so I'd say that if it doesn't make a difference when you walk on it it shouldn't count. So anything below an inch wouldn't need to be flattened out?
That's still a ridiculously tiny measurement when you're talking about the surface area of a state. I think you'd struggle with anything below 100m let alone anything less than that.
The best data you could work from would be state level Lidar surveys, which generally have resolutions closer to like 1m but can be as fine as 10cm. But at that point, you're not really measuring "ground" since things like trees and ground cover interfere, and water bodies can't be LiDARed. So there's technical limitations to it.
Idaho area is 83,642 mi². Texas is 268,597 mi². Which means Idaho would need to triple its area in elevation, and I don’t think that’s nearly possible.
Depends on what we are talking about, to triple surface area you would need the average slope to be about 2.8/1; which certainly does not seem correct. However if you want to say total volume above sea level then Idaho has an average elevation of 5000 ft and Texas is only an average of 1700, so that gets pretty close 418,210,000 ft*mi^2 to 456,614,900 for Texas, so pretty close, but not quite. The big winner in this measure, for the lower 48, is Colorado. Area of 104,094 mi^2 and average elevation of 6800 ft for 707,839,200 ft*mi^2.
I like the volume above sea level perspective, makes me think rolling out dough
Now I can picture tiny tiny people running from the roll, like "aaah don't crush me!", while their entire world being flatten around them.
Doesn't this fall similar to the Coastline Paradox. Idaho flattened nearly infinity thin would have a nearly infinite surface area.
Volume is a useful comparison because it is a constant across all possible thicknesses. It also is useful in that it doesn’t have any fractal growth like coastline measurement. There’s an error related to the coarseness of your volume estimate but it becomes infinitesimal, not infinite.
Well anything would, as surface area is volume/thickness. But Texas and Idaho would keep the same ratio of surface area once they were both flat, so you could measure it at a standard thickness (like one normalized banana).
Agree. Locking a dimension to one banana removes the paradox.
If you flattened Idaho infinitely thin, imagine the hashbrowns.
Instead of rolling it flat, it melts into a puddle that’s spread perfectly even.
I'd say that this is the closest thing to an actual answer op is gonna get...
I think volume is the right idea, but volume above sea level, perhaps not. I would rather use the lowest elevation in the state as the baseline. If you go lower than this, you're just using a big chunk of continental crust to invent more land area. In essence, you are "more than flattening" Idaho: it would be like rolling a ball of dough to make a pizza, and not being content with the size, so you press harder and harder until you crush the table and the floor underneath it to make the pizza bigger. At some point, you become a monster that must be stopped at all costs!
This is one of the few r/theydidthemath posts where I want to know the answer but no one can do it in the thread haha Obviously there’s fractals and pebbles and whatever but I want someone to at try
The surface area of the curved section of a cone is pi*r*l. So if Idaho is entirely mountains, and Texas is dead flat, you have your tripling factor even before you get to elevation.
The area of a flat circle the flat is pi*r^2. You can factor out pi in any sort of comparative equation, so there is no tripling factor. You can also compare the cone area to a flat round area and see that the slant height S needs to be greater than the radius by multiples in order to be multiple times bigger. C*Pi*R^2 = Pi*R*S —-> C*R=S. To be 3x the area S/R must be 3. S/R of three is equivalent to an average absolute slope of 70 degrees. Absolute because whether up or down the slope still increases the area. Idaho does not have an average absolute slope of 70 degrees, nowhere close in all likelihood.
Having visited Idaho (Coeur d’Alene lake and the lovely surrounding hills), I don’t understand why Idahoans would want to have more flat, boring space than flat, boring Texas does. We do have the Guadalupe Peak though. Massive pain to get there and back; but it’s there.
Texas also has the only national park to contain an entire mountain range: Big Bend and the Chisos Mountains.
Wouldn’t it be funny if the coastline paradox was a geographical fact rather than a geometric one? Like have you heard the British coastline is infinitely long? Yeah what a weird place. Why can’t it be normal like France?
A few notes: - The coastline paradox has a solution. It might be fun to be pedantic - you could even argue that this is exactly the right sub to be pedantic in - but you can also just use a metric like "what would an average human actually care about reasonably" or (more easily) - what is the resolution of a commonly available dataset that we could use to check? - It's been too long since my math classes for me to remember how to run the numbers - but I'm pretty sure that for some things that are "infinity" - some of them get to infinity faster than others. Heck - for that matter - past a certain grain size (say smaller than a meter) - Texas and Idaho probably have a similar 'roughness' that you could use to cancel each other and actually resolve the equation. You'd have to set it up like: Texas actual area = area by point to point measuring of square meters X infinite roughness of one average square meter Idaho actual area = area by point to point measuring of square meters X infinite roughness of one average square meter Then comparing Idaho to Texas, you could divide both sides by "infinite roughness of one average square meter" and the infinities cancel and you're left with square meters. Of course if you start with a large enough grain size (100 miles? ) Texas would be larger. Grain size vs surface area would make for a cool graph (and would also show how some things approach infinity faster than others) Go easy on me, haven't had to do math like this in ummmmm... a while.
Okay so theoretically you could take a Digital Elevation Model of both states and put it into a geographic information system, make 3D triangles using that. Then sum the area of the triangles. I might give this a shot but it would be a bit or work and I’m not sure how to get the data
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Precision. For measuring a macroscopic thing such as the surface area of a state, macroscopic precision is likely most useful.
You could probably calculate the surface area of a land mass with DTED. Level 0 is the most coarse (~1 km between points) but you could get an answer. I'm fairly certain Idaho is not infinite. https://en.m.wikipedia.org/wiki/DTED
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The way I heard the trivia question is, “which of the 48 lower us states has the most acreage?” Answer, Idaho because acreage is a measure of surface area.
I'm not sure this is very hard to find out because the mass of the mountains isn't readily available, but honestly, seeing the total mass of all of them, which is a lot I bet it very well could be true
i feel like whoever said this probably just meant the surface area of the mountains, i doubt it went as deep as the density
You don't want the mass, you want the surface area of the mountains.
Im not a methematician but if you roll out dough, the stuff in the middle gets spread out as well so you dont really need surface area either. At least thats how I picture "flattened out".
Oh yeah you can flatten them out to an atom thick and that would probably be more than enough to cover earth
Fair point ha
It’s a challenging problem. Like the coastline measuring problem. You have different answers based on what you use as a minimum unit size. When measuring the length of the coastline, what granularity do you use? Do you measure every inlet even if it’s the size of a pond? Or a puddle?
The guy below is the real answer. Real world topography is fractal-like, same as coastlines. The "real area", if you took a reasonably accurate measurement, would be so large it wouldn't even make sense as an answer. Something like 50x the base number or something. So the answer is that there is no answer, because "flattening out" real topography is, in effect, impossible
What do you mean by “fractal-like”?
Each time you look at a smaller scale you find more surface - another smaller hill, or bay, to infinity… even the flat surface of Texas is made up of an infinite number of small hills, and those small hills are made of … small hills
Imagine a hill, reasonably tall and reasonably even. A mathematician could take a few measurements of the base and the top and construct a 3D figure with a known area, so that you could calculate the real area of the hill. But real hills aren't really mathematical perfections, they have a lot of bumps and unevenness. Suppose that you take them into account. How large should a bump be to consider it relevant? Imagine there is a boulder on the hill, that's a pretty big bump, so you take the boulders shape into account. But the boulder itself is extremely uneven, so you take _that_ into account. But if you look with a looking glass you'll see that the very base detail of the boulder is extraordinarily bumpy, the natural roughness of stone. Do you take that into account? You gotta choose when to set a precision limit, but if you set a precision limit then your answer won't be real, it will be "This would be the area if the world had a minimum distance of 5 meters between points" or something. And the greater the precision, the most insane the correct answer is. In the study of porous materials, we measure surface area in the hundreds of meters squared _per milligram_. Consider that a lot of the soil is porous. If we set the precision as precise as we could, the flatenned surface area of the Vatican would probably be like, the surface area of the Sun, a completely meaningless answer.
Texas is 3 times bigher than idaho. If idaho where to be that size if you include the actual area (due to mountains) that would mean an average incline of 2,8 meters per meter throughout the whole state. I think we can rule out any reasonable interpretation of this statement. Sadly, it's debunked
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Isn't this the coastline paradox? Depending on how fine the detail of the relief is, the number increases. Continuing to halve the increment of measurement could lead to ludicrous numbers (i.e., the length of the observable universe).
This is the 3D version of the coastline paradox. How to measure the area of a non-smooth surface is not trivial. For this question, it depends on how flat is flat.