@@standupmaths Now we have another challenge for you, which also strikes at the issue of geoids. Take the flight distances between A: London, B: New York, C: Tokyo, D: Johannesburg, E: Melbourne and F: Rio De Janeiro. [1] Show the distances for {A,B,C,D} make a tetrahedron of *positive* volume (hence they are not co-planar and the Earth is not flat). [2] Show the distances for {A,B,C,D,E} can *not* exist in *any* Euclidean geometry of *any* number of dimensions! (Instead, they lie in a 3+1 dimensional geometry - a Minkowski geometry). [3] Account for the Earth's curvature by assuming all flight distances are actually arcs along circles of diameter d and use the corresponding chords instead (with the conversion Chord = d sin(Arc/d)). Prove that: [3a] for d ranging from (2/pi x the longest flight distance) to a fixed value, the chords for {A,B,C,D,E} can only exist in 4 dimensions, [3b] for d ranging beyond this fixed value, the chords for {A,B,C,D,E} can only exist in a 3+1 dimensional Minkowski geometry and not in Euclidean space, [3c] for a very specific d, the chords for {A,B,C,D,E} can be embedded in a 3 dimensional Euclidean geometry. [3d] what is d? [4] Do the same with all 6 cities {A,B,C,D,E,F} and show: [4a] for d ranging from (2/pi x the longest flight distance) to a fixed value #1, the dimension required is 5 (signature +++++ in the 5 coordinates), [4b] for d at the value #1, the dimension is 4 (signature ++++0), [4c] for d ranging from value #1 to a second fixed value #2, the dimension is 4+1 (signature ++++-), [4d] for d ranging from value #2 to a third fixed value #3, the dimension is 4+1 (signature +++-+), [4e] for d at value #3, the dimension is 3+1 (signature +++-0), [4f] for d beyond value #3 as well as for flat geometry where d = infinity, the dimension is 3+2 (signature +++--) - an anti-deSitter geometry, [4g] at d = value #2, attempts to fit the chords of {A,B,C,D,E,F} *diverge* (the signature approaches +++00 but at least two of the cities must diverge to infinite null vectors before the signature is reached), [4h] value #2 is almost exactly equal to the diameter found in [3]. [5] Use the results of 4 to show that the flight distances of {A,B,C,D,E,F} do *not* fit on a sphere at all but require an irregularly-shaped geoid, like an ellipsoid. Find the ellipsoid that allows the flight arcs to be embedded in a 3D Euclidean space, and use it to also determine the *latitudes* of the cities! This requires converting from *elliptical* arcs to chords, which requires elliptical functions. [6] Add in a 7th city, G: Nome. Repeat all of the above with {A,B,C,D,E,F,G} assuming a sphere, then assuming an ellipsoid. Compare the results to those of [4] and [5]. In particular, does it fit the same ellipsoid that was found in 5, with the *same* absolute latitudes and relative longitudes for the subset {A,B,C,D,E,F}?
So if we can’t measure coastline properly because we have a 1d shape in 2d space, and we can’t measure area with topography because that’s a 2d shape in 3D space, then clearly the sane thing to do is start measuring the volume of countries.
@@mozarteanchaos I hope this charade will be over soon. My guess is this will ultimately result in Britain rejoining once people realize what they did to themselves. So lets get it over and done with, because there are people who have to suffer from all this uncertainty.
Internally I was like: "Father, here I command my soul to the eternal infinity of the heavens", when he mentioned the coastline problem... But when mentioned that the coastline problem could be extended to 3D, I felt as if I had become god itself that created reality itself just so that line would be delivered with a close-up shot on the crevices of that wall. Complete mathematical enlightenment: The surface area of every country is infinity. 🤣
Matt: As I always do when something about the UK irritates me, I went to Australia Me: Ah. Matt did the British thing and sent his troubles to Australia
@@timbeaton5045 On vacation in Australia, Douglas Adams realized that England is a cold, rainy, dreary, and cramped place, Australia is a sunny, beautiful, wide open place, and England had the bright idea to use Australia as a prison. No wonder there's a special smile Australians reserve for the English.
@@tparadox88 That special smile on an Aussie bowler's face at the GABBA when the next English batsman is coming to the crease when they are 5 for 70, perchance? 🤣
That’s academia for you! “I extracted elevation data from the Google Earth engine, and then I combined this with the shape file from the global administrative database. I used a function within the R statistical language, which is based on trigonometry, to calculate the surface area taking into account the terrain; and I also did this at several different spatial resolutions to have a look at the effect that that has on the accuracy of the number. ...” ... for free.
@@makatogonzo no, i do too, hes saying online information is so diverse and available, that college is not that necessary to have the knowledge to make something happen
@@artratengo I'm pretty sure OP is amazed at how willing this person was to donate time and effort to this cause which OP(maybe erroneously) attributed to them being a part of academia. I don't understand how you ended up at your conclusion.
As a geospatial scientist from Melbourne, Australlia I was hoping to relax with a fun maths video. But instead this sounds EXACTLY like the queries I deal with at work everyday.. LOL I literally just spent the last three hours building a coastal digital elevation model from LiDAR. Thanks for the great video Matt and Bec! Would be great to see you cover differential GPS, where we use ground stations/mobile networks and a lot of maths to obtain ultra-accurate locations (sub-millimetre).
@@robertstuckey6407 nah, modern datasets are almost all from survey planes. (One of the standard global elevation datasets however was made by the space shuttles synthetic aperture radar experiments)
@@RobertSzasz A whole host of methods are used to generate geospatial datasets. Including GPS (measurement of asset locations/heights), surveying/lidar (land/property), aerial imagery/elevation from planes, but its also very common to use raster datasets derived from constellations of remote sensing satellites (ie SPACE, think > LandSat, Digital Globe, PlanetLabs, etc) to map/model our environments with all kinds of variables (we can even determine soil moisture levels "from space"). Would be great to see Matt cover differential GPS, where they use ground stations/mobile networks and a lot of maths to obtain ultra-accurate locations.
@@apocalypsepaul Harry Parker and the Mathematician's stone. Harry Parker and the Vector Space of Secrets. Harry Parker and the Theorem of Azkaban. Harry Parker and the Goblet of Fractals. Harry Parker and the Order of Operations. Harry Parker and the Half-Blood Primes. Harry Parker and the Mathly Hallows.
I, personally, take great pride in the fact that since I stopped shaving myself and started to cultivate broccoli, I contribute to the size of my country in an efficient and meaningful way.
@@murpledeer A fractal just needs to have detailed structure at arbitrarily small scales. If you want it to look the same all the way down at different scales, that's a Mandelbrot set.
In a physical chemistry class I took once, we ran an experiment where we had to measure the internal surface area of zeolites, which are extremely porous solids meant for having narrow channels and very high internal surface areas, by various methods. One method, BET, uses measuring gas pressure and volume to extrapolate surface adsorption and then divide by the surface area of the gas particle size. We noticed that the surface area per gram changed depending on the gas used, and I found a paper about fractal dimension and zeolite surface area. I wrote my paper for the project about the size of the gas particles relating to the resolution of surface area measurement and my professor told me he had never heard of that connection before! It is truly fascinating how fractal dimension can be relevant in some surprising places.
So to measure the surface area of each country, build an impermeable barrier around each, fill it with hydrogen, then do a bunch of mathematical calculations...
And if you were to gather results for such a project to measure the areas of countries, you'd find they tend to infinity, and probably strongly correlate with physical properties of the countryside's materials. Not to mention the errors from the hydrogen getting absorbed or reacted by things.
If you could have a chamber where you can pump gas in and out to achieve constant pressure, with circulation to provide convection, and a heating system for the gas, would it be possible to measure some fractal metric by selecting a pure gas, then changing the temperature and measuring the outflow of gas? Obviously you'd need to factor out chemical reactions and account for adsorption, but this should allow you to test the sizes of gaps and crevices continuously, as hotter, less dense particles might avoid reaching into those crevices more than expected. You could sort of imagine the particles "acting bigger" when at the same pressure and lower density. This would drive out more gas than should be driven out than temperature increase alone should provide (remember the system has constant pressure), as the small crevices are not packed as efficiently because the particles are "acting bigger." A really clever model could perhaps also look at the cohesion of the gas (and how that cohesion changes with temperature), but the mathematical relations for that could be a respectable nightmare. With or without such extra modelling, this effect would perhaps be nearly impossible to measure, and the errors would still drive out any meaningful data, but it's fun to think about. Some large (particle size) inert gas and a very tight (average bonding distance) material object could make some of these considerations more possible. The large particle size and tight material in the solid might decrease adhesion, while having large particle size also limits resolution to ignore whatever wierd physics shenanigans might happen if particles get into very enclosed spaces while the particles are very close to the solid's surface. Making the gas inert should also pretty much eliminate cohesion, I think.
Peter LeRoy Barnes there’s lots of interesting ideas in here! Using pressure is only realistic on smaller scales, especially because things like reactions but also because different materials adsorb gas on the surface at different concentrations so you would essentially get a free variable out the other end that would be difficult to manage-not to mention the difficulties of such an experimental setup!
I think people are underestimating the Netherlands. Anyone who has ridden any substantial distance in the Netherlands know the county has hills. The lowest point in the Netherlands is about 7 meters below sea level while the highest point is a bit more than 320 meters, for a total variation of about 327 meters. According to Wikipedia, the flattest country is the Maldives, with a total variation of about 1.5 meters. (As a result, you can see why the Maldives are concerned about rising sea levels as a result of global warming.)
And @6:39 (and elsewhere!) we also found the limits of Matt's camera's Autofocus settings. Panasonic G5, possiblement?* *Ahh. The Panasonic Pony of Hope
I got a very similar question from my twelve year old son when we went mountain hiking. "Mum, did the GPS take the slope into account when it calculated the distance we hiked?"
Ah, the brilliance of children. not quite smart enough to know the answers but smart enough to know how to ask good questions, and then they know the answers.
my dads strategy was to just promise that the "hütte" (restaurant on or near a mountain top in europe usually supplied by helicopter, cable car or on foot, i dont know if thats a thing in america) was right around the next corner, or across that hill, basically just out of sight.
It's not. You built a house vertically, or with a floor whose plane vector matches the vector of the force of gravity (if you use good enough builders). That means that for each floor you build you have consumed the 2D footprint of it in real-estate. Agriculture now... That's a whole different thing...
@@ExMachinaEngineering Also you should never mix up 2D footprint of your roof (which is important calculating the amount of rain water) and actual surface area of your roof (which is important when ordering new rooftiles).
Look at the % increases given on screen. They match the written version, which also are intuitively more credible. You people are wasting your time looking at a maths channel.
Dr Laura Graham. Living proof that real heroes don't wear capes. Great work to answer the biggest questions here, Dr Laura, and kudos for coming on to such a huge TH-cam channel.
Matt: "Conceivably Denmark... if we included its terrain..." Every Swede or Norwegian, laughing patronisingly: "Come on, Denmark's tallest 'mountain' is like 100m high, it HAS no terrain"
Our tallest peak is 170.86m unless we count monuments on top, in which case there's one at 183m and several other structures reaching even higher. Which brings up the general question if topographic area should include the exact shape of various buildings, including balconies and broken windows that extend the surface into the inside rooms.
Well, Netherlands is only 1,4% larger than Switzerland whereas Denmark is 4,4% larger (according to Wikipedia). Both countries are pretty flat (although with hills). Also, it doesn't matter how tall the tallest peak is, just how much it goes up and down.
I'm a surveyor. When we survey land, the area we calculate is the horizontal area at the elevation of some point in the land. So the way to compute the area of a country is to break it into little pieces, raise or lower each to the elevation of the land there, and add up the horizontal areas of the pieces, then take the limit as the diameter of the pieces tends to zero. (Not the area, since then you could make all of them long thin stripes.) Also, we use conformal, not equal-area, projections. Surveys indicate the scale factor between the map projection and the horizontal ground distance. Projecting a *geoid* is prohibitively difficult. What we project is the *ellipsoid*. The vertical distance between the geoid and the ellipsoid goes into the elevation scale factor.
16:01 If the entire population of Lichtenstein unsubscribed from Matt for that affront to their national pride, his subscriber count would only drop by about 6.7%. On the other hand how dare you.
I've done this calculation a few years back on the levant area. I basically took the greyscale data from the USGS website, found the elevation range for each greyscale image and normalized them (each image is different, sadly... lots of work), then compiled them into one big image. Next, import the image into Blender and map the image to a plane, offsetting the z axis by color, according to the normalized elevation. Blender then calculated the total area of the plane for me. It was a fairly significant difference, definitely over 1%. I'd love to see the impact this has on Peru or Nepal.
I'm really pleased to see that this video is only 7 months old. I remember watching it quite a while ago and thought it was actually years ago, and now I'm relieved to realize that time doesn't always fly by as fast as expected.
16:51 "I think we can all agree that the Netherlands is completely flat." --Matt Parker, 2020. You heard it here folks. The Netherlands has only two dimensions. No buildings. No trees. Just perfectly smooth bedrock.
Interesting surveying anecdote; one of my surveying professors in college was taken to court by his father in law after performing a land survey for the FIL. The claim was that my professor stole (or disappeared) several acres of land from his property. The difference came from the original survey counting the topography in the land area, whereas the new survey was done conventionally with only the horizontal portion of surface area being counted. The two men still hate each other decades later.
As an Australian, whenever I see a drone fly-up shot, I always check the grass colour. "Oh, it's brown, yeah, that's Australia. Oh, now it's switched to another shot... green grass, that'll be the UK."
@William White Agreed!! Not an expert, but I find India too, having climate varying place to place. I'm sure Australia is same. I visited Melbourne once and the weather was changing everyday.😄 Awesome place!!
Great intro sequence for a great question! Please keep this epic music for your videos!! Where can I download this epic track in full length without voice-over?
I think you missed an interesting tidbit. You see, the earth's geoid, which the Australian source referenced, is none of the above. It's not flat, nor is it a sphere, nor does it follow the terrain. Instead, it follows, roughly, where sea level would be in the given area. You see, mountains have gravity and the earth isn't uniformly dense. Oceans, for example, are much less dense than granite. So you see the ocean depress in the deepest and furthest out regions of the pacific. Meanwhile, the gravity of a large, and dense, mountain range would pull ocean toward it, making the geoid rise. In the end, you get something that neither follows the oblong sphere that we think of as the shape of the earth, nor does it follow the terrain, yet is affected by both.
Well he did mention that it was corrected for the _local_ shape of the geoid and that it's not the same over the whole globe. He just didn't explain why that is, and the mention was just a side-note, so it's easily missed.
@@sourcererseven3858 Also it mentioned they use the Australian Geoid, which is probably slightly different to geoids designed to minimize error for the whole globe (Eg WGS84 used by GPS) as many are local standards that only care about accuracy within a given region usually a country.
As someone who has never had a class in geoscience but has for professional reasons gone down the rabbit hole of projections, I chuckled at how flummoxed Bec was by Geoscience Australia's explanation which for better or for worse made sense to me. I was like, ok, sounds like they use an Albers projection with datums that minimize the distortions for Australia. The fact that Matt couldn't get a straight answer for UK was puzzling, but for this American, the US Geological Survey also uses Albers (different datums). And if you ever have to calculate land areas like me, you find out pretty quick that everyone does it by projecting it to a 2D plane first, not some calculation based on the surface area of a sphere.
What have we learned? A country can have infinite area, and an infinite border, but maintain a finite volume. I started thinking about this a couple weeks ago, and it gave me a headache to say the least.
The natural next question is: What's the volume of UK? The surface might be a fractal, and so the area infinite, but surely the volume again converges to a finite value. More precisely: What's the volume between the surface of UK and the mean sea level?
I would imagine that's probably unknowable realistically. You'd have to measure every single building, every person, every animal, every tree, it would be absurd. And not just one by one, but simultaneously in an instant, or else all your measurements would be wrong again by the time you finished.
@@TheRealHungryHobo I think he just means the land volume... It shouldn't be too hard to calculate depending on how the topographical data is recorded. The land area is already calculated from land at mean tide.
@William White Take a 2d fractal. Give the fractal depth by giving it a finite thickness, say 1m. You now have a 3d object with infinite surface area (infinite linear perimeter x 1m is infinite area) and finite volume.
@William White 1) You say draw a box that's x by y miles, well you have to include z as well since we're including the 3rd dimension. 2) I can draw a 2D box around the island that has a finite perimeter, but the 1D fractal perimeter of the coast is still infinite. I can draw a 3D box around the island with a finite area, why does that imply that the 2D fractal area is finite?
@@macizogalaico I think all that was missing was context. You can easily get confused by the long name of the dataset if you don't know it beforehand. If you know that "GEODATA COAST 100k 2004 data" is to be read as just one entity, you're trying to insert some kind of break inside there (as it's usually unlikely that a single term is made of so many words, and our brains work on likelyhood). But then you wonder where's the break? _Obviously_ there's some punctuation missing to make that break clear ;) And honestly, punctuation _would_ help, though it's not grammatically necessary. Replace the "and" by a period and make it two sentences. Or replace the "and" by ", so" to show that the latter part is the conclusion and all of the former part is one subsentence. The sentence was written by someone who was rightfully assuming that the reader would know what "GEODATA COAST 100k 2004 data" would mean and wrote the sentence with that in mind, but it was read by someone who didn't. The only one at fault was Matt for not providing the required context ;)
I was watching this wondering how far into the video we’d get before fractal geometry came up. I’d love to see more geography videos on this channel - I find the intersection between geography and mathematics fascinating.
Exactly! I have actually previously wondered the 1D->2D equivalent of this, namely: does Google Maps quote you the "flat" distance when travelling, or does it take into account topography. Never went into enough of a rabbit-hole to find a definite answer, and also never considered expanding it to a 2D->3D problem!
@@EcceJack Hmmmm I never thought of that either, but I would assume Google Maps uses the topographical distance, since I would assume they get their distance data from surveys of the roadways, which I would assume are done by someone driving along the roadway and measuring every so often. That's a lot of assuming though, lol.
No, not wasting my time on this question doesn't make me feel silly because taking topography into account makes no sense when the calculation seems to be intended as a means for comparing the relative size of countries. If I dig a large hole in my backyard and pile up the dirt, thereby increasing the surface area of my yard, would it make sense to say the size of my property increased relative to my neighbors? In a strong field, this is the strangest Matt Parker video I've seen and the first one I've turned off because the premise is lost on me.
My favorite part of stand up maths is just how you can tell that matt has come from old TV with even just a 17 minute video having a TV show intro! 😍 love it. Really glad I can finally get my high video quality maths kick!
Exactly. And I even live in Friesland which is, apart from Groningen, petty much the flattest landscape possible. When I use my bike the only times I have to cycle uphill or downhill is when there is a man-made height difference for a bridge or underpass. Any other road is as flat as it could be xD
For the sentence at 3:20... Probably the comma was meant to go here: "This area calculation is based on the 2D polygons in the GEODATA COAST 100k 2004 data, and topography is not taken into account." The suggested comma placement, "This area calculation is based on the 2D polygons in the GEODATA COAST 100k, 2004 data and topography is not taken into account," would be a comma splice.
Well, the topography part is intuitive and for many countries curvature is irrelevant. Therefore you are wrong and could also cause smirks. So be more careful next time :)
I was confused at first where I'd heard the sweeping title music before, if it's something in the Incompetech library. Then a few minutes later I realized it was just an epic orchestral arrangement of the regular channel theme
Since the fractal surface leads to an infinite area, let's calculate a country's volume. It's not a practical thing to know, but who cares? I'd propose a countrys volume is either... a) The volume between its actual surface and a surface at sea level. b) The volume between its actual surface and a section of a sphere with the country's average height as a diameter. a) goes to the country with the biggest average height times idealized surface area. b) however advantages country's with the biggest elevation changes.
Considering many countries have mines and such I think mineable earth should be included, which theoretically would include all the crust which is about 18 mi thick over the continent’s (deepest borehole is 7.6 mi)
The Danes thought that a hill that's 147 m was their highest peak until 1847. Said hill is called "The Sky Mountain" in Danish. I just find that hillarious.
The Vatican might as well be called the flatican, despite being built on a hill. Although of course for the Vatican, the topography of the buildings would actually make a difference. Hey I think I just stumbled onto a related problem, I suppose it would only really be a thing for the Vatican, Monaco and maybe Singapore, but those little countries are basically completely full of urban development, and buildings are about as extreme as topographic features get.
@@AlRoderick if you're interested, the data used here seems to be DTMs (digital terrain models) of various resolutions rather than DSMs (digital surface models). A DTM represents the elevation of the bare earth whereas a DSM represents the surface including any buildings etc. So, it's unlikely the buildings in the Vatican were taken into account this time.
@@TheAlison1456 I'm not entirely certain but, from what I understand, DTMs are taken from DSMs with building heights and vegetation etc. subtracted. A common method for scanning elevation is using LiDAR. It's like radar except using lasers instead of radio. With LiDAR, you can tell how far something is but also what sort of thing you're detecting. So I guess it can be determined if a height is of a building or land and then the rest is figured out somehow.
This seems like a very relevant question for countries that have land holders on their mountains for farming (growing coffee/tea/ect). It would account for why there seems to be more land ownership than area in the country.
At 14:03 Matt says 9 instead of the displayed 7 in the third digit, which is it? (Just wondering because that’s a fairly large difference) The 0.94% change gives a 7, so I will assume that is correct for now
Apparently Parker Square can also refer to some sort of arithmetic operation he screwed up in that computation as well as an NxN square of numbers with some special property.
It's entirely plausible that he deliberately places mistakes into his work, just like in his book humble pi. Either that or I diagnose early onset dementia
By the same logic, if you flattened our England, the ranking would be the same as it was. There is a lot of stored area in the Pennines, Lake District, and various wolds.
"The Netherlands is completely flat" Well, as you said the area depends on the raster size of the measurements you used for calculation. The entirity of the Netherlands has been measured with laser altimetry for a public dataset (AHN3). This dataset has 6 to 10 points per square meter and a standard deviation in the height measurements of 5cm. Because of how fine that grid is we might just beat Switzerland again in land area. Interactive height map of the Netherlands: www.ahn.nl/ahn-viewer
Wow, 6 to 10 points per square meter is impressive. Not sure if it is sufficient againts the one of Switzerland with resolution of 0.5m though: shop.swisstopo.admin.ch/en/products/height_models/alti3D
The americas as a whole would be interesting to see how much of a difference each has given that both North and South America’s have rather sprawling countries that have distinctive mountian ranges to them.
I feel like when we ask for land area, 90% of the time what we really want is what you call the "flat footprint" (with corrections for spherical curvature for large countries) anyway. If we're trying to estimate population density, for example. Well, people don't stand tilted when they are on a steep incline, they stand straight up and down. If we want to estimate how much floor space we have for buildings: those are built straight up and down as well. You don't get any extra floor space when building a one-story building on the side of a hill. (Though you will save money on support material if the building has more than one story.) Want an estimate on crops a country can grow? We need to know how much sunlight it gets. Hills don't get any more sunlight just because they're tilted. (Though mountainous areas will affect cloud-cover, I suppose.) Want an estimate of how much rain they collect? Other than mountains' effect on clouds, having a larger topographical area doesn't increase your catch. Because rain falls, wait for it, straight down. Or... does it? *throw to VSauce*
Hey Vsauce, Michael here. Stadiums and cinemas have tilted floors, can't they fit more people because of that? What about multiple floors? In my building, there are 4 families stacked on top of each other on the same flat footprint. If a terrain in the southern hemisphere is sloped northwards, it will get more sun than the flat footprint predicts. And for the rain, you would have to take into account the prevailing wind direction and speed to know how much rain would fall on that land.
Oh Matt, Matt, Matt... The whole production of this video is brilliant! I adore all the drone shots, the film-style graphics and the orchestration of the theme music! I've had to switch to watch it on my 4k TV to appreciate the video in full glory (instead of my phone)! Keep up the good work!
The more important question for the area of The Netherlands is: do you include the area of the Wadden sea, IJsselmeer and Markermeer? Because that changes the area from 33.500 to 41.865 km2, a 25% increase!
Since the dutch can make land from anywhere, we should just include every bit of water that can be feasibly made into land. The Netherlands is now the biggest country on earth.
I would include the IJsselmeer and Markermeer, as they don't have a real connection to the ocean, so are functionally inland lakes. The Wadden sea is a sea, not a lake. Fun fact: water area changes the largest province: Gelderland if you exclude lakes, and Friesland if you count lakes as land area.
according to two google searches what the highest points of The Netherlands and Denmark are, I have to conclude Denmark is flatter than flat, because your highest 'mountain' is about half the height of the highest Dutch 'mountain'
@@joeytje50 That's mostly because Limburg extends into the Eifel foothills. Though I assume it's similar for Denmark as well, having just one or a few really high points. It's much better to look at how flat the terrain is on average rather than what the highest point is, though I'm not sure whether there's good data available for that.
Tim the Traveler went to both. Sadly the highest mountain here is the old garbage dump with roughly 42-49m (measurements differ) The highest natural peak is 32.5m
If one goes sufficiently high resolution, one could account for the undulating features of the brickwork, the rocks, the soil particles. even bark on the trunk of a tree, or the features leading to the chloroplasts through the stomata of the leaf. The area would be increased by orders of magnitude.
As you skimmed over the flat/spherical question I think you may have missed a significant point. The Australian area figure in particular states that it's based on the Australia geoid. This is NOT the spherical surface, nor the ellipsoid one. The geoid is the gravitationally flat "sea level" surface of the Earth, which has its own terrain due to the strength of gravity varying from place to place as determined by variation in density and depth of the Earth. So the the Australian area (at least) does take terrain into account, just not the visible one!
thinking back a long ways to when I took a surveying class I realized when we took 2D area measurements we didn't take into account slopes (but we did for 3D topographical measurements) And I think I know why, buildings are almost always built on a flattened bit of land, so either earth is removed or added to bring things to a level area. So, if my thinking is correct, the reason countries only care about flat measurements for area is because that's the usable area for structures. I am curious how that all works out for farms and ranches, because if a farmer buys a hilly plot as one acre, and it's actually, for example, an acre and a half they'd need more seed to plant it, and if a rancher with livestock bought the same land they'd theoretically be able to raise more livestock per acre. I live in the Mississippi Delta area, which for all purposes is flat, so all the farmers I know don't have to deal with topography so I can't just ask one of them haha
For an area to increase by 50%, it would need to be on a slope of greater than 45°. So quite hilly, and probably marginal for many farming systems. Not really cropping land. However, it's worth asking whether the land would have a greater potential productivity than an adjacent equivalent flat area. The major factors are energy inputs and water. Clearly the energy input from the sun doesn't change just because the land is tilted - it's the same total energy per unit time, spread out over a greater area. There might be some minor benefit in terms of energy conversion rates, due to the plants being spread out more. On the other hand, the slope creates challenges for water management. And having grown the pasture (or crop), we are faced with less efficient conversion rates due to the livestock using up extra energy climbing up and down the hill (or due to harvesting the crop on a steep slope).
3:15 pretty sure it's supposed to be something like "This area calculation is based on 2D polygons in the 'GEODATA COAST 100K' data from the year 2004, and topography is not taken into account."
@@HelgiWaag It's not arbitrary you dingus, Limburg in itself is an arbitrary piece of land that remained to the Netherlands which shouldn't have given it ended under Belgium for years.
Matt: "The terrain where I live is reasonably..." Me as a Scot: "That looks reasonably flat, yes." Matt: "...hilly." Me: "Wait, what?" But I'd really like to know how the area change differs in Scotland from the rest of the UK. Not because of any kind of nationalistic mentality, but because Scotland is fundamentally different from a geological (and topological) perspective. Scotland has the highest 18 peaks in the UK, and out of 120 hills over 600m, Scotland has 81. Edit: Sorry, in the British Isles. Sorry Ireland! :(
I did break it down by country (at each spatial resolution) - have a look at the table near the bottom of this: github.com/laurajanegraham/surface_area/blob/master/surface_area.md In short, Scotland has the highest change (1.8%, based on the 90m resolution data), followed by Wales, NI, then England.
@@lauragraham6149 have you looked at this data set? www.ahn.nl/ahn-viewer I really wonder what difference the height map would make at this extremely high resolution.. and what would a graph of the percentual change as a function of the height map's resolution look like?
I haven't watched yet so I don't know how it's really done, but I can't believe I always took for granted (not even that, it never really crossed my mind, it's something I did unconsciouslly) that everything is measured as flat.
There was some whispering about the "epic orchestral arrangement of the regular channel theme" and how we require a full length standalone version for our aural enjoyment as well as a bit of enlightenment. Sadly, it seems to have been kept _from_ us.
Good day Matt! I was once traveling and heard a local meteorologist say "With rain over 20% of the area" instead of what I believe to be more commonly "With a 20% chance of rain." Perhaps these statements are different, perhaps they're analogous. I'd love to see what you find interesting about a 20% chance of rain over 100% of the land area vs. 100% chance of rain over 20% of the land area. Implications? Thank you, and I'm thoroughly enjoying Humble Pi :-)
And what does 20§ chance of rain even mean? Does it mean it will rain about 20% of the time? Or is it the chance of 20% that there will be rain at least once in the mentioned time period?
Daniel Bohrer, they run the computer models for the forecast under many different assumptions, then average out what proportion of them said rain would happen to 10% or 20% increments.
![General Relativity: Top 05 Mishaps [inc INTERSTELLAR]](http://i.ytimg.com/vi/Z4oy6mnkyW4/mqdefault.jpg)
![General Relativity: Top 05 Mishaps [inc INTERSTELLAR]](/img/tr.png)
The dislikes are from the entire population of Liechtenstein.
Hi 👋 I just need need weekend
hahahahaha I need to show my uncle this video, he's from Liechtenstein
@@sebastianpeheim8851 Let us know the results 😉
All 9 of them!
Liking just for that intro music
This feels like something a giant iron could solve pretty quickly.
I've seen this anime before
Are you the evil genius that's going to do it purely for wanting to flatten the Earth to make the flat-Earthers right?
@@Axartsme Was it both Fooly and Cooly?
O
Hey! I watch you're videos
Man, these videos just keep getting better.
Awh shucks.
BOOOO music too short BOOOO ENCORE!
Where's the rest?
Two teachers I greatly respect... Nice.
🍉
I love seeing educational TH-camrs commenting on the videos of other educational TH-camrs.
@@standupmaths Now we have another challenge for you, which also strikes at the issue of geoids. Take the flight distances between A: London, B: New York, C: Tokyo, D: Johannesburg, E: Melbourne and F: Rio De Janeiro.
[1] Show the distances for {A,B,C,D} make a tetrahedron of *positive* volume (hence they are not co-planar and the Earth is not flat).
[2] Show the distances for {A,B,C,D,E} can *not* exist in *any* Euclidean geometry of *any* number of dimensions! (Instead, they lie in a 3+1 dimensional geometry - a Minkowski geometry).
[3] Account for the Earth's curvature by assuming all flight distances are actually arcs along circles of diameter d and use the corresponding chords instead (with the conversion Chord = d sin(Arc/d)). Prove that:
[3a] for d ranging from (2/pi x the longest flight distance) to a fixed value, the chords for {A,B,C,D,E} can only exist in 4 dimensions,
[3b] for d ranging beyond this fixed value, the chords for {A,B,C,D,E} can only exist in a 3+1 dimensional Minkowski geometry and not in Euclidean space,
[3c] for a very specific d, the chords for {A,B,C,D,E} can be embedded in a 3 dimensional Euclidean geometry.
[3d] what is d?
[4] Do the same with all 6 cities {A,B,C,D,E,F} and show:
[4a] for d ranging from (2/pi x the longest flight distance) to a fixed value #1, the dimension required is 5 (signature +++++ in the 5 coordinates),
[4b] for d at the value #1, the dimension is 4 (signature ++++0),
[4c] for d ranging from value #1 to a second fixed value #2, the dimension is 4+1 (signature ++++-),
[4d] for d ranging from value #2 to a third fixed value #3, the dimension is 4+1 (signature +++-+),
[4e] for d at value #3, the dimension is 3+1 (signature +++-0),
[4f] for d beyond value #3 as well as for flat geometry where d = infinity, the dimension is 3+2 (signature +++--) - an anti-deSitter geometry,
[4g] at d = value #2, attempts to fit the chords of {A,B,C,D,E,F} *diverge* (the signature approaches +++00 but at least two of the cities must diverge to infinite null vectors before the signature is reached),
[4h] value #2 is almost exactly equal to the diameter found in [3].
[5] Use the results of 4 to show that the flight distances of {A,B,C,D,E,F} do *not* fit on a sphere at all but require an irregularly-shaped geoid, like an ellipsoid. Find the ellipsoid that allows the flight arcs to be embedded in a 3D Euclidean space, and use it to also determine the *latitudes* of the cities! This requires converting from *elliptical* arcs to chords, which requires elliptical functions.
[6] Add in a 7th city, G: Nome. Repeat all of the above with {A,B,C,D,E,F,G} assuming a sphere, then assuming an ellipsoid. Compare the results to those of [4] and [5]. In particular, does it fit the same ellipsoid that was found in 5, with the *same* absolute latitudes and relative longitudes for the subset {A,B,C,D,E,F}?
So if we can’t measure coastline properly because we have a 1d shape in 2d space, and we can’t measure area with topography because that’s a 2d shape in 3D space, then clearly the sane thing to do is start measuring the volume of countries.
Oh no, what have you done
So for simplicity i suppose we start at average sea level. Do the Netherlands then end up at negative volume?
@@arandacil you extrude the board down to the center of the earth
And then how about when the volume changes due to erosion, construction, or land reclaimation?
Now there are 4 dimensions.
Also fractal. As you increase resolution you get to elementary particles, and these have undefined (fractal) volume.
"Not only do they not match, no one took responsibility for them."
Yeah, that sounds like the UK government at the moment.
Are they finally out now?
@@A.Lifecraft last i heard, no, but it's possible something happened super recently and we're suddenly Out now
@@mozarteanchaos I hope this charade will be over soon. My guess is this will ultimately result in Britain rejoining once people realize what they did to themselves. So lets get it over and done with, because there are people who have to suffer from all this uncertainty.
That's been the UK for the past 200 years probably more
That's EVERY government.
Matt parker: "we can all agree that the Netherlands is perfectly flat"
Me: "what do you mean we have speed bumps."
And Limburg
Vergeet de dijken niet
@@mimi3570 ah, yes, dutch
Netherlands have bedrock above terrain
dremples, if I'm not mistaken
"where do we stop?" Just about 1 meter past the microphone RF distance.
This comment is underrated
Exactly what I thought.
Wow, I didn't even notice at first
Yes, Matt, your mic range is bounded
Hoooly smokes is that an orchestrated version of the StandUpMaths theme?!
It's goofy and epic at the same time!
Wait a second, I recognize you...
it supposed to be soundtrack for parker wars,
but it turned out that spreadsheet is not very good way for rendering a movie.
I am hoping for an ost
@@Irondragon1945 No seriously, I'm subscribed to you... I must know you somehow
I just have to say that your transition music is PERFECT for your show
Hey Andrew, didn't expect to see you here!
They have the original song available on the website, but can’t find the remixes
What hey andrew
I hate it, I cringe every time I hear it lol
@@MBKill3rCat that's weird
That moment when you realize, as the video approaches 7:00 and you feel the dread in your stomach; you realize that the coastline problem is near....
I was surprised he made it that far without mentioning it
Not only would the 2d coastline problem appear, but then a much worse 3d terrain fractal-ish problem would appear and ruin everyone's day
Skanderbeg not with volume!
@@skanderbeg152 Yes, that is mentioned in the video.
Internally I was like: "Father, here I command my soul to the eternal infinity of the heavens", when he mentioned the coastline problem... But when mentioned that the coastline problem could be extended to 3D, I felt as if I had become god itself that created reality itself just so that line would be delivered with a close-up shot on the crevices of that wall. Complete mathematical enlightenment: The surface area of every country is infinity. 🤣
Matt: As I always do when something about the UK irritates me, I went to Australia
Me: Ah. Matt did the British thing and sent his troubles to Australia
I like that someone once said, "In the 18th century, the British should have all moved to Australia and left the criminals behind!
@@timbeaton5045 On vacation in Australia, Douglas Adams realized that England is a cold, rainy, dreary, and cramped place, Australia is a sunny, beautiful, wide open place, and England had the bright idea to use Australia as a prison. No wonder there's a special smile Australians reserve for the English.
@@tparadox88 That special smile on an Aussie bowler's face at the GABBA when the next English batsman is coming to the crease when they are 5 for 70, perchance? 🤣
That’s academia for you!
“I extracted elevation data from the Google Earth engine, and then I combined this with the shape file from the global administrative database. I used a function within the R statistical language, which is based on trigonometry, to calculate the surface area taking into account the terrain; and I also did this at several different spatial resolutions to have a look at the effect that that has on the accuracy of the number. ...”
... for free.
Facts
Is it sad that I actually understand what was said???
@@makatogonzo no, i do too, hes saying online information is so diverse and available, that college is not that necessary to have the knowledge to make something happen
@@artratengo I'm pretty sure OP is amazed at how willing this person was to donate time and effort to this cause which OP(maybe erroneously) attributed to them being a part of academia. I don't understand how you ended up at your conclusion.
@@ruukinen well if you read the last part, the for free at the end of OOP' s post tells the same thing as i said more compactly
As a geospatial scientist from Melbourne, Australlia I was hoping to relax with a fun maths video.
But instead this sounds EXACTLY like the queries I deal with at work everyday.. LOL I literally just spent the last three hours building a coastal digital elevation model from LiDAR.
Thanks for the great video Matt and Bec! Would be great to see you cover differential GPS, where we use ground stations/mobile networks and a lot of maths to obtain ultra-accurate locations (sub-millimetre).
That sounds pretty cool. It's like you're an old timey surveyor *BUT FROM SPACE*
Matt just has a lot of questions and doesn’t want to bombard you
@@robertstuckey6407 nah, modern datasets are almost all from survey planes. (One of the standard global elevation datasets however was made by the space shuttles synthetic aperture radar experiments)
@@RobertSzasz A whole host of methods are used to generate geospatial datasets. Including GPS (measurement of asset locations/heights), surveying/lidar (land/property), aerial imagery/elevation from planes, but its also very common to use raster datasets derived from constellations of remote sensing satellites (ie SPACE, think > LandSat, Digital Globe, PlanetLabs, etc) to map/model our environments with all kinds of variables (we can even determine soil moisture levels "from space").
Would be great to see Matt cover differential GPS, where they use ground stations/mobile networks and a lot of maths to obtain ultra-accurate locations.
@@fieldo85 I some how missed the first couple words of your post .. 😳 sorry bout that
Orchestral theme of Stand-up maths means only one thing
Stand-up Maths : The movie
The movie? Nah, the trilogy!
And judging by the amazing landscape pictures in this video, the first one is going to be:
The Fellowship of the Torus.
@@wolframstahl1263 American π
@@apocalypsepaul
Harry Parker and the Mathematician's stone.
Harry Parker and the Vector Space of Secrets.
Harry Parker and the Theorem of Azkaban.
Harry Parker and the Goblet of Fractals.
Harry Parker and the Order of Operations.
Harry Parker and the Half-Blood Primes.
Harry Parker and the Mathly Hallows.
@PAUL BAILEY Fermat Wars: The Last Theorem? Alice in Fractal Land? Matt Parker and the Philosopher’s Geoid?
i'm ready for the Parker Cinematic Universe
I, personally, take great pride in the fact that since I stopped shaving myself and started to cultivate broccoli, I contribute to the size of my country in an efficient and meaningful way.
Lmaoo
...are there any places with any fractal buildings?
@@TinyDeskEngineer Yes, every building. Look at stone or paint under a microscope.
@@karlhendrikse that wouldn’t be a fractal
@@murpledeer A fractal just needs to have detailed structure at arbitrarily small scales. If you want it to look the same all the way down at different scales, that's a Mandelbrot set.
Honestly, I'm surprised Tom Scott hasn't covered this already.
"And that is something you might not have known"
I don't think there's a red tshirt that measures 247,719km²
Parker Tom Scott
It IS a thing that I may not have known
Numberphile did touch on the coastal area problem before!
Particular Kudos for the NationalAntheming of the Stand Up Maths music.
NationalAntheming is my new favourite verb
I think it was more CountryFile-ing, tbh, but I lolled!
That intro music just kills me.
Everyone
Everyone around is
That intro caught me off guard lol
9:30 - As an academic, I fully expected the sentence to go "... because they're an academic, they replied 9 months later."
In a physical chemistry class I took once, we ran an experiment where we had to measure the internal surface area of zeolites, which are extremely porous solids meant for having narrow channels and very high internal surface areas, by various methods. One method, BET, uses measuring gas pressure and volume to extrapolate surface adsorption and then divide by the surface area of the gas particle size. We noticed that the surface area per gram changed depending on the gas used, and I found a paper about fractal dimension and zeolite surface area. I wrote my paper for the project about the size of the gas particles relating to the resolution of surface area measurement and my professor told me he had never heard of that connection before! It is truly fascinating how fractal dimension can be relevant in some surprising places.
So to measure the surface area of each country, build an impermeable barrier around each, fill it with hydrogen, then do a bunch of mathematical calculations...
And if you were to gather results for such a project to measure the areas of countries, you'd find they tend to infinity, and probably strongly correlate with physical properties of the countryside's materials. Not to mention the errors from the hydrogen getting absorbed or reacted by things.
If you could have a chamber where you can pump gas in and out to achieve constant pressure, with circulation to provide convection, and a heating system for the gas, would it be possible to measure some fractal metric by selecting a pure gas, then changing the temperature and measuring the outflow of gas?
Obviously you'd need to factor out chemical reactions and account for adsorption, but this should allow you to test the sizes of gaps and crevices continuously, as hotter, less dense particles might avoid reaching into those crevices more than expected. You could sort of imagine the particles "acting bigger" when at the same pressure and lower density. This would drive out more gas than should be driven out than temperature increase alone should provide (remember the system has constant pressure), as the small crevices are not packed as efficiently because the particles are "acting bigger."
A really clever model could perhaps also look at the cohesion of the gas (and how that cohesion changes with temperature), but the mathematical relations for that could be a respectable nightmare. With or without such extra modelling, this effect would perhaps be nearly impossible to measure, and the errors would still drive out any meaningful data, but it's fun to think about.
Some large (particle size) inert gas and a very tight (average bonding distance) material object could make some of these considerations more possible. The large particle size and tight material in the solid might decrease adhesion, while having large particle size also limits resolution to ignore whatever wierd physics shenanigans might happen if particles get into very enclosed spaces while the particles are very close to the solid's surface. Making the gas inert should also pretty much eliminate cohesion, I think.
In WWI they tried it with Chlorine gas
Peter LeRoy Barnes there’s lots of interesting ideas in here! Using pressure is only realistic on smaller scales, especially because things like reactions but also because different materials adsorb gas on the surface at different concentrations so you would essentially get a free variable out the other end that would be difficult to manage-not to mention the difficulties of such an experimental setup!
Meanwhile in the Netherlands:
"Does the map assume the country is flat or use geographic data"
Netherlands: Yes
@@garrysekelli6776 'inland water counts as land'
Dutch: 'hold my Ijselmeer'
@@baskoning9896 _Waddeneilanden has entered the chat_
If you want to verify, there's elevation data available at half a meter resolution: downloads.pdok.nl/ahn3-downloadpage/
I think people are underestimating the Netherlands. Anyone who has ridden any substantial distance in the Netherlands know the county has hills. The lowest point in the Netherlands is about 7 meters below sea level while the highest point is a bit more than 320 meters, for a total variation of about 327 meters. According to Wikipedia, the flattest country is the Maldives, with a total variation of about 1.5 meters. (As a result, you can see why the Maldives are concerned about rising sea levels as a result of global warming.)
@@baskoning9896 "We're not trapped in by the ocean, the ocean is trapped in by us" - The Dutch presumably
There is a joke is Arkansas that the Ozark Mountains are so steep, that realtors can sell both sides of the same acre.
Greetings from Conway!
That’s the way you do it
Just have to watch out for Gowrow.
11:55 We found the range of Matt's wireless microphone.
I thought the cord on my headphones was broken. XD
xenontesla122 same lol
And @6:39 (and elsewhere!) we also found the limits of Matt's camera's Autofocus settings.
Panasonic G5, possiblement?*
*Ahh. The Panasonic Pony of Hope
I thought it was the cheap HDMI cable I bought.
We need an orchestral release of the standupmaths theme NOW!
Well, i have good news!
@@LeventK What's the good news! Tell us!
Well, I have good news!
I got a very similar question from my twelve year old son when we went mountain hiking. "Mum, did the GPS take the slope into account when it calculated the distance we hiked?"
Ah, the brilliance of children. not quite smart enough to know the answers but smart enough to know how to ask good questions, and then they know the answers.
my dads strategy was to just promise that the "hütte" (restaurant on or near a mountain top in europe usually supplied by helicopter, cable car or on foot, i dont know if thats a thing in america) was right around the next corner, or across that hill, basically just out of sight.
GPS can include altitude data when connected to more than 4 satellites, so, depending on the software, that's a real possibility
@@CMDR_Hadion Yes. My telephone (app Geo Tracker) includes altitude but I think the distance is calculated as a projection in x-y only.
My Garmin watch can track distance and spee in 2D or 3D.
Me when increasing the accuracy of my geodata: "it's free real estate".
It's not. You built a house vertically, or with a floor whose plane vector matches the vector of the force of gravity (if you use good enough builders). That means that for each floor you build you have consumed the 2D footprint of it in real-estate.
Agriculture now... That's a whole different thing...
@@ExMachinaEngineering woosh
@@ExMachinaEngineering Also you should never mix up 2D footprint of your roof (which is important calculating the amount of rain water) and actual surface area of your roof (which is important when ordering new rooftiles).
Thank you, fellow Dutchman. See my comment on removing seawater with Dutch canals to gain land area in the Netherlands.
@@A.Lifecraft what’s the difference?
The production quality of this video is stupid and I love it.
14:04 ish) Oops! Not sure if that was a typo or a speako...Screen says 247,719 but Matt says 249,719 😘
ha! I saw that too
Look at the % increases given on screen. They match the written version, which also are intuitively more credible. You people are wasting your time looking at a maths channel.
He puts these in to see if we’re paying attention.
Dr Laura Graham. Living proof that real heroes don't wear capes. Great work to answer the biggest questions here, Dr Laura, and kudos for coming on to such a huge TH-cam channel.
I'm feeling the need to point out his theme done in that semi-orchestral way was actually really good. Like, it works so well that way.
It was an epic opening theme
@@neurofiedyamato8763 epically awesome
Matt: "Conceivably Denmark... if we included its terrain..."
Every Swede or Norwegian, laughing patronisingly: "Come on, Denmark's tallest 'mountain' is like 100m high, it HAS no terrain"
I know right. I'm literally from the Netherlands and we have a higher "mountain" than Denmark.
@@Leyrann Really? I used to think that the Denmark is a bit more hilly than the Netherlands but I admit that I've never visited Denmark.
Technically, both Netherlands and Demark have their highest point in overseas territories
Our tallest peak is 170.86m unless we count monuments on top, in which case there's one at 183m and several other structures reaching even higher. Which brings up the general question if topographic area should include the exact shape of various buildings, including balconies and broken windows that extend the surface into the inside rooms.
Well, Netherlands is only 1,4% larger than Switzerland whereas Denmark is 4,4% larger (according to Wikipedia). Both countries are pretty flat (although with hills). Also, it doesn't matter how tall the tallest peak is, just how much it goes up and down.
I'm a surveyor. When we survey land, the area we calculate is the horizontal area at the elevation of some point in the land. So the way to compute the area of a country is to break it into little pieces, raise or lower each to the elevation of the land there, and add up the horizontal areas of the pieces, then take the limit as the diameter of the pieces tends to zero. (Not the area, since then you could make all of them long thin stripes.)
Also, we use conformal, not equal-area, projections. Surveys indicate the scale factor between the map projection and the horizontal ground distance.
Projecting a *geoid* is prohibitively difficult. What we project is the *ellipsoid*. The vertical distance between the geoid and the ellipsoid goes into the elevation scale factor.
16:01 If the entire population of Lichtenstein unsubscribed from Matt for that affront to their national pride, his subscriber count would only drop by about 6.7%.
On the other hand how dare you.
I love the implication that the entirety of Liechtenstein was, prior to this video, subscribed to Stand Up Maths.
I've done this calculation a few years back on the levant area. I basically took the greyscale data from the USGS website, found the elevation range for each greyscale image and normalized them (each image is different, sadly... lots of work), then compiled them into one big image. Next, import the image into Blender and map the image to a plane, offsetting the z axis by color, according to the normalized elevation. Blender then calculated the total area of the plane for me. It was a fairly significant difference, definitely over 1%. I'd love to see the impact this has on Peru or Nepal.
**and the larger budget enters the room**
*drones everywhere*
**...and it's so large that Matt has to exit the room and film this video outside.**
Obnoxiously so.
Still managed a Parker Microphone moment, though.
There is a PhD thesis topic in this somewhere...
I've been watching your content all day and was not expecting you here too!
@@MrBroady02 He *is* the official TH-cam Nerd Representative for Australia, so of course he's here.
Always nice to see somewhere in comments mate!
@@chemputer oh man, there are lots of nerdy Australian TH-camrs. Heck, Both Matt Parker and Brady Haran are partly from Australia.
added difficulty: use DaveCAD.
I'm really pleased to see that this video is only 7 months old. I remember watching it quite a while ago and thought it was actually years ago, and now I'm relieved to realize that time doesn't always fly by as fast as expected.
Your comment made me so happy until I saw your comment was made one year ago 😂😫
Sadly it now says 2 years ago...
"It's time we deal with the fractal in the room"
I absolutely giggled with joy honestly.
16:51 "I think we can all agree that the Netherlands is completely flat." --Matt Parker, 2020.
You heard it here folks. The Netherlands has only two dimensions. No buildings. No trees. Just perfectly smooth bedrock.
So that's where Flatland takes place. I always thought it was here in Florida
Wrong, Sir, WRONG !
You need to count all the damned dams in the Netherlands !
Rock, in the Netherlands? It's all sand, peat and clay, my friend.
Wait until sea level rises and they start to dome their entire country. In the 22nd century they will have the biggest submarine navy in the world.
@@HappyBeezerStudios We (dutch) would just call them cars
Interesting surveying anecdote; one of my surveying professors in college was taken to court by his father in law after performing a land survey for the FIL. The claim was that my professor stole (or disappeared) several acres of land from his property. The difference came from the original survey counting the topography in the land area, whereas the new survey was done conventionally with only the horizontal portion of surface area being counted. The two men still hate each other decades later.
That’s hilarious
As an Australian, whenever I see a drone fly-up shot, I always check the grass colour. "Oh, it's brown, yeah, that's Australia. Oh, now it's switched to another shot... green grass, that'll be the UK."
The grass is always greener...
@@ErwinPommel ...on the other side of the planet.
You must never have been to Tassie, or Gippsland...
You guys need to visit India
@William White Agreed!! Not an expert, but I find India too, having climate varying place to place. I'm sure Australia is same.
I visited Melbourne once and the weather was changing everyday.😄
Awesome place!!
most epic title sequence for a maths problem in history? this is fantastic
Great intro sequence for a great question! Please keep this epic music for your videos!!
Where can I download this epic track in full length without voice-over?
@@engywuck85 I would like to know that too!
For real, this was like watching a short film.
The orchestral arrangement of the stand up maths theme was absolutely breathtaking.
as soon as I read the title of the video I was like "why did I never wonder that before?!?"
also: oh my gosh I love the intro
Epic music upgrade!
Yes!
Stand Up Maths: The Movie
I think you missed an interesting tidbit. You see, the earth's geoid, which the Australian source referenced, is none of the above. It's not flat, nor is it a sphere, nor does it follow the terrain. Instead, it follows, roughly, where sea level would be in the given area. You see, mountains have gravity and the earth isn't uniformly dense. Oceans, for example, are much less dense than granite. So you see the ocean depress in the deepest and furthest out regions of the pacific. Meanwhile, the gravity of a large, and dense, mountain range would pull ocean toward it, making the geoid rise.
In the end, you get something that neither follows the oblong sphere that we think of as the shape of the earth, nor does it follow the terrain, yet is affected by both.
Well he did mention that it was corrected for the _local_ shape of the geoid and that it's not the same over the whole globe. He just didn't explain why that is, and the mention was just a side-note, so it's easily missed.
@@sourcererseven3858 Also it mentioned they use the Australian Geoid, which is probably slightly different to geoids designed to minimize error for the whole globe (Eg WGS84 used by GPS) as many are local standards that only care about accuracy within a given region usually a country.
The "solid" ground being measured also deforms constantly due to tidal force, just like the ocean, but without the large delay due to sloshing.
Thanks for this comment! I didn’t think about different densities
There is no point, you would get a percent at most from that
As someone who has never had a class in geoscience but has for professional reasons gone down the rabbit hole of projections, I chuckled at how flummoxed Bec was by Geoscience Australia's explanation which for better or for worse made sense to me. I was like, ok, sounds like they use an Albers projection with datums that minimize the distortions for Australia. The fact that Matt couldn't get a straight answer for UK was puzzling, but for this American, the US Geological Survey also uses Albers (different datums). And if you ever have to calculate land areas like me, you find out pretty quick that everyone does it by projecting it to a 2D plane first, not some calculation based on the surface area of a sphere.
I hope the orchestral version of the Stand-up Maths theme will also be uploaded to bandcamp at some point? Looking forward to it
Same!
We need a marching band version as well.
So epic. I love it!
bardcore it!
Oh wow, that new intro music is effing _EPIC!_
I loved the bit where you said the more accurate the elevation data is, the more slopes you find so the higher the area is
What have we learned? A country can have infinite area, and an infinite border, but maintain a finite volume. I started thinking about this a couple weeks ago, and it gave me a headache to say the least.
The natural next question is: What's the volume of UK? The surface might be a fractal, and so the area infinite, but surely the volume again converges to a finite value. More precisely: What's the volume between the surface of UK and the mean sea level?
I would imagine that's probably unknowable realistically. You'd have to measure every single building, every person, every animal, every tree, it would be absurd. And not just one by one, but simultaneously in an instant, or else all your measurements would be wrong again by the time you finished.
@@TheRealHungryHobo I think he just means the land volume... It shouldn't be too hard to calculate depending on how the topographical data is recorded. The land area is already calculated from land at mean tide.
@William White The perimeter of a 2D fractal is infinite. It seems the area of a 3D fractal would similarly be infinite, no?
@William White Take a 2d fractal. Give the fractal depth by giving it a finite thickness, say 1m. You now have a 3d object with infinite surface area (infinite linear perimeter x 1m is infinite area) and finite volume.
@William White 1) You say draw a box that's x by y miles, well you have to include z as well since we're including the 3rd dimension. 2) I can draw a 2D box around the island that has a finite perimeter, but the 1D fractal perimeter of the coast is still infinite. I can draw a 3D box around the island with a finite area, why does that imply that the 2D fractal area is finite?
Bec: *has a mental breakdown while reading a poorly punctuated sentence*
Matt: “...Thanks for that reading, Bec”
idk it doesn't seem weirdly punctuated to me, what's missing?
a comma before and? I don't think it's necessary
@@macizogalaico I think all that was missing was context. You can easily get confused by the long name of the dataset if you don't know it beforehand. If you know that "GEODATA COAST 100k 2004 data" is to be read as just one entity, you're trying to insert some kind of break inside there (as it's usually unlikely that a single term is made of so many words, and our brains work on likelyhood). But then you wonder where's the break? _Obviously_ there's some punctuation missing to make that break clear ;)
And honestly, punctuation _would_ help, though it's not grammatically necessary. Replace the "and" by a period and make it two sentences. Or replace the "and" by ", so" to show that the latter part is the conclusion and all of the former part is one subsentence.
The sentence was written by someone who was rightfully assuming that the reader would know what "GEODATA COAST 100k 2004 data" would mean and wrote the sentence with that in mind, but it was read by someone who didn't. The only one at fault was Matt for not providing the required context ;)
some italics would help
I was watching this wondering how far into the video we’d get before fractal geometry came up.
I’d love to see more geography videos on this channel - I find the intersection between geography and mathematics fascinating.
This is one of those questions that makes you feel silly for never having thought of it.
No,it just makes me amazed that their are such curious people in the world.
Exactly! I have actually previously wondered the 1D->2D equivalent of this, namely: does Google Maps quote you the "flat" distance when travelling, or does it take into account topography. Never went into enough of a rabbit-hole to find a definite answer, and also never considered expanding it to a 2D->3D problem!
@@EcceJack Hmmmm I never thought of that either, but I would assume Google Maps uses the topographical distance, since I would assume they get their distance data from surveys of the roadways, which I would assume are done by someone driving along the roadway and measuring every so often. That's a lot of assuming though, lol.
No, not wasting my time on this question doesn't make me feel silly because taking topography into account makes no sense when the calculation seems to be intended as a means for comparing the relative size of countries. If I dig a large hole in my backyard and pile up the dirt, thereby increasing the surface area of my yard, would it make sense to say the size of my property increased relative to my neighbors?
In a strong field, this is the strangest Matt Parker video I've seen and the first one I've turned off because the premise is lost on me.
@@tompaine4044 "thank you for coming to my ted talk"
Wow the production quality of this video is outstanding! Great job!
My favorite part of stand up maths is just how you can tell that matt has come from old TV with even just a 17 minute video having a TV show intro! 😍 love it. Really glad I can finally get my high video quality maths kick!
"Shared their slides" is geek-speak for flirting.
The Netherlands is already flat so it doesn't matter 🇳🇱🇳🇱
Speed bumps are the Dutch mountains.
Well, it goes into the negatives. Even your airports are below sealvl
Exactly. And I even live in Friesland which is, apart from Groningen, petty much the flattest landscape possible.
When I use my bike the only times I have to cycle uphill or downhill is when there is a man-made height difference for a bridge or underpass. Any other road is as flat as it could be xD
@@duncanhw exactly, Saba contains our highest point
*Flat earth vibe intensifies*
For the sentence at 3:20...
Probably the comma was meant to go here: "This area calculation is based on the 2D polygons in the GEODATA COAST 100k 2004 data, and topography is not taken into account."
The suggested comma placement, "This area calculation is based on the 2D polygons in the GEODATA COAST 100k, 2004 data and topography is not taken into account," would be a comma splice.
Flat earthers: *We don't understand the problem here*
Waited for this.
Well, the topography part is intuitive and for many countries curvature is irrelevant. Therefore you are wrong and could also cause smirks. So be more careful next time :)
Very dangerous to make measurements near the edges.
Many surveyors have fallen off,,, never to be seen again.
@@Gladaed *gasp*, smirks? How awful
I always wonder, do they think that Mars (etc) is flat too?
I was confused at first where I'd heard the sweeping title music before, if it's something in the Incompetech library. Then a few minutes later I realized it was just an epic orchestral arrangement of the regular channel theme
Yes. Where's the full length standalone version? Came to rewatch the view for the music, not the math.
I am so glad to get an answer to this… I thought about this when running up a steep hill and haven’t been able to get it out of my head since!
Since the fractal surface leads to an infinite area, let's calculate a country's volume. It's not a practical thing to know, but who cares?
I'd propose a countrys volume is either...
a) The volume between its actual surface and a surface at sea level.
b) The volume between its actual surface and a section of a sphere with the country's average height as a diameter.
a) goes to the country with the biggest average height times idealized surface area.
b) however advantages country's with the biggest elevation changes.
How about c) the volume between its actual surface and a section of a sphere with the country’s average border height as its diameter?
@@wsshambaugh That would be fun, too!
I guess once you got the data to answer one of those questions, it'd be a simple task to answer the others.
Considering many countries have mines and such I think mineable earth should be included, which theoretically would include all the crust which is about 18 mi thick over the continent’s (deepest borehole is 7.6 mi)
Each country goes to the center of the earth. So therefore the actual relief of each country is negligable. Only land area and latitude matter then.
Wouldn't b) make the volume 0? There will be as much of the country above and below it's average height, after all...
Just wanted to mention that Denmark's highest point is 171 meters, while Netherlands is 322 meters … #prouddutchie
The Danes thought that a hill that's 147 m was their highest peak until 1847. Said hill is called "The Sky Mountain" in Danish. I just find that hillarious.
I wouldn’t be proud of beating the danish. They don’t have many things. Let them have this
@@Spoon80085 Whoah mate! Unnecessary shade much?
@@kisteglad Well, I mean, they don't even have the Danish pastry. Lego is good stuff, though.
@@fanbuoy9234 They’re best traditional food is just bread, or sandwich in other langusges
How about the Vatican, source of many weird statistics due to a lot of old men living in a tiny country
The Vatican might as well be called the flatican, despite being built on a hill. Although of course for the Vatican, the topography of the buildings would actually make a difference.
Hey I think I just stumbled onto a related problem, I suppose it would only really be a thing for the Vatican, Monaco and maybe Singapore, but those little countries are basically completely full of urban development, and buildings are about as extreme as topographic features get.
@@AlRoderick if you're interested, the data used here seems to be DTMs (digital terrain models) of various resolutions rather than DSMs (digital surface models). A DTM represents the elevation of the bare earth whereas a DSM represents the surface including any buildings etc. So, it's unlikely the buildings in the Vatican were taken into account this time.
How are DTMs generated?
@@TheAlison1456 I'm not entirely certain but, from what I understand, DTMs are taken from DSMs with building heights and vegetation etc. subtracted.
A common method for scanning elevation is using LiDAR. It's like radar except using lasers instead of radio. With LiDAR, you can tell how far something is but also what sort of thing you're detecting. So I guess it can be determined if a height is of a building or land and then the rest is figured out somehow.
Okay Matt, you NEED to get that orchestrated theme tune on Spotify!
This seems like a very relevant question for countries that have land holders on their mountains for farming (growing coffee/tea/ect). It would account for why there seems to be more land ownership than area in the country.
At 14:03 Matt says 9 instead of the displayed 7 in the third digit, which is it? (Just wondering because that’s a fairly large difference)
The 0.94% change gives a 7, so I will assume that is correct for now
Apparently Parker Square can also refer to some sort of arithmetic operation he screwed up in that computation as well as an NxN square of numbers with some special property.
I assumed 7 because it didn't seem like the measurement would jump by 200.
I'm guessing 7. He has done this in a few other videos lately, too, where he said the wrong number but the correct number was displayed on screen.
@@tomtrask_YT It's the fabled Parker Number!! :)
It's entirely plausible that he deliberately places mistakes into his work, just like in his book humble pi. Either that or I diagnose early onset dementia
30 years ago, my brother bring back from a language exchange a mug that state "if Wales were flattened out, it would be bigger than England".
Not that anyone should try that, wales are a protected species!
XD
By the same logic, if you flattened our England, the ranking would be the same as it was. There is a lot of stored area in the Pennines, Lake District, and various wolds.
I remember having this question in geography class right after math when i was about 14 i believe, finally i get the answer.
"The Netherlands is completely flat"
Well, as you said the area depends on the raster size of the measurements you used for calculation. The entirity of the Netherlands has been measured with laser altimetry for a public dataset (AHN3). This dataset has 6 to 10 points per square meter and a standard deviation in the height measurements of 5cm. Because of how fine that grid is we might just beat Switzerland again in land area.
Interactive height map of the Netherlands: www.ahn.nl/ahn-viewer
Wow, 6 to 10 points per square meter is impressive. Not sure if it is sufficient againts the one of Switzerland with resolution of 0.5m though: shop.swisstopo.admin.ch/en/products/height_models/alti3D
@@niemandwirklich Then we might lose out, Although not completely flat, the Netherlands is likely to have much gentler slopes than Switzerland.
Okay, that data is detailed. You can see individual plants in someone's garden!
Funny thing is that the highest elevation in NL is higher than in DK ;)
Wow the standard deviation is 5cm? Netherlands really IS flat!
Oooooooooooh you meant the SD of the error not the dataset itself??
14:12 You say 249719, but the graphics show 247719
I saw that, 14:04 precisely. Think he needs to do more proofing before publishing.
Other than that good research as per usual.
HAPPY DAYS.....
He acknowledges this in the description- the on screen numbers are correct.
Probably what's written is correct seems to fit with the % maybe he didn't wanna redub the audio.
It's the Parker Area of the UK
@@Varil81lol
Enjoying the music going epic for the wide shots (helps me think about the big land instead of picturing the tiny drone filming it)
I've been wondering this exact problem for 10 years.
Nepal and Clile seem to be ideal candidates for investigation. . .
The americas as a whole would be interesting to see how much of a difference each has given that both North and South America’s have rather sprawling countries that have distinctive mountian ranges to them.
I would expect all three of Chile, Peru, and Equador to gain quite a bit, all of them with their mountainous areas near the coastline.
Very interesting video! Thank you Matt! I would consider Nepal as a contender for the biggest gain when compared to it's flat size.
Imagine if we included the surface area of the trees.
Imagine if you calculated the surface of the _leaf cells_
*R E S O L U T I O N I N T E N S I F I E S*
*computer science intensifies*
Or cave systems
FRACTALS!
I feel like when we ask for land area, 90% of the time what we really want is what you call the "flat footprint" (with corrections for spherical curvature for large countries) anyway.
If we're trying to estimate population density, for example. Well, people don't stand tilted when they are on a steep incline, they stand straight up and down.
If we want to estimate how much floor space we have for buildings: those are built straight up and down as well. You don't get any extra floor space when building a one-story building on the side of a hill. (Though you will save money on support material if the building has more than one story.)
Want an estimate on crops a country can grow? We need to know how much sunlight it gets. Hills don't get any more sunlight just because they're tilted. (Though mountainous areas will affect cloud-cover, I suppose.)
Want an estimate of how much rain they collect? Other than mountains' effect on clouds, having a larger topographical area doesn't increase your catch. Because rain falls, wait for it, straight down. Or... does it? *throw to VSauce*
Hey Vsauce, Michael here.
Stadiums and cinemas have tilted floors, can't they fit more people because of that? What about multiple floors? In my building, there are 4 families stacked on top of each other on the same flat footprint. If a terrain in the southern hemisphere is sloped northwards, it will get more sun than the flat footprint predicts. And for the rain, you would have to take into account the prevailing wind direction and speed to know how much rain would fall on that land.
Im proud of myself i anticipated the coast line fractal problem in higher dimensions from title. Actually learning something from uni
Oh Matt, Matt, Matt... The whole production of this video is brilliant! I adore all the drone shots, the film-style graphics and the orchestration of the theme music! I've had to switch to watch it on my 4k TV to appreciate the video in full glory (instead of my phone)! Keep up the good work!
The more important question for the area of The Netherlands is: do you include the area of the Wadden sea, IJsselmeer and Markermeer? Because that changes the area from 33.500 to 41.865 km2, a 25% increase!
Since the dutch can make land from anywhere, we should just include every bit of water that can be feasibly made into land. The Netherlands is now the biggest country on earth.
I would include the IJsselmeer and Markermeer, as they don't have a real connection to the ocean, so are functionally inland lakes. The Wadden sea is a sea, not a lake.
Fun fact: water area changes the largest province: Gelderland if you exclude lakes, and Friesland if you count lakes as land area.
I love how he changes his distance from the camera. Keeps it really interesting visually
As a Dane... We have Himmelbjerget. We'd be huge with terrain taken into account. - That mountain's like 147 metres above sea level!
according to two google searches what the highest points of The Netherlands and Denmark are, I have to conclude Denmark is flatter than flat, because your highest 'mountain' is about half the height of the highest Dutch 'mountain'
@@joeytje50Isn't it true, that the tallest point in Denmark is the top of some really tall bridge?
@@joeytje50 That's mostly because Limburg extends into the Eifel foothills. Though I assume it's similar for Denmark as well, having just one or a few really high points. It's much better to look at how flat the terrain is on average rather than what the highest point is, though I'm not sure whether there's good data available for that.
Tim the Traveler went to both.
Sadly the highest mountain here is the old garbage dump with roughly 42-49m (measurements differ)
The highest natural peak is 32.5m
Love the discussion guys but just to be clear my initial comment was meant as a joke :P
I have to say, this is unequivocally a good investment of Patreon money, research, music and filming included :) Cheers, Matt!
If one goes sufficiently high resolution, one could account for the undulating features of the brickwork, the rocks, the soil particles. even bark on the trunk of a tree, or the features leading to the chloroplasts through the stomata of the leaf. The area would be increased by orders of magnitude.
But when the leaves fall off in autumn ...
Absolutely love the orchestral theme music version, would love a full version to be released!
"Switzerland has the largest change in area percentage"
Nepal - "Hold my beer."
Exactly ! He didn't even mention Nepal!
Although high, isnt there a huge plateu in nepal, which is flat (ish)
@@paulquaife7974 fair point, there is sizable lowlands on Nepal's southern borders. Perhaps Bhutan or Lesotho may be better contenders?
@@davegrimes3385 my guess was chile
I have to imagine Chile and Argentina are up there too.
As you skimmed over the flat/spherical question I think you may have missed a significant point. The Australian area figure in particular states that it's based on the Australia geoid. This is NOT the spherical surface, nor the ellipsoid one. The geoid is the gravitationally flat "sea level" surface of the Earth, which has its own terrain due to the strength of gravity varying from place to place as determined by variation in density and depth of the Earth. So the the Australian area (at least) does take terrain into account, just not the visible one!
"Hang on a sec...we've introduced a whole new problem now." Ah, maths, where problems arrive faster than they can be solved!
Forklift17 very few subjects solve problems faster than creating them
That's the nature of problem solving.
thinking back a long ways to when I took a surveying class I realized when we took 2D area measurements we didn't take into account slopes (but we did for 3D topographical measurements)
And I think I know why, buildings are almost always built on a flattened bit of land, so either earth is removed or added to bring things to a level area. So, if my thinking is correct, the reason countries only care about flat measurements for area is because that's the usable area for structures.
I am curious how that all works out for farms and ranches, because if a farmer buys a hilly plot as one acre, and it's actually, for example, an acre and a half they'd need more seed to plant it, and if a rancher with livestock bought the same land they'd theoretically be able to raise more livestock per acre.
I live in the Mississippi Delta area, which for all purposes is flat, so all the farmers I know don't have to deal with topography so I can't just ask one of them haha
For an area to increase by 50%, it would need to be on a slope of greater than 45°. So quite hilly, and probably marginal for many farming systems. Not really cropping land.
However, it's worth asking whether the land would have a greater potential productivity than an adjacent equivalent flat area. The major factors are energy inputs and water. Clearly the energy input from the sun doesn't change just because the land is tilted - it's the same total energy per unit time, spread out over a greater area. There might be some minor benefit in terms of energy conversion rates, due to the plants being spread out more.
On the other hand, the slope creates challenges for water management. And having grown the pasture (or crop), we are faced with less efficient conversion rates due to the livestock using up extra energy climbing up and down the hill (or due to harvesting the crop on a steep slope).
3:15 pretty sure it's supposed to be something like "This area calculation is based on 2D polygons in the 'GEODATA COAST 100K' data from the year 2004, and topography is not taken into account."
Denmark's highest point is 170m, so can't get much flatter.
Wow, never knew the Netherlands were (slightly) higher.
@@rogerwilco2 almost double
Yeah there's a reason people in the Netherlands and Denmark like cycling so much
The state of Florida tops out at 345ft (105m)
@@dustinbearden150 So you're saying sea level rise isn't all bad then?
The Netherlands has a "mountain" of 300m high, it's not completely flat.
Denmark, on the other hand, is entirely lowland, as the highest point is less than 200 m above sea level.
@@ragnkja A perfect country for cycling😂
If you exclude Limburg, the highest point in the Netherlands is 109m
@@uitham and what is the higest point of Denmark if we arbitrarily exclude 2000km2 of the highest elevation area?
@@HelgiWaag It's not arbitrary you dingus, Limburg in itself is an arbitrary piece of land that remained to the Netherlands which shouldn't have given it ended under Belgium for years.
I can imagine Chile, Nepal, and Bhutan would also have significant differences in area with topology included
Matt: "The terrain where I live is reasonably..."
Me as a Scot: "That looks reasonably flat, yes."
Matt: "...hilly."
Me: "Wait, what?"
But I'd really like to know how the area change differs in Scotland from the rest of the UK. Not because of any kind of nationalistic mentality, but because Scotland is fundamentally different from a geological (and topological) perspective. Scotland has the highest 18 peaks in the UK, and out of 120 hills over 600m, Scotland has 81.
Edit: Sorry, in the British Isles. Sorry Ireland! :(
I did break it down by country (at each spatial resolution) - have a look at the table near the bottom of this: github.com/laurajanegraham/surface_area/blob/master/surface_area.md
In short, Scotland has the highest change (1.8%, based on the 90m resolution data), followed by Wales, NI, then England.
@@lauragraham6149 have you looked at this data set? www.ahn.nl/ahn-viewer
I really wonder what difference the height map would make at this extremely high resolution.. and what would a graph of the percentual change as a function of the height map's resolution look like?
I haven't watched yet so I don't know how it's really done, but I can't believe I always took for granted (not even that, it never really crossed my mind, it's something I did unconsciouslly) that everything is measured as flat.
You have one of (if not THE) best theme musics for your channel!
Inb4 Switzerland submitting a UN resolution to officially change the way land areas are calculated in order to climb up a rank.
I would have included Nepal in the consideration for most mountainous countries.
@William White True, though Matt would've probably excluded it based on area, like Liechtenstein.
my first thought too
Bhutan is probably even more creased than Nepal, as it has basically no large flat areas at all.
@@emptyshirt I shall be on Google Earth later - educate myself!
@@emptyshirt p.s. love the word 'creased'! could another be 'pleated'?
There was some whispering about the "epic orchestral arrangement of the regular channel theme" and how we require a full length standalone version for our aural enjoyment as well as a bit of enlightenment. Sadly, it seems to have been kept _from_ us.
Bec Hill's readings have convinced me to listen to the podcast.
Good day Matt! I was once traveling and heard a local meteorologist say "With rain over 20% of the area" instead of what I believe to be more commonly "With a 20% chance of rain." Perhaps these statements are different, perhaps they're analogous. I'd love to see what you find interesting about a 20% chance of rain over 100% of the land area vs. 100% chance of rain over 20% of the land area. Implications? Thank you, and I'm thoroughly enjoying Humble Pi :-)
And what does 20§ chance of rain even mean? Does it mean it will rain about 20% of the time? Or is it the chance of 20% that there will be rain at least once in the mentioned time period?
Daniel Bohrer, they run the computer models for the forecast under many different assumptions, then average out what proportion of them said rain would happen to 10% or 20% increments.
in that statement he clearly say 20% of the area. so he is certain that 20% if the area is rainy (asuming the area is flatenned)