Beware the Runge Spikes! 

the "8 inches per mile squared" isn't an invention of the flat earthers, they just took it from land surveyors and large scale engineering (like building long bridges, where you have to account for curvature), where it is used as an approximation.

Actually, this wierd 8*d^2 formula comes from the Taylor expansion of the cosine finction. And the approximation is so damn good because the argument of a cos(.) is tiny as we divide by the Earth's radius

I was not prepared for anyone, much less Matt Parker, to say "Well done, flat earthers" without a trace of irony in his voice, but here we are! I'm actually impressed by how accurate that approximation is for measuring the drop of the earth behind the horizon for most practical distances. It's not perfect, of course, but it's accurate to three significant digits out to hundreds of square miles, and I can respect that kind of accuracy for napkin math.

I love that the first 6 minutes is literally just rediscovering taylor series from first principles, that was fun

“With four parameters I can fit an elephant, and with five I can make him wiggle his trunk” - Johnny von Neumann

My understanding is that 8 inches per miles squared is actually used as a quick shorthand approximation for engineers when working over relatively short distances, like the scale of buildings. I believe that was where the flat earthers got it.

1:51 spat out my tea laughing at Matt being a baller.

“That’s still over the horizon, if you will. Or, indeed, won’t.” 😂😂😂 That’s some baller humor right there.

I just have to say the baller glasses made of spreadsheets made my evening. Just the right amount of hilarious detail.

Hey, I just learned while working on a project recently that a shortcut for making an absolute cell reference in excel/sheets is pressing F4 while it's selected. At roughly 3:30, you select cell D1 with and it looks like you manually insert the dollar signs. While you have D1 selected, you can press F4 and it will automatically change it to an absolute cell reference with the dollar signs. I'd say it's a gamechanger but really all it does is keep me from pressing the arrow keys to edit the text field while actually moving a cell to the left.

Time to send this video to senior management, who didn't believe I could replace their very expensive software with some polynomial approximations in Excel.

The gangsta shades made of excel is just pure genius!

I'd like to point out the difference between Taylor series (used by the flatearthers to approximate the drop (cos(x)=1-0.5*x^2)) and Lagrange interpolation polynomial which you use to connect points with a polynomial (used later in the video).

This brings back some good memories. In 1989, I was frustrated at the poor performance of floating point operations in Turbo Pascal and I started optimizing them. One of optimizations I did was to use Chebyshev polinomials to approximate transcendental functions instead of the simple Taylor expensions that Turbo Pascal used. In 1990, my work got even published in a German computer magazine.

Gotta love watching people suffer using imperial units :D

4:00 Hold on. It’s not a FE approximation. It’s as you’ve shown very accurate for a few 10s of miles. In my Ordnance Survey textbook of 1925, it’s stated ‘for distances that are practicable’. In land surveying one rarely deals with observations longer than a few 10s of miles. And 8” x mi^2 is archaic. In pencil and paper calculation, pre machine calculation, it’s how it was done. In todays software world 8” x mi^2 is not used. I still do a lot of pencil and paper calculation.

the 8in/mi^2 is a surveyor's approximation so it makes sense that its very close

It also works in civilized units! It's 8(ish) cm for each squared kilometer It's 7.8456 to be exact, but I'd say it's close enough. It works because the ratio of mile/km squared is close to the ratio of inch/cm (mile/km)^2 = 1.609^2 = 2.589, inch/cm = 2.54

gotta say overleaf is probably the most appropriate ad I've seen in a maths video. Thanks Matt and overleaf!

I kept on waiting for you to expand the graphs past the point that you did the fit to, to see how badly they blow up. To my mind the biggest problem with a polynomial approximation is that it tends to go wild once you leave the region of interest that you did the fit in.

The 8d^2 approximation does actually get used in practice fairly often - if you're on a boat and see the tops of the masts of a ship over by the horizon, how far away is the ship? You could do this by doing trig, or you could just take a reasonable guess of how tall it is, divide it by 8, and take the square root.

I had to deal with the Runge Phenomenon for a regression task and i wish i knew about Chebyshev Spacing. Luckily it wasn't too much of a trouble, i used Generalized Additive Models (GAM) and it literally saved me.

Very excited to have this show up, as it is one of the things I'm studying right now, exam in two days even. One note: Chebyshev nodes do not guarantee no blow-up of the approximation error; One can construct a continuous function such that the error blows up. BUT for functions at least one time continuously differentiable (i.e. in C^1), the error always converges, which is amazing. Also any potential (and rare) blowup is logarithmic, so pretty tame for polynomials. There is even a sad theorem that for any choice of interpolation nodes, one can find a function such that the (polynomial) approximation error blows up with more and more nodes. So for an 'A priori' choice of interpolation nodes, Chebyshev nodes will likely remain the best choice forever. Hope I got everything right, this is one of my favourite topics, thanks for covering it!

Yes. I believe the quadratic approximation to the earth dropping away is very commonly used in surveying and in artillery, and since if you want to survey a single distance or fire an artillery shell, it's effectively always less than .6 times the radius of the earth away, it's a safe assumption.

Anyone who's dealt with splines in solidworks has had the delight of discovering Runge Spikes. It is very common to make a small adjustment to a spline and end up with a disgusting mess. It is really cool to learn just how CAD software actually handles the maths behind splines.

I really, really enjoyed seeing the input into the spreadsheet, searching google, et cetera! I felt vividly connected to your thought process watching that segment. 😘

Matt talking to his dog: "Yeees, you would if you had to collaborate on a scientific dogument."

Literally every academic I know uses overleaf. It's a life-saver.

You're making clear why all my attempts at vector graphic design failed miserably to match what I intended.

The name Runge brings memories of my student days when I was only starting with numeric methods

2:19 - I'm reminded of Weird Al's - White & Nerdy =)

No barleycorns were harmed during the computation of the units conversions.😂

I'm amazed, the flat Earthers actually found out the first term of the taylor series for the trigonometric function! Well done!

I love the idea at the beginning of the video implying that in order to understand a flat-earther's mind you'll need to apply reverse engineering 😂

Love how my brain shouted use Chebyshev spacing, the moment I saw the equally spaced points. Guess numerics I wasn't complete and utter useless for me.

Feels weird to see a sponsorship of something that I have personally been using a LOT in the past couple of years! Overleaf is incredibly powerful, and has helped my studies very much.

I actually stumbled on both the effect and the solution accidentally a couple of years ago when visualizing the ellipticity of different orbits. Started with evenly spaced points on the Major axis but the joined ellipses looked horrible around the apsides. Re worked it to start with evenly spaced points on the circumference of a circle and boom, beautiful arcs at the apsides. Never thought to apply it to function fitting though, so thank you, I learned something new = )

I think the pup should be in every episode going forward.

I was wondering about those rungebot tweets... Turns out problem was that rungebot always tried to plot -1 to +1 even if function is not defined on the entirety of the interval...

As soon as I saw "polynomial approximation" in the title I knew you were gonna talk about chebyshev nodes, since it's one of the coolest results in approximation theory to me

3:55 The frustration you feel when the Flat Earth Math is actually kind of sorta correct

I hated math as a kid. I used to skip it and tell my art teacher my next class was a free period. Eventually my math teacher came down to the art room and hauled me out. But eventually I grew to appreciate math.... eventually. Thanks Matt, loved your book by the way.

Note that Kerbin from Kerbal Space Program is more like 85 inches per mile. That's the sort of thing where it becomes really very noticeable not just that it's not flat but that it's significantly smaller than Earth, although still immense.

I actually was on a forum with flat-earthers in HughesNet many decades ago. This would have been around 1993. And I came up with the same formula for the approximate distance to the Horizon by first driving it via the cosine relationship you showed and then using the first two terms of the Taylor expansion of cosine. And then of course making all the substitutions and conversions back from the metric system to the imperial system. I wonder if I accidentally wound up giving this formula to the flat-earthers that I was debating with.

You can get the formula with the distance squared with the help of the small angle approximation.

I'm constantly blown away by the power of Taylor series, even when I use the almost every day while I get my PhD. Math is just so heckin' rad!

Learned this the hard way at work trying to use polynomial to interpolate missing weather data. Original data was hourly, but some gaps were weeks long. Ended up with temperatures above the surface of the sun 😅

I would like to know why Chebychev nodes work so well as they do, and why the spikes appear in the first place

Great to see you on Mastodon!

Would love to see a comparison of this with Taylor approximations and best approximations (the whole projection into polynomial vector space stuff) too.

um, not sure they came up with this approximation, but I was taught this 20 years ago in surveying class, or sticking with imperial it was 1 inch every 660 ft. Which is 10 survey chains. This was knowledge built into the US public land survey system. Which is why you have principal meridians and such to deal with the errors from curvature and terrain.

The same approximation comes up for beam deflection under a constant moment (equal moments applied at both ends). Needless to say, for steel beams, the radius of curvature is usually very large compared to the length of the beam. For an engineer like myself, anything within 1% is good enough. In my early career, when I worked as a welder, I used to use the same approximation for presetting/cambering beams.

Polynomial fitting is very handy in forward error correction. Nice and easy to follow. Except that you have to do it in a finite field. At least Ronge won't bother you there.

The chebyshev spacing brought me back to my mechanical engineering mechanisms classes as this is used in creating a function generating 4 bar mechanism.

The Auto-generated captions are amusing me with all the talk about "Chevy Chef" distributions on a "Jojoba" file.

Was the Semicircle distribution the mathematically best distribution or was it just the best "most general" distribution? I guess my question is if one could fine tune a distribution for any given plot that maximizes the accuracy of the approximation.

Your editor deserves a raise

Hi Matt & Lucie. So good to see you together on RU-vid. I'd totally forgotten that you two are a pair.

Samuel Rowbotham in his first version of Zetetic Astronomy included the Encyclopedia Britannica article on levelling. That gives the derivation of a surveyors approximation of 8.008 inches times the square of the distance in miles. The drop is the radial drop, not the perpendicular to the tangent.

+1 For freedom respecting social media, like Mastodon

Slightly related, from what I have seen online and in the US, I think something that holds back numerical literacy today is that many people were not taught how to properly handle units. I was taught unit/dimensional analysis for the first time in 10th grade (UK year 11 equivalent) in my physics class, which was not required by my district. I imagine a vast majority of adults in my area were never introduced to that way of thinking and when presented information that is strange or just blatantly nonsensical, they weren't given the basic tools to understand that.

I'm having finals in two days and that's one of the things I didn't understand fully. Matt, you were sent down by the God to help me pass.

Matt's video topics seem to overlap more and more with subjects at my uni, I love it!

I remember seeing the a graph approximated Chebyshev-style and I was like why are some point so close together whaaat??and this video just really cleared it up! thanks a bunch!

Not sure if this was covered in the video since I didn't have time to watch all of it, but there's a really obvious reason as to why a quadratic works so well out to even a few hundred miles. The exact formula is given by r*(1 - cos(d/r)). We know that cos(d/r) = 1 - (d/r)^2/2 + (d/r)^4/24 + ..., so the exact formula equals d^2/(2*r) - d^4/(24*r^3) + ... In short, we have a small angle approximation that looks quadratic for maybe the first 10 degrees of rotation around the earth.

this reminds me of the Gibbs phenomenon: when you try to get a square wave pulse, the sharper you make the edges the more "spikes" you get on them. It has something to do with fourier transform but I can't remember exactly how it relates. Also is it just me or does the polynomial aproximation shown looks exactly like sinx/x ?

My first thought was "that's not dimensionally correct". 3:42 When I was in the Scouts we had some maps that were 1:63360 scale. I mean it's 12 x 3 x 220 x 8, it makes perfect sense. As Newton said, "If I haven't seen further, it's because I'm standing on my head."

Changing the spacing for your approximation sometimes feels like black magic and is super useful, especially in numerical integration! For some work I did, I used to do radial numerical integration on a equidistant grid and then switched to a Gauß-Legendre grid - the number of abscissas needed went down by so much (Like more than a factor of 10 iirc)! Furthermore, I switched some angular integrals to use points on a Gauß-Chebychev grid and without using more abscissas, the accuracy was much, much better, especially close to kinematically critical spots! It is very nice, seeing a popular youtuber putting some attention to this :)

I like to use rational functions as approximations. I rarely feel the need to go above third order.

4:31 The Flat Earthers are not stupid on math. They are capable of making a projection of what their opponents' theory would imply, intellectually. The _real_ problem is, the kind of drop they measure below your horizon is, the kind of horizon you see if your eyes are down on the ground of a totally flat landscape. Part of the reason you actually see further is, you might be standing on something elevated in the landscape. But another part is, you are arguably standing, when you look for the horizon. This changes the equation, since you are yourself elevated.

I'm so glad I was great at maths in school. I'd hate to have missed out on all of the great content on your channel, Matt. Thank you! 👍👌

The name Chebyshev comes up in amateur radio as a common type of filter that may be designed either as a pass-band or stop-band filter. The advantage of this filter type is that it has sharper band edges (cutoff) than some other filter designs, but they suffer from some ripple in the pass- or stop-band.

Im really tired of flat earthers getting the press they do, in general, but I have really enjoyed a lot of the neat different proofs I have seen for showing the earth is not flat. At least something positive has come from all of this. Thanks Matt! I have seen a video that actually showed a visual drop of the base of a structure that should actually be visible if the earth were flat. A cool cad program showing how a building large enough would have to compensate for the curvature of the earth and tons of stuff.

For the Earth drop formula, is this not just an example of a small-angle approximation. With those you can drop the trig functions by assuming that Sin A = A when A js a small angle.

My physics professor assigns us Latex problems on overleaf. Can confirm, it's wonderful.

Hey Matt! Thanks a lot. I stumbled upon this exact problem with this exact function while researching problems to give to students of mine, and since then I was too lazy to research what it was and how to fix it.

I have used the Chebyshev Spacing before, but I came up with it on my own, no heavy math, just kind of manually. I was not aware that it even had a name, now I know super cool.

Please do a video on the more arcane trig identities, like secant, cosecant, versine, etc. My memory might be wrong on the details, but I think even in quantum mechanics they will use 1/cos x rather than use the secant of x, which I feel misses out on what the maths is actually telling us (plus it just looks more elegant). I learned the trig identities when I began debunking flat Earthers many MANY moons ago, and was amazed to learn what they were telling me mathematically about some very basic geometry. I think it's a shame we've mostly lost them not just from the classroom, but also obviously from advanced academia. They're very beautiful when you get to realise what they are and how they work. I mean, we use the haversine formula to calculate the shape of the Earth, and barely anyone would be able to tell you how it works or what half a versed sine even is. I just think that's a bit of a shame. Like we're losing something important in the maths by not acknowledging these identities enough. Or maybe I'm just a hopeless maths romantic? 🤣 One thing I can tell you is that debunking flat Earthers made me fall in love with maths - specifically geometry and trigonometry - again, to the point I'm pretty sure I've invented a theorem that didn't exist before about how the baseline of a right angled triangle grows as the apex angle increases whilst the height remains constant. Pretty useless, and I'm not surprised if nobody's done it before, because it's only use seems to be to debunk flat Earthers, but it was a lot of fun to figure out anyway. And if I am by some weird miracle to be the first person to come up with it, there's a mischievous part of me that enjoys my legacy being coming up with a mathematical theorem that is totally useless and means nothing to anyone in the greater scheme of things 😉🤣

That is why I always approximate pi to 3 and pi^2 to 10...

really like the improvement in production value, videos are much more fun now :)

While this is an awesome empirical formula to use (and as another commenter pointed out, it also works for cm/km too!) If you want a unitless approximation formula: Take the cosine approximation, cos(th) = 1 - th^2 / 2 And plug into the equation derived in the video, h = r [ 1-cos(d/r)] And you get, after simplifying, h = d^2 / 2r Which is a pretty nice result as well!

Could you also do a video explaining how to use these polynomial approximations to approximate integrals? I really love the topic

This is a fantastic demonstration of why things like Newtonian Physics worked for us for centuries before Einstein's much more complicated corrections came along.

Some flat earthers accidentally claimed that their community has members from "all around the globe".

Just wanna remind everyone that Matt did a kickstarter in April 2022 that took £100,000, and we haven't had an update since December even though it's now almost six months overdue from the estimated delivery time. Don't let him forget all the people waiting for this. Really feels like it's been swept under the rug.

I used Overleaf in my engineering undergrad for most of my group papers. It takes two hours to learn and saves dozens over using Word. One thing you didn't mention that makes it so great is it has live updates. So you see in real time what your collaborators are doing vs Word where you have to update every so often manually.

love the switches from light to dark theme as you search

Right on. Thanks for sharing.

I look forward to more collabs with doggo and wife

I used Overleaf for my PhD thesis.

a really good paced video

Matt: I'm a baller. Me: One might even say, a shoot caller.

As an engineer who tries to draw with the spline tool often, I have an intuitive sense for this phenomenon.

The Flat Earthers do some great science! Problem is they when the results say the earth isn't flat they toss it out. I think many of them are just trolling us. I watched a documentary where the flat Earthers proved the earth was sphere-ish twice.

I love your units. Very creative and unique

9:30 Ahh, the inverse quadratic radial basis function. Warms my heart.

This video is baller 😎

I've been using Overleaf ever since I started my Bachelor in Physics 4 years ago, *I absolutely love it!*

Don't forget that the Chebyshev nodes are also the pointa uses for Gaussian quadrature.

"You don't want to do trig on the fly" - So true, Matt.