A Strange Map Projection (Euler Spiral) - Numberphile
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- เผยแพร่เมื่อ 12 พ.ย. 2018
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The Euler Spiral paper Hannah mentions: arxiv.org/pdf/1202.3033.pdf
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Poster and sticker based on this video: teespring.com/en-GB/euler-spiral-world-map
Bought the poster and framed it, I love it so much ❤️
I really love this channel. I'm glad that you're all doing this.
Can we see this mapped out properly on a computer model image please? That would be pretty cool!
@@lxdimension Mathematica should be able to handle it.
Earth is flat. Like a dinner plate.
Oh no, wait until the flat orangers see this video...
Oh no
There’s probably not a great deal of them watching Numberphile.
My mind went to the team from the marble races instead of poking fun at flat earthers.
Hirudin
So globbies can no longer say a sphere cannot be put on a flat surface
Brek Martin
I’m a flat earther and a scientist and i watch them
This is spiralling out of control.
Your joke might have fallen flat...
Orange you happy with the result?
That orange pun was terrible! I bet helix windows.
gottem
I appreciate that pun
Normal people: the earth is round
Flat earthers: the earth is flat
Mathematicians: the earth should be a spiral
The Earth is a doughnut
the earth is hollow
The earth is a Potato
Earth is a planet
Only satanist christians think the world is round. Especially the capitalistic Christians from USA that love Trump the demon lord Nurgle. Ha ha I needed to write that sorry.
"In 3 hundred meters, make a loop around the earth, and then turn right"
The Euler Spiral map both horrifies and intrigues me.
We need a history of the world day by day map on this projection.
That's the beauty of math
i am ONLY utterly horrified at the time waste, as it has 0 benefit
how about that math -- cost, (trends to) infinite, and benefit, (trends to) 0.
HISTORY YEAR BY YEAR ON THIS MAP
Marcus Aurelius it has a certain amusement value and introduces the concept of the Euler spiral. An old saying about knowing the cost of everything but the value of nothing comes to mind.
I love Hannah's presenting "style", relaxed but enthusiastic at the same time. I've always been a bit of nerd when it comes to peeling oranges. One of my favourites is to make a little lantern out of it. I also remember seeing (in my grandpa's magic book) a way of peeling an orange that allows you to remove the orange but the peel stays as a sphere that can expand to get the orange out but keeps the overall shape. I've forgotten how to do it though. I'll need to see if I can find out how to do it again. I doubt it would work as a map though, but it looks cool.
Hey, a globe is the best kind of map we have! :)
@@GreeneyedApe Yup, hands down! Except when you want something you can fold up and put in a drawer or a glove box.
Fred
@@ffggddss Inflatable globe! However, globes are rather impractical for typical driving directions.
A spiral would work, but I guess it was a more compact shape?
I wanna hear more about your grandpa's magic book
My favorite map of the globe was designed by Buckminster Fuller. It projects the earth's surface onto an icosahedron, distributing the distortions across 20 triangles. It also radiates the continents out from the north pole, displaying the world as basically one large land-mass surrounded by one continuous ocean. As much as possible, the cuts are in the ocean.
they use that map too
I don't know if what I enjoyed more - Hannah's narration or the mathematics. Both are delightful
Hello, I am from the Flat Orange Society. Mind if we have a word?...
The Earth is Flat. These are bunch of projections. This is not our real map. This is a small map of small part of Earth. Who knows what they're doing in Antarctica.
@@ancientindianguru1714 Hahahaha
@@ancientindianguru1714 if we end the federal government we can take back nasa and area 51 and we can find out whats really going on.
@@alfredodominguez2799 Yes, I heard they are developing some serious high tech pasta!!!
Thank your lucky stars you're not from the Flat Easy Peeler Society, I want to know why there's a conspiracy against my satsumas.
I've learned about gaussian curvature when the Klein Bottle professor explained to me how to correctly hold a pizza slice!
Clive Stoll's video about it is great!
Cliff is very... enthusiastic :D
Yes! I was thinking of that video too.
*Klein bottle.
@@epajarjestys9981 thanks, edited ;)
Hannah Fry: “We’ve only got this room for an hour. What should we do?”
Obviously cut up a globe into an Euler spiral
My reply would have been inappropriate!
😂😂
Lock the door
@@JLHunter61 can incels just get outta here?
@@americantoastman7296 lol dude the original comment was an obvious setup for lewd jokes. stop getting offended on behalf of other people, dickface.
"I think we should prioritize mathematical beauty over geographical practicality." - Hannah Fry
I can't tell you how much I love this statement.
You sound incredibly pompous
@@alandouglas2789 ?
It's just a little joke.
@@alandouglas2789 I will devour your mother.
New Hannah Fry video?!?!? It's like christmas to me
Merry Christmas
To me to
In Poland >30 years ago you would’ve seen oranges ONLY on Christmas
I was thinking the same thing!
Same here !!
Agreed :)
Imagine folding the Euler Spiral map in the car.
Simon Moore No problem. You are not folding the spiral, you are folding a piece of paper with the flattened spiral on it. So it's like a nice long map that has one or two folds in the short direction and an easy harmonica in the other.
Nooooooooo...
@@johnfrancisdoe1563 yeah.. better than what I remember.
You could reel it up on two spindles.
@Multi Mason So basically like a new Torah?
If hannah fry was my maths lecturer, I wouldn't miss a class
And neither would we really ever learn anything.
I came to the comments section for this.
TH-cam: Numberphile in title
My brain: HannahFryphile
I'm glad I'm not the only one with those embarrassing lapses in focus.
What? Sorry I wasn’t paying attention.
I'm so lazy, I probably still would.
"Mathematically beautiful, if Geographically impractical" 😂
We're using this to assemble big spherical concentrators in space to make really low mass solar power systems. So, we find it technically practical.
Shut your dirty mouth Hannah Fry is perfection
Gotta get the T Shirt!
@@williammook8041
That sounds very interesting. Who is the 'we' in that comment?
@@williammook8041 but still not geographically practical :(
As a geographer, I really loved this video and learning about this new projection. This projection shares a feature with the Mercator projection. On the Mercator projection there is one place where there is zero distortion, which is the equator, where the cylinder would touch the sphere. The Euler spiral enables you to create a similar line of zero distortion that encompasses the whole globe. Of course, in both cases the true line of zero distortion is a one dimensional line, which is why you have to go to infinity in the spiral to get there, but I completely see the beauty in this. I love it.
thanks, this helped me understand what they meant by "no distortion"!
gussnarp could you explain this further for me?
@@brendonholder2522 when you have a curved surface mapping it in 2d you get some distortion
but when you make the strips since you can make them really tiny stripes
while you still get distortion it will be smaller
practically the centre line of each stripe will have no distortion
which in practice means you will have more points of no distortion of your end map
if you theoretical you can get stripes the size of line you will end with no distortion at all
ANIKHTOS that’s really interesting! I’m writing an paper (an Internal Assessment Math investigation for the IB Diploma Program) and I’m thinking of using an Euler spiral to make a map such that you can move in vectors on the map in a manner similar to that of a Mercator projected map. Any suggestions on how to go about investigating this?
@@brendonholder2522
well i have not seen any map
but the most serious question is how the coordinate system would look like??
the longitude lines wil be inside the strip going up left to down right depending how you cut the spirals from the globe
while the latitude will be almost vertical line at the strip
i imagine the goals is the spiral will match the latitude lines to be one big line???
besides how weird it will look like
since the map lets see reaches perfect of the globe that means you have 2d representation of a 3d surface
so even if you make vectors in the map
you need to have 3d geometry to calculate their values
and that will be for the 2 circular parts
what about the line that connects them??
some points (area of the globe ) will not be in either the 2 circular parts put in the line connecting them
so you will have to split the formula for the circular part and the line part
but we need a visual of this projection to see where everything has moved in
is the spiral actually form a circular area or not?? or there is tiny gaps in between ??
or you will make a tine distortion there and make it a circular part?
after all you will not be able to cut infinite stripes so you will introduce some distortion in the end
love hannah
dingaia yes, “talk to”
4:05 “There are a few different options here, but none of them are going to get you completely around this problem.”
I see what you did there.
Cutting earth spirally?
Yeah. Why not?
@Noel Coward "...get *completely around* the problem" ;)
I never get tired of the deadpan humor on this channel, great work as always guys.
I think it's important to say why/how Mercator is useful for navigation. A straight line in Mercator is not a straight line in real life, however, if you navigate with a compass, your compass will remain pointing to the same direction throughout your line.
The video isn't specifically about the Mercator projection, dude
Would you happen to know if a reference that explains this in detail?
@@XenoghostTV Dr. Fry talks about Mercator and why it is used so commonly, especially in navigation.
But a straight line (a geodesic) in the real world always corresponds to a curved line on a Mercator map. For example, the shortest path between Mumbai and New York passes through western Russia, Swender, Norway and Iceland. If you looked at the Mercator map you'd think it went through Arabia and North Africa, and those are quite far away.
What lumer2b wrote is correct. Mercator is useful when navigating with a compass.
@@Banzybanz Okay but that's not the damn point of the video...
It’s called the rhumb line. It’s not far off the great circle route, with the advantage that you can steer a constant heading.
Are you telling me there is no Euler Spiral map of the earth generated by computer with n=9999 anywhere on the internet?
yet.
I trust that by writing this comment I will be notified when this happens
I would like to be added to this list.
As would I
RemindMe! 2 days
And yes this is the first entry in an impromptu petition to implement Reddit's RemindMe bot on TH-cam. Spread the word.
Your subscribers are now 3.2 million but I am wondering who was the Pi millionth subscriber 🤔
There couldn't have been! Don't be irrational.
That number does not belong to the set of countable numbers.
There wasn’t one. The queue for that title is infinite, so the person queuing for it, would have had nowhere to stand......
@@legislativequeery There is at least the 3,141,592nd subscriber (or 3,141,593rd if you round it up).
@@dwc1970 Yes, if π≠π such that π ∈ ℚ→ ∃ π*10⁵ ∈ ℕ
But that would break math
It doesn't actually have to be geographically impractical if you're trying to travel in a straight line... We have different projections for small pieces of the globe that are minimally distorted. So if you adjust the "poles" of the projection to a place that'll allow your course to fall along the spiral, you can have a nearly undistorted map of your course the whole way!!
Google maps doesn't use Mercator anymore. It's a globe now.
It never used Mercator projection either. The projection that was used called Web Mercator.
Well, sorta…it still uses it for mobile :)
Interesting! You know, I don't think I've ever zoomed out enough to notice that
"Web Mercator" is just a shittier Mercator that's easier to compute.
@@AleksyGrabovski omegalul. "It's not English, It's British English!"
A Euler Spiral map would be a great piece of Numberphile merch.
Now you're playing with power!
Cristobal Jorje I prefer playing with surds to be honest.
Only if Hannah does the cutting.
"When I went to university, this is not how I imagined my life would turn out"
Same...
That's because university is just a continuation of the public school indoctrination. Real scholars can do as well or better apart from the university...especially now that we have internet, and all the knowledge on earth is easily accessible to many millions.
@@rickbluecloud531 Stop bringing politics up where it doesn't belong. We get it, school bad because it teaches you about slavery. Shut up already
@@ChangedMyNameFinally69 there are political aspects to this issue that need to be considered if it's going to be correctly understood. And I'll continue sharing whatever I'm led to share by the Spirit. I have no regard for arrogant demands by rude people.
@@rickbluecloud531 What political aspects? Them teaching you that slavery happened? I'm genuinely curious.
What Spirit?
@@ChangedMyNameFinally69 your comment is rather incoherent anyway. Maybe you should proofread and make corrections after you sober up.
Bradey: What is it doing that other maps aren't doing?
Hannah: ... It's an Euler spiral Bradey, what more do you want?
Exactly what I was thinking! haha
*Yooler*
Send the orange man to the hydraulic press channel, and we'll see if he'll still have a positive gausian curve number
Yes!
The trick is knowing when to stop...
:-/
Cataphractos Contrafactum this may be a joke, but it will.
"Velcome to heeoodlraawlik plress channel...."
Or "velcome to beyond depressed" (Beyond the Press, their second channel)
"Vat de faak!?"
it will have tears and wrinkles because it has to go somewhere
8:51 "Turns out the world is really big"
[citation needed]
The example given evinces an interesting ingrained geometrical and psychological bias that, if bypassed, increases the utility of using an Euler spiral. As can be seen in the physical globe ball that was cut up, the area of greatest utility (at least for use in mapping) occurs at the centers of each spiral (the start and end cuts). Conversely, the these are the areas of least utility for most uses on a map. But it is oddly ingrained psychologically to think we need to start the process at the poles. But clearly this is not the case. If, instead, one starts the cut in the center of North America, or the Eurasian land mass, and pick a point precisely so that portion of the cutting that becomes the long connecting arm between the spirals rests in the middle of the ocean (or some other arbitrarily chose point of least interest). one gets an Euler spiral projection of greater utility.
Rediscovering some numberphile videos is a reminder of how much of an inspiration classic TH-cam used to be. Numberphile and computerphile really did help me realise I could still learn new things in my late 20s and early 30s 💕
You think you learned something, but it's just trivia. Popular science (or math) is not college science (or math).
@@Raison_d-etreof course one learns
I thought it was pronounced "oiler"?
Jay Benton [🇺🇸-er]
It is.
It was his name, so it's pronounced the way he pronounced it, and it rhymes with "oiler".
Eh-yu-lehr
"oiler" is right.
Map-matically beautiful projection. Love all the Hannah Fry vids. Keep 'em coming.
Thank you.. Great fun and great presentation! I'm 62 with two degrees from university.. but I did not major in mathematics or physics.. now I find myself wanting to start over.. this is wonderfully engaging.
Frank from Boulder, Colorado, USA
As a retired US Navy officer, this is pretty interesting. It looks like this projection is what you would get by cutting along a rhumb line, which is the line you get by taking a constant compass course from one pole to the other. Or in other words, you cross every meridian at the same angle. The biggest virtue of the Mercator Projection, as Hannah noted is that every rhumb line on a Mercator Projection is a straight line. One of the faults of the Mercator Projection is that great circles (shortest distance between two points on the sphere) are not. I believe that, except for the meridians and the Equator, they are all sine waves on the Mercator projection.
Would this projection also have great circles as straight lines, if chopped up straight lines?
maigretus1 Intuitively, I think any great circle would need to go through the long arm between the spirals, and so would not be a straight line.
You'd get a single line that is infinitely thin for an infinitely long line so it wouldn't preserve angles at all, so great circles couldn't be calculated very easily. The position of an object on the surface of the line (if it has a finite width, so not at the limit where it goes to infinity etc) would be a function of the width of the line and a periodic function. You could produce a version of this with different limits exactly how you describe, you're an engineer, you know this stuff, the limit being that your compass direction to produce an infinite line would be exactly east or west starting at exactly the north or south Pole. Given that this is a right angle, it would take some time to travel it.
That is because the earth is flat and they project it onto a 3d surface
which is then projected (with a different transform) onto a 2d surface
OSSSSHHHH The Earth is not flat.
My upcoming math papers are about Euler spirals and transverse Mercator projections, so of course I clicked on this.
Mercator is NOT projected from the center. That would magnify too much along the meridians. You project from the South Pole, then take the logarithm of the resulting y-coordinate.
Awesome on Tomorrow's World tonight Hannah. Such great memories of a real favourite childhood programme.
If Hannah Fry was my math lecturer, I would take extra lessons.
Jam Kon much...
Euranges. Yummmmm.
Eurth
Eurquator
Speurl
I'm using a Euler to measure this sentence
Acella, Multiverse Dweller
Do you know how to pronounce Euler?
But everything you’ve said makes no sense.
interestingly my mum always pre-peeled my oranges for school in such a spiral, only peeling the orange rind and leaving the pith intact, and leaving the rind just attached by a little bit at the base and wrapped around to preserve the freshness. I also commonly see people peel apples in spirals like this to bake - I think it just feels right as somehow there is less resistance to the peeling motion when you follow this path because it lies flat against the fruit instead of bending up and fighting against you
Charming and very practical! Spirals from different starting points provide multi-plex code potential placing place names opposite other place names to make word list. Change angle of attack and point of first scissoring into the blow up globe and you can develop unique word order sequence. Code key would be 4 items: Long-Lat of start, angle of attack, choose your first place word, such as "Chicago", and finally which way to amble, either left or right (poleward or spinward are obviated by initial attack and can't be used as constant)
Great video! I'd like to see another one like this, explaining Euler's spiral in the context of Fresnel diffraction.
Cheeky 1:09! The position of the stalk is just perfect 🍊 😂
Naked Orange Peel man is same proportions as naked Orange Man according to Stormy...
Nice video on map projections. Of course cartographers have several others used over-the-years, and for various applications. Thanks for the arXiv link as well.
Euler is pronounced: Oiler. Preferably with an Australian accent.
Just for Your informationtion.
Thomas Borgsmidt And Fresnel is pronounced “Freynel”, IIRC.
Thanks. I was pronouncing it as "Ew-ler" xD
Which proves that Australia does not exist. Or so I learned somewhere on the Internet..
Yeah, exactly what I was gonna say.
oiahlah
I read about this ages ago. Its been on my mind when I had a conversation with someone about globe maps in a flat plane. thanks!
YES, one of my favorite people on this channel!
Before just now the Dymaxion map was my favorite projection, but now the Euler Spiral projection takes the cake.
Dan Whiteman I prefer the Winkel-Tripel projection (or however you spell it); way less distortion than Mercator, but still familiar enough to be easily used.
Her dad did it because that's the kind of things dads do. Childhood is a magical time, and encouraging that sense of wonder in your kids is a fantastic thing.
*Playing with trash on the floor*
"It's worth it for the mathematical beauty!"
This helps explain is the reason why countries closer to the equator appear smaller compared to those closer to the poles
"yuuler"
:[ Hannah why
You missed the "Dymaxion Map," the Fuller Projection Map - the only flat map of the entire surface of the Earth which reveals our planet as one island in one ocean, without any visually obvious distortion of the relative shapes and sizes of the land areas, and without splitting any continents. It was developed by R. Buckminster Fuller who "By 1954, after working on the map for several decades," finally realized a satisfactory deck plan of Spaceship Earth.
Ordered one off amazon a couple days ago, I love it, I like flattening it out imagining the migration of humans out of Africa ~60,000 years ago
We got ourselves a new mathematical object: New Zealand-preserving map :D
Hannah is amazingly intelligent and super lovable. Amazing math.
You spelled mouth wrong :o
Pawel Kita looool
Check out Buckminster Fuller's Dymaxion map or Fuller map is a projection of a world map onto the surface of an icosahedron, which can be unfolded and flattened to two dimensions. The flat map is heavily interrupted in order to preserve shapes and sizes.
actually there is still distortion. The distortion is minimized and localized to the centroid of each facet in the icosahedron. It's flatter, but not on the same scale an this euler coil.
Although this is to be expected, it was neat to see [in practice] that the Equator is contained along the path between each of the two spirals. Super cool!
Hannah's voice is perfect for ASMR.
OOh bro, it's off to the Gulags with you.
@@dixztube Like. I think IQ videos are supposed to animate us to life. Nobody should be thinking ASMR on a Geophysics video.
I looked into this question of getting the Mercator projection by projecting a light - in order to do it you'd have to do some funny business on the map to the cylinder. The problem is that the Mercator projection involves a natural log for the y-coordinate, while projecting light rays is all intersections of lines with spheres and cones, which can only get you algebraic maps.
Absolutely right - what Hannah describes would give you the perspective cylindrical projection, not the Mercator.
Living in Canada and having access to roads that travel directly north to as near the pole as possible, there are deviations in the route which are called correctionals. They are the euhler equivalent of the orange peel or strip map created in the video. I prefer the Mercator map. It spreads out the imperfections evenly and appears to give a lesser distortion.
Thank you for your efforts to explain a difficult subject in an entertaining way
I wish y’all showed a simulated product with a ton of spirals, but thanks so much!!! Awesome video
This is wonderfully done. It demonstrates that every type of map has it's pluses and minuses.
I bet you could do a great presentation of how the requirement for spherical trig in celestial navigation, yielding a great circle route, is a proof of the round Earth. Plane geometry, the option for a flat earth fails; it would get you lost if not killed. It would be enlightening to discuss how each can be treated as a testable hypothesis which models a world view.
If you squish a 4D sphere to 3D sphere, does this rule still apply?
Rapth Yes.
Valid question
How do you cut up time into a Spiral?
@@StarryNightSky587 the question implies four spatial dimensions
I want a video about what the animator has to go through lol
10 minutes of inverse kinematics
Very nice!!
Great job Hanah!!! Thanks
Very well done, and down-to-Earth. Lively and fun, and technically accurate. Euler, as in Beuller! That's great!
2:51 That form of projection was on one of the US evening news shows in the 1960s and 70s. I think CBS.
Thanks, Hannah, your mispronunciation has reminded me that I’ve got to get an Eul change for my car today…
A yuul change
I really love the way you present and the fact that you love what you do - I could listen for ages. Week don e
With guidelines in the longitudinal direction for visual queues and cutting it at the equator to make two circles and exact spacing of the gaps between the spirals this would be a fairly functional map. That is pretty awesome. You should totally use a computer program to make one perfectly and see how it presents.
"youler spiral" oh no
The video brought to you by Anglo gang.
wheeler
1066 worst day of my life
freznel
@@chigginheadD mathS ;-)
This was the purest, and honestly beautiful thing I have seen on TH-cam in quite a while. I was captivated.
Love it. Great look at radical map projections..
Great video, math rules! Just finished reading "Hello World" and completely enjoyed it. I'm sure the book will inspire some young people to study math and pursue math careers. Thank you.
4:01 I feel so bad for him.
Very informative! I love map projection mathematics.
That's pretty cool in that it separates into the two hemispheres quite nicely
Great video, thank you for sharing your idea with us. On a side note, I as well, always used to peel my grapefruits a in a spiral way!
1:09 oh look at his little "stalk" 😍
A mathematician pronouncing Euler as "Euler" is tantamount to a computer geek pronouncing LateX as "LateX".
Hmm. That is better said than written.
Why cant we develop a system such that we spell and write as we proununce? (In English).
@@rajeshwarsharma1716 you mean like the hungarian language? :)
@@rajeshwarsharma1716 Italian language my friend
@@rajeshwarsharma1716 International Phonetic Alphabet my friend.
@@rajeshwarsharma1716 That's how we do it in the Balkans.
Love the way you are explaining.
Fascinating! Thanks for the video!
Math question: what's the probability the stem position at 1:10 is purely coincidence given its surface area relative to the orange? A) less than 0.01, B) 0.01 to 0.03 C) 0.03 to 0.05, or D) over 0.05
I'm pretty sure the "shining a light" description is not how the Mercator projection is defined. According to that, the poles would be an infinite distance from the equator, and from the Mercator maps I've seen, you can see too much of Antarctica and the northern boundaries of Canada for that to be the case.
It's a simplification the video is short
It is cut at some areas around poles, so it doesn't get to the "infinite distance" part. It just gets to the "far enough that the distortion is really bad" part
@@doku7335 Nope. I actually looked up the definition of the Mercator projection since this discussion. See my previous response that describes the difference in the trigonometry.
At my kids school they have a map of the world which is flat, but it also got a logarithmic scale at the bottom too calculate the real distances based on the curvature.
So at the equator it's more or less 1:1 and at the north or south pole you've got quite different ruler to use.
The scissor cut you made on the globe follows a rhumline; a line that intersects lines of longitude at a constant angle. The Euler strip does not have straight edges, they are curved, and the cardinal compass points vary in orientation depending on location, but the distortion is minimal. On a Mercator projection, staight lines plot as rhumlines and the cardinal points are orthogonal everywhere. Mercator has little distortion at the equator and lots of distortion toward the poles. Mercator is best for navigation up to about 60 dgrees latitude. Navigation in polar regions is done on Gnomic projections. A Gnomic projection casts shadows from the centre of the globe onto a flat plane that is tangent to the Earths surface at one point near your location.
YES HANNAH FRY!!!
You had me at “mathematically beautiful” ❤️
I like the phrasing "peel the surface of the Earth to make a map". The orange analogy is super easy to understand!
“We’ve only got this room booked for another hour.”
Steady my trembling soul.
"As a map projection, this is mathematically beautiful if geographically impractical." Brilliantly elegant. 11:30
I never would have figured that I would get one of my best laughs in a long time by watching a math video.
i really enjoyed this one. thanks for that
I love that nerdy grin at 7:58!
Today I discovered the pattern I've always doodled is actually a beautiful mathematical shape.
I've long harboured an urge to do exactly this. Hannah has saved me a lot of time and eventual disappointment with the outcome. I'm not sure if I feel relieved or cheated.
bob street well you could probably manage a much neater (flatter) one if you’re willing to devote the time to cutting the strips with infinitesimally narrow widths!
For me this is the most beautiful video about science I know.
The Earth is big, but practical navigation usually involves distances that are relatively a lot smaller. At the scale of a typical voyage between places not too close to the poles, say from Sydney to Hobart which is about 630 nautical miles, it is OK to draw a straight line on a Mercator projection from Sydney to Hobart and sail down that line. Ideally, the shortest course would be the great circle route, but that is inconvenient, and the difference is small. The Admiralty Navigation Manual publishes traverse tables which provide the solutions for plane triangles and they have been getting ships from A to B very nicely.