A Strange Map Projection (Euler Spiral) - Numberphile

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Numberphile : Poster and sticker based on this video: https://teespring.com/en-GB/euler-spiral-world-map

fishy paw : 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.

EmperorTigerstar : The Euler Spiral map both horrifies and intrigues me.

jalabi99 : Hello, I am from the Flat Orange Society. Mind if we have a word?...

Cataphractos Contrafactum : Send the orange man to the hydraulic press channel, and we'll see if he'll still have a positive gausian curve number

Useless GTA V : *sexiest mathematician ever*

Danny : I've learned about gaussian curvature when the Klein Bottle professor explained to me how to correctly hold a pizza slice!

e4r : No oranges were hurt during the shooting of this video

temujin9000 : "orange man bad!"

Hirudin : Oh no, wait until the flat orangers see this video...

TheCordlessOne : The sexual tension in the room is unbelievable.

Simon Moore : Imagine folding the Euler Spiral map in the car.

lumer2b : 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.

Luis Mijangos : Hannah is amazingly intelligent and super lovable. Amazing math.

auskott : This is all fake. CGI trickery. Oranges are flat.

Shruggz Da Str8-Faced Clown : Isn't Euler phonetically pronounced "Oiler", not "Yoo-ler"?

maigretus1 : 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?

Kentnstay : A Euler Spiral map would be a great piece of Numberphile merch.

Sonny Mattsson : 5:34 Not anymore though...

Samsul Hoque : Just wait until the Flat Orange Society finds out about this video.

Mr Chains : but isn't e pronounced oiler and not yuler

Pierre Abbat : 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.

Michael Wisslead : Lots of comments about Euler being mispronounced but what about Fresnel?

Adam Charvát : New Hannah Fry video?!?!? It's like christmas to me

Darwin Schuppan : 1:14 uuhhhmmm what is that thing between the the orange man's legs..?

Tomasz Gałkowski : Google Maps no longer uses Mercator, they moved to showing 3D globe when you zoom out far enough.

Nisarg Jain : The amount of rage people have here about youler and oiler makes me repel the mathematical community over youtube.

McLurr : This is a grapefruit!

geryon : Google maps doesn't use Mercator anymore. It's a globe now.

James : Cheeky 1:09! The position of the stalk is just perfect 🍊 😂

Batfan1939 : Does this mean a mathematically perfect Euler spiral is infinitely long, despite its finite area?

ManifoldSky : 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.

John Harley : Are you telling me there is no Euler Spiral map of the earth generated by computer with n=9999 anywhere on the internet?

Pixie Panda Plush : Mathematically beautiful, geographically impractical.

Chongee : Even though I knew about these, I've still enjoyed the video ;)

Kasper Joonatan : 0:05 so cute <3

agmessier : 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.

Dev Agarwal : This is spiralling out of control.

Jeff Erickson : Gah! Ack! Pffftht! Please! It's pronounced "oiler", not "yooler"! A mnemonic fact to help remember this: The first generalization of Leonhard Euler's polyhedral formula to polyhedra with holes/handles/genus was proposed by the French mathematician Simon Antoine Jean l’Huillier, whose last name literally means "the oiler". (And it's pronounced roughly "l'weeyay".) Don't get me started on Frez-nell.

radastir : Honor the noble sacrifice of the hero beach ball, martyr of science!

The Hellene Archduke : YES HANNAH FRY!!!

Max Johnson : Orange man have glorious pubis. Very nice!

Gijs van Dam : I see that Numberphile has prioritized beauty as well.

poshnomolbralpku : It is pronounced Oiler not yuuler

Michael Anuzis : 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

Henry Segerman : 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.

panchamkauns : The most adorable Numberphile video in a long time!

Gabriel Dos Santos : Or you could have the map drawn on a sphere

[username not found] : You got the construction of the Mercator projection wrong. Let φ denote the latitude of the point that is to be mapped and let y denote the vertical coordinate the corresponding point on the map has. (The equator is at y = 0.) In the projection described in the video: y ∝ tan(φ) In the actual Mercator projection: y ∝ arsinh(tan(φ)) These are quite close to each other for small |φ| resp. |tan(φ)|, i. e. close to the equator, because arsinh'(0) = 1. However, the closer you get to the poles, the worse this approximation works. Also, the projection from the video is not conformal, as opposed to the Mercator projection.

SteelSkin667 : Not great for navigation, but probably great for data storage.