This reminds me of a discovery I made while vacationing in the Corn Islands, Nicaragua -- There was a giant cube buried in the middle of nowhere with one corner poking out of the earth. A plaque nearby said it was the 'soul of the world' and I looked it up to find this funny old organization (http://www.souloftheworld.com) had planted the cubes around the planet, saying that their positions were the vertexes of a cube that was positioned in such a way (and only possible in this way) that every corner touched land. Apparently it has a certain spiritual significance, which is funny to me, but I thought it was really interesting that someone went through the trouble to create them.
After an exhaustive investigation, we found, surprisingly, that there was only one possible combination on the whole planet that would permit the eight vertices of the cube to emerge on solid land. (The results of the investigation made with the AMQ were later corroborated in a study carried out by highway engineer D. David Fernándes-Ordóñez.)[0]
I wonder how difficult it would be to replicate this finding (or a similar finding). I'm naturally skeptical given the extreme remoteness of the islands (Points include the Cocos Islands (Australia), the Corn Islands (Nicaragua), and the Hawaiian Islands (USA))[1] and the not-perfect-sphere shape of the earth, but I also know that I don't have a good natural intuition about the shape of the earth. This would be an interesting project!
Well a good starting approximation might be to take a WGS ellipsoid model of the earth and see if their existing coordinates forms something close to a cube.
>After an exhaustive investigation, we found, surprisingly, that there was only one possible combination on the whole planet that would permit the eight vertices of the cube to emerge on solid land.
Not really that interesting IMHO. After enough searching your're going to find a shape that meets the criteria. If a cube had more than one configuration, they likely would have just move up a notch, and if a cube had no solution they would have moved down a notch.
Brings back many memories. Back in around 2000/01, the website had complete contact information for Gene Ray including phone number. So I called him up and he rambled at me for about 20 minutes before he had to go because his grandkids were coming over.
I don't believe so. I know there is one on the island of Molokai in Hawaii, which I always wanted to visit when I used to live in Honolulu, but otherwise its not a super tourist-friendly place to visit and I opted to visit other islands instead. The Corn Islands, however I highly recommend to anyone!
On a similar 'frequency', this sort of 'spiritual significance' question has been around for quite a while. It appears that three Temples — of Parthenon (438BC), Poseidon (440BC), and Aphaea/Aphaia (500BC) — are located at the vertices of a triangle. AKA 'Holy Triangle of antiquity'.
That's not unusual. Any three points can form the vertices of a triangle. :)
The first site claims that these three temples form an isoceles triangle, which really just means that the distances between two pairs of points are roughly equal. The second claims that they form an equilateral triangle, which would be more unusual -- but is clearly false.
> "A really smart mathematician, Fra Luca Pacolini, demonstrated mathematically, that the four regular solid bodies: the Tetrahedron, the Cube, the Octahedron, and the Icosahedron, correspond respectively to the four elements: fire, earth, water, and air."
They use height data rather than land/water data, and assume that there's just a cutoff at sea level. I think this gives them the correct answer for the water path, but it gets things wrong for the land path because there are a lot of lakes and rivers above sea level and also some dry land below sea level.
If you don't allow the path to cross any lakes or rivers at all then I think the land path has to be much shorter, since rivers makes it impossible to make any progress at all in most places. If you trace their path it definitely goes through several lakes. The optimal path is probably across a desert, my guess would be Antarctica.
If you do allow the path to cross lakes and rivers then I think there's a longer path than the one they give, starting in Liberia and ending near Fuzhou, China. They probably didn't spot this one because it passes too close to the Dead Sea, which is below sea level (and crosses the Suez canal, which is at sea level).
> They probably didn't spot this one because it passes too close to the Dead Sea, which is below sea level
This is adressed in the paper. The relevant part of the problem statement:
"the longest distance one could drive for on the earth without encountering a major body of water"
and about the Dead Sea:
"Guy Bruneau of IT/GIS Consulting services calculated [5] a path from Eastern China to Western Liberia as being the longest distance you can travel between two points in straight line without crossing any ocean or any major water bodies. However, the path crosses through the Dead Sea (which can be considered to be a major water body), and hence does not satisfy the constraints originally set out."
Oh, I didn't spot that they had already considered that path.
However I don't accept their defence.
1) Depending on environmental conditions, my path can cross the Lisan Peninsula. https://en.wikipedia.org/wiki/Lisan_Peninsula (EDIT: In fact there's enough clearance to just go completely south of the Dead Sea)
2) Their path crosses the Volga River, which is much larger in total surface area than the Dead Sea. And at the point at which they cross it, the Volga is just as wide as the Dead Sea.
A global map with resolution
of 1.85 kilometers ...
1.85 km is roughly one Nautical Mile, which is roughly one arc-minute of a great circle. As such it's a natural unit of measure.
The original definition of the meter was 1 ten-millionth of the distance from the North Pole to the Equator via Paris, making the Earth's circumference 40 million meters. Divide by 360 degrees per circle, then by 60 arc-minutes per degree, and you get (40 x 10^6) / (360 x 60) which is about 1852 meters.
I was wondering how one would make a global map with a resolution of a nautical mile. Spacing dots a nautical mile apart on the equator is simple, but you can’t continue with a rectangular grid up north or south (by the time you’re at 60 degrees, the dots would be only half a mile apart)
Equatorial circumference is 40075 km. One arc-minute of equatorial longitude is about 1.855 km.
People who do crazy things with measurements cause a significant fraction (at least 15/360) of the many pains in my ass. Degrees need to die a painful death, and so can grads. There is nothing at all "natural" about them as a unit of measure, as they are arbitrary divisions of one rotation, by 360 and 400 respectively.
But I admit, even if not natural they may be more convenient for some calculations than other numbers.
Given that you are already using degrees, arc-minutes, and arc-seconds in your specifications of latitude and longitude, it makes sense to use a unit of distances that meshes with them. Using meters makes less sense because it doesn't work well with the size of the Earth.
Difficulties with measurements and conversions are like a dead cat under the carpet - no matter how you push it about there's always an inconvenient lump and a bad smell. The Earth is inconveniently shaped, and you simply have to deal with it. The existing system of measurements may offend your sense of taste, but it has evolved over time to be useful to those who have to use it. Attempts to devise systems a priori and without taking into account the extensive experience of those who actually use them have always failed.
I think "natural" was meant in the sense "it naturally follows" or "it's natural to assume", where it's meant to imply a natural path given your preexisting knowledge and conditions, not natural as in "this comes about in nature".
With that in mind, 360 is a fairly natural way to segment a circle for modern Humans.
You have twelve joints on the fingers of one hand. (Use the thumb to count.) You have five fingers (including the thumb) on the other hand to track multiples of 12 with. Voila, 60!
360 and 60 are quite handy, by virtue of having a lot of integer divisors. For a civilization where long division is unknown or limited to a few scribes, being able to divide things by common fractions easily is an important criterion in a system of measure.
“The question now is: who will be the first to make these journeys, when, and how?”
These have the potential to become quite important journeys... with the potential for many “firsts”... and also much contention; how much did they deviate from the path, how much deviation from the path is acceptable?
Same here, that’s what I thought as well :-)
I’m sure that the big documentary channels will pick this up and we’ll see both journeys in a few years.
Seems to me that the land journey will be harder to complete. When you’re sailing it’s easy to keep a “straight line” without having to consider mountains, rivers, permissions, politics and any other obstacles associated with crossing land.
For the land route a realistic approach would be to plot major checkpoints along the true great circle path and just navigate between those checkpoints in whatever way (preferably on land) is most feasible. That would still be interesting.
Yeah I was thinking about how hard it would be to stay “true” to the path as possible if a race or challenge was by wind power only. Also in competition, points could be given or taken based on the amount of deviation from the line etc.
"A global map with resolution of 1.85 kilometers has over 230 billion great circles. Each of these consists of 21,600 individual points, making a total of over five trillion points to consider."
What? Is there something I'm missing here, or did they just decide to include "individual points" as an utterly useless way of inflating the difficulty of a brute-force approach?
at a 1.8 km resolution (without checking my maths) 21,000 points along a great circle is basically every 1.8km so i guess you are checking at each 1.8km if you just drove into water (or vice versa)
Your point that you only need to fail once is fair - there is a lot of boundary conditions one can apply quite quickly
Going from Pakistan to Russia seems like the wrong way around Cape Horn. Today you've got 30 kt winds gusting up to 50 kts. Who wants to beat into that?
If we're going to split hairs, an actual straight line would be tangent to Earth at the point of the boat, and you could sail forever, so long as you do it at the speed of light and don't hit the land masses on some other celestial body.
There is some definition of "straight line" that includes a course of a constant bearing, or a rhumb line. It's straight when plotted on a Mercator projection.
By that definition, keeping to a true east or true west bearing would be a straight line.
Latitude and longitude integer degree intersections around the world: http://confluence.org/
The integer degrees is arbitrary, but it provides a random sampling of landscapes. I like the idea of going out to find some defined spot, like geocaching without any caches. It also provides a page of antipodes photos, places exactly on the other side of the earth from each other: http://confluence.org/antipodes.php
I don't have a globe handy, but can't you sail indefinitely round and round Antarctica? Or doesn't following a line of lattitude count as a "straight line".
that would certainly not be a geodesic (the only sensible defintion of "straight line" on a sphere), more like a circle. It would be just like running circles around your house.
You would have to keep turning slightly South to maintain a bearing parallel to the lines of latitude - the only one that's "straight" (it curves downward, but not North or South) is the Equator.
No line of latitude is straight, except arguably the equator.
Think about the line of latitude at 89.9999°. It will trace a circle around the pole of a few meters radius. 60° south is the same such circle, only bigger.
I was just thinking this. If it is allow to sail over the same places more than once, they must be a route that is just a circle. Maybe along one of the southern parallels?
From the maps they show, it’s pretty clearly a great circle (the obvious meaning for “straight line” on the surface of a sphere), not constant bearing.
What sort of constant bearing - magnetic bearings will vary along a map-based bearing. There's an arbitrary choice involved. We're probably not looking at changes in sea-level that put us off line - is a necessarily idealised system in which the question makes sense.
I was curious one day after reading yet another comment contrasting driving in Europe vs driving in the U.S., distance-wise, and I decided to measure Seattle->Key West and compare that to Europe. Decided that the equivalent would be Paris->China (almost, I think the terminating point was eastern Kazakhstan).
I wonder if the answer changes if you consider actual geodesics instead of great circles (which are geodesics on a sphere, but the Earth is not a sphere). Even great ellipses (closer in length to actual geodesics) can deviate laterally from the geodesic by kilometers. See e.g. https://geographiclib.sourceforge.io/html/greatellipse.html#...
Looks like another cool app: Click somewhere on the coastline and it'll take you to what's directly on the other side of the sea. Results may be surprising:
This is cool. Though as you take into account more of the detail of the shape of the surface, your straight line can get a lot longer. I would be interested to see one that takes into account the large scale ocean surface topography, to see if that can change it by much.
Technically, you need neither. Branch and bound works by solving a series of relaxations with fixed integer variables in order to better search the discrete space. Now, imagine we're solving a minimization problem. Finding a feasible solution to the problem may be difficult, but if we were to find one, discrete variables and all, we'd have an upper bound on how good the solution could be. Simply, it's feasible, but not optimal, so the objective value is higher. Now, since integer variables are hard to work with, we can relax them into continuous. For example, instead of a binary variable that's {0,1}, we could relax it into a continuous variable bounded between 0 and 1, [0,1]. If we do this relaxation and find the globally optimal solution, then we have a lower bound as to how good the solution is. In branch and bound, if we can show that the lower bound for one branch is not as good as the upper bound of another branch, we don't have to explore that branch.
Now, convexity comes into play because it allows us to correctly determine these lower bounds because we can guarantee a globally optimal solution to the relaxed problem. If we lack convexity, it's hard to find such solutions. Does that mean branch and bound requires convexity? No. You can still perform branch and bound using locally optimal solutions and even though you don't have a guaranteed global bound, there's relatively good information about where we should search next. This can lead to good, but not provably globally optimal solutions.
But the article was making a strong claim that this was the provably optimal solution. In this case it is, because the objective function (in the way I'm assuming they formulated the problem) is convex.
Why not start at the most southern point of South America and go due east? Maybe even 0.01 degree South. Couldn't you do loops around the Earth before hitting Antarctica?
I once flew from Chicago to Beijing and the thing that astonished me is that you are over land the entire flight except in the vicinity of the Bering Strait.
If your plane flies the shortest great-circle path, you'll fly over the arctic sea, which is definitely water, though sometimes solidified. IAH or DFW would be better examples :).
Ideally they'd show a map with a projection based on having the line drawn on it as a straight horizontal or vertical line in the middle of the map. So basically, a Mercator projection with the line being illustrated as its 'equator'.
That would require a custom rendering for each line though.
A single map using that projection calls a lot of attention to the quirks of projecting a sphere onto a plane. A different projection or multiple maps centered on segments of the line would call attention to the straight path you'd take as you graze landmasses.
Why are these comments full of people trying to make what is an interesting and easy to understand problem into an insanely complicated problem? This is hacker news. We're not actually going to sail around the world. It's interesting because someone used a computer to find an answer.
Absolutely. Particularly under sail, where top speed is typically going to be under 15mph (often under 8mph), a 2mph side current can impact course over ground quite a bit over a day or two when you are only doing 200 miles a day.
This reminds me of a discovery I made while vacationing in the Corn Islands, Nicaragua -- There was a giant cube buried in the middle of nowhere with one corner poking out of the earth. A plaque nearby said it was the 'soul of the world' and I looked it up to find this funny old organization (http://www.souloftheworld.com) had planted the cubes around the planet, saying that their positions were the vertexes of a cube that was positioned in such a way (and only possible in this way) that every corner touched land. Apparently it has a certain spiritual significance, which is funny to me, but I thought it was really interesting that someone went through the trouble to create them.
Looks like it's true - I threw this together over lunch and it looks pretty close:
http://callumprentice.github.io/apps/soul_of_the_world/index...
This seems like a very interesting question.
After an exhaustive investigation, we found, surprisingly, that there was only one possible combination on the whole planet that would permit the eight vertices of the cube to emerge on solid land. (The results of the investigation made with the AMQ were later corroborated in a study carried out by highway engineer D. David Fernándes-Ordóñez.)[0]
I wonder how difficult it would be to replicate this finding (or a similar finding). I'm naturally skeptical given the extreme remoteness of the islands (Points include the Cocos Islands (Australia), the Corn Islands (Nicaragua), and the Hawaiian Islands (USA))[1] and the not-perfect-sphere shape of the earth, but I also know that I don't have a good natural intuition about the shape of the earth. This would be an interesting project!
[0] http://www.souloftheworld.com/genesis.html [1] http://www.souloftheworld.com/work.html
Well a good starting approximation might be to take a WGS ellipsoid model of the earth and see if their existing coordinates forms something close to a cube.
https://en.wikipedia.org/wiki/World_Geodetic_System
>This seems like a very interesting question.
>After an exhaustive investigation, we found, surprisingly, that there was only one possible combination on the whole planet that would permit the eight vertices of the cube to emerge on solid land.
Not really that interesting IMHO. After enough searching your're going to find a shape that meets the criteria. If a cube had more than one configuration, they likely would have just move up a notch, and if a cube had no solution they would have moved down a notch.
Someone just did that in this thread!
"I have cubed the earth, with 4 simultaneous _corner_ days in 1 rotation of Earth!"
-- Dr. Gene Ray, Cubic and Wisest Human.
How can you quote him and not link to Time Cube? http://timecube.2enp.com/
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Brings back many memories. Back in around 2000/01, the website had complete contact information for Gene Ray including phone number. So I called him up and he rambled at me for about 20 minutes before he had to go because his grandkids were coming over.
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Wow those are going to confuse the hell out of future beings (if there are any left). I wonder if they're all "built" already?
I don't believe so. I know there is one on the island of Molokai in Hawaii, which I always wanted to visit when I used to live in Honolulu, but otherwise its not a super tourist-friendly place to visit and I opted to visit other islands instead. The Corn Islands, however I highly recommend to anyone!
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Given the tectonic nature of the planet's surface that has to be the worlds longest running piece of performance art.
On a similar 'frequency', this sort of 'spiritual significance' question has been around for quite a while. It appears that three Temples — of Parthenon (438BC), Poseidon (440BC), and Aphaea/Aphaia (500BC) — are located at the vertices of a triangle. AKA 'Holy Triangle of antiquity'.
* http://www.goddess-athena.org/Museum/Temples/Aphaea/
* https://eyesofaroamer.com/2016/09/06/the-holy-triangle-the-p...
That's not unusual. Any three points can form the vertices of a triangle. :)
The first site claims that these three temples form an isoceles triangle, which really just means that the distances between two pairs of points are roughly equal. The second claims that they form an equilateral triangle, which would be more unusual -- but is clearly false.
Cool.
Found a pic of the one in Nicaragua:
https://retirenicaragua.wordpress.com/2014/03/05/the-soul-of...
> "A really smart mathematician, Fra Luca Pacolini, demonstrated mathematically, that the four regular solid bodies: the Tetrahedron, the Cube, the Octahedron, and the Icosahedron, correspond respectively to the four elements: fire, earth, water, and air."
Does anyone have any idea what that even means?
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That was it! There was nothing around the area other than cows and some shanty houses. It was really cool to stumble upon.
Cool idea. Totally new to me.
They use height data rather than land/water data, and assume that there's just a cutoff at sea level. I think this gives them the correct answer for the water path, but it gets things wrong for the land path because there are a lot of lakes and rivers above sea level and also some dry land below sea level.
If you don't allow the path to cross any lakes or rivers at all then I think the land path has to be much shorter, since rivers makes it impossible to make any progress at all in most places. If you trace their path it definitely goes through several lakes. The optimal path is probably across a desert, my guess would be Antarctica.
If you do allow the path to cross lakes and rivers then I think there's a longer path than the one they give, starting in Liberia and ending near Fuzhou, China. They probably didn't spot this one because it passes too close to the Dead Sea, which is below sea level (and crosses the Suez canal, which is at sea level).
> They probably didn't spot this one because it passes too close to the Dead Sea, which is below sea level
This is adressed in the paper. The relevant part of the problem statement: "the longest distance one could drive for on the earth without encountering a major body of water"
and about the Dead Sea: "Guy Bruneau of IT/GIS Consulting services calculated [5] a path from Eastern China to Western Liberia as being the longest distance you can travel between two points in straight line without crossing any ocean or any major water bodies. However, the path crosses through the Dead Sea (which can be considered to be a major water body), and hence does not satisfy the constraints originally set out."
Oh, I didn't spot that they had already considered that path.
However I don't accept their defence.
1) Depending on environmental conditions, my path can cross the Lisan Peninsula. https://en.wikipedia.org/wiki/Lisan_Peninsula (EDIT: In fact there's enough clearance to just go completely south of the Dead Sea)
2) Their path crosses the Volga River, which is much larger in total surface area than the Dead Sea. And at the point at which they cross it, the Volga is just as wide as the Dead Sea.
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Is that a straight line? Maybe they allow crossing rivers but not seas/lakes.
In case people are wondering:
1.85 km is roughly one Nautical Mile, which is roughly one arc-minute of a great circle. As such it's a natural unit of measure.
The original definition of the meter was 1 ten-millionth of the distance from the North Pole to the Equator via Paris, making the Earth's circumference 40 million meters. Divide by 360 degrees per circle, then by 60 arc-minutes per degree, and you get (40 x 10^6) / (360 x 60) which is about 1852 meters.
Just in case people were wondering ...
I was wondering how one would make a global map with a resolution of a nautical mile. Spacing dots a nautical mile apart on the equator is simple, but you can’t continue with a rectangular grid up north or south (by the time you’re at 60 degrees, the dots would be only half a mile apart)
A perfect solution doesn’t exist, and AFAIK no exact solution for the simpler “place N dots on a sphere in maximizing the minimum distance between dots” exists, but decent approaches exist. See http://web.archive.org/web/20120315152121/http://www.math.ni..., https://www.maths.unsw.edu.au/about/distributing-points-sphe...
See also gradian: https://en.wikipedia.org/wiki/Gradian
"one grad of arc along the Earth's surface corresponded to 100 kilometers of distance at the equator; 1 centigrad of arc equaled 1 kilometer."
i.e. both nautical miles and kilometres are derived from the size of the Earth. Plain old statute miles are just a mess, and best avoided.
Yup, so that will be the best thing once everyone switches to using grads for latitude and longitude.
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Polar circumference. Earth is an ellipsoid.
Equatorial circumference is 40075 km. One arc-minute of equatorial longitude is about 1.855 km.
People who do crazy things with measurements cause a significant fraction (at least 15/360) of the many pains in my ass. Degrees need to die a painful death, and so can grads. There is nothing at all "natural" about them as a unit of measure, as they are arbitrary divisions of one rotation, by 360 and 400 respectively.
But I admit, even if not natural they may be more convenient for some calculations than other numbers.
> As such it's a natural unit of measure
As natural as the number 360, and 60.
I would argue the meter seems more natural, since it is related to 10 fingers ...
Given that you are already using degrees, arc-minutes, and arc-seconds in your specifications of latitude and longitude, it makes sense to use a unit of distances that meshes with them. Using meters makes less sense because it doesn't work well with the size of the Earth.
Difficulties with measurements and conversions are like a dead cat under the carpet - no matter how you push it about there's always an inconvenient lump and a bad smell. The Earth is inconveniently shaped, and you simply have to deal with it. The existing system of measurements may offend your sense of taste, but it has evolved over time to be useful to those who have to use it. Attempts to devise systems a priori and without taking into account the extensive experience of those who actually use them have always failed.
There's probably a reason for that.
I think "natural" was meant in the sense "it naturally follows" or "it's natural to assume", where it's meant to imply a natural path given your preexisting knowledge and conditions, not natural as in "this comes about in nature".
With that in mind, 360 is a fairly natural way to segment a circle for modern Humans.
60 is a very natural number.
You have twelve joints on the fingers of one hand. (Use the thumb to count.) You have five fingers (including the thumb) on the other hand to track multiples of 12 with. Voila, 60!
360 and 60 are quite handy, by virtue of having a lot of integer divisors. For a civilization where long division is unknown or limited to a few scribes, being able to divide things by common fractions easily is an important criterion in a system of measure.
> since it is related to 10 fingers
If your parent comment is correct. 10 finger thing will merely be an afterthought.
If you like this sort of things, you absolutely should read "Which lines of longitude and latitude pass through the most countries?".
https://nwhyte.livejournal.com/2929721.html
Through Greece and I didn't read the other one because the author is a terrible writer.
Central America (?)
I loved the final question in the article:
“The question now is: who will be the first to make these journeys, when, and how?”
These have the potential to become quite important journeys... with the potential for many “firsts”... and also much contention; how much did they deviate from the path, how much deviation from the path is acceptable?
So many interesting questions to ask.
That final question struck me as odd, the next question for my mind was could the technique be generalised to a map on a torus?
Same here, that’s what I thought as well :-) I’m sure that the big documentary channels will pick this up and we’ll see both journeys in a few years. Seems to me that the land journey will be harder to complete. When you’re sailing it’s easy to keep a “straight line” without having to consider mountains, rivers, permissions, politics and any other obstacles associated with crossing land.
It won't be easy to sail in anything resembling a straight line through the Drake Passage unless your ship is very big.
Speaking of big ships, you might also want to avoid icebergs.
For the land route a realistic approach would be to plot major checkpoints along the true great circle path and just navigate between those checkpoints in whatever way (preferably on land) is most feasible. That would still be interesting.
Yeah I was thinking about how hard it would be to stay “true” to the path as possible if a race or challenge was by wind power only. Also in competition, points could be given or taken based on the amount of deviation from the line etc.
“The question now is: who will be the first to make these journeys, when, and how?”
I wouldn't volunteer for the water journey any time soon considering you're basically launching into a pirate haven.
Non stop circumnavigations have already been done.
For example: https://en.m.wikipedia.org/wiki/Wilfried_Erdmann
Can you clarify this comment, how is circumnavigation relevant to this considerably different journey / path?
"A global map with resolution of 1.85 kilometers has over 230 billion great circles. Each of these consists of 21,600 individual points, making a total of over five trillion points to consider."
What? Is there something I'm missing here, or did they just decide to include "individual points" as an utterly useless way of inflating the difficulty of a brute-force approach?
The longest path needs to start and end somewhere along a great circle. So those individual points are needed.
The only relevant points are those that make up the coastlines - there is no need to test paths that start and/or end in the middle of sea or land.
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at a 1.8 km resolution (without checking my maths) 21,000 points along a great circle is basically every 1.8km so i guess you are checking at each 1.8km if you just drove into water (or vice versa)
Your point that you only need to fail once is fair - there is a lot of boundary conditions one can apply quite quickly
Going from Pakistan to Russia seems like the wrong way around Cape Horn. Today you've got 30 kt winds gusting up to 50 kts. Who wants to beat into that?
https://www.windy.com/-57.136/-71.191?-37.370,-80.244,3,m:3V...
I windsurf in 30 knot winds at sea. Yeah, it's rough, but it's not hurricane level madness that rips ships apart.
That sounds like a lot of fun :)
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For the water challenge couldn't you just sail the 60th parallel south forever? https://en.wikipedia.org/wiki/60th_parallel_south
Also could we challenge flat earthers to the same challenge and see who travels the furthest?
Straight line here is taken to mean geodesic, the shortest possible path on the surface between two points. Parallels on the sphere are not geodesics!
If we're going to split hairs, an actual straight line would be tangent to Earth at the point of the boat, and you could sail forever, so long as you do it at the speed of light and don't hit the land masses on some other celestial body.
There is some definition of "straight line" that includes a course of a constant bearing, or a rhumb line. It's straight when plotted on a Mercator projection.
By that definition, keeping to a true east or true west bearing would be a straight line.
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You’d have to constantly turn your boat to the right to stay on course.
That's not a straight line (it only looks like that on this map).
There are no actual straight lines.
And we'd need a frame of reference in any case.
The problem seems ill-defined.
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I love geographic puzzles such as this one. Here are links to some other interesting ones.
Longest straight lines of sight, modulo some atmospheric refraction (has been discussed on HN before): https://beyondhorizons.eu/lines-of-sight/
Furthest points from the sea/land: https://en.wikipedia.org/wiki/Pole_of_inaccessibility
Latitude and longitude integer degree intersections around the world: http://confluence.org/
The integer degrees is arbitrary, but it provides a random sampling of landscapes. I like the idea of going out to find some defined spot, like geocaching without any caches. It also provides a page of antipodes photos, places exactly on the other side of the earth from each other: http://confluence.org/antipodes.php
Must also mention earth sandwiches: http://www.zefrank.com/sandwich/
Fun fact: there are parts of the Pacific ocean that are opposite each other, therefore the Pacific ocean spans (not covers) half the globe.
I don't have a globe handy, but can't you sail indefinitely round and round Antarctica? Or doesn't following a line of lattitude count as a "straight line".
that would certainly not be a geodesic (the only sensible defintion of "straight line" on a sphere), more like a circle. It would be just like running circles around your house.
You would have to keep turning slightly South to maintain a bearing parallel to the lines of latitude - the only one that's "straight" (it curves downward, but not North or South) is the Equator.
Source? My gut tells me this is not true, but I'm willing to be convinced.
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No line of latitude is straight, except arguably the equator.
Think about the line of latitude at 89.9999°. It will trace a circle around the pole of a few meters radius. 60° south is the same such circle, only bigger.
FYI, the PDF of the source paper is https://arxiv.org/pdf/1804.07389.pdf
Isn't "straight line" somewhat arbitrary? How about "the longest circle you could sail without hitting land" or some other reasonably smooth shape?
I was just thinking this. If it is allow to sail over the same places more than once, they must be a route that is just a circle. Maybe along one of the southern parallels?
That's pedantic. Most people will realise that "straight line" means constant bearing.
From the maps they show, it’s pretty clearly a great circle (the obvious meaning for “straight line” on the surface of a sphere), not constant bearing.
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What sort of constant bearing - magnetic bearings will vary along a map-based bearing. There's an arbitrary choice involved. We're probably not looking at changes in sea-level that put us off line - is a necessarily idealised system in which the question makes sense.
But, unless you are passing over a pole, a great circle doesn't use a constant bearing.
Life is arbitrary. Your question is equally interesting, except for the fact that you haven't offered up an answer to it.
Yes, "straight" is misguiding.
Google maps seems unable to find a route between Jinjiang, Quanzhou, Fujian, China and Sagres, Portugal :
https://www.google.com/maps/dir/Jinjiang,+Quanzhou,+Fujian,+...
If anyone is interested in a 7000 mile trek, this is the approximate path of the land route:
http://www.gcmap.com/mapui?P=prm-JJN
Thanks. Maybe I'll try that over lunch next week.
I was curious one day after reading yet another comment contrasting driving in Europe vs driving in the U.S., distance-wise, and I decided to measure Seattle->Key West and compare that to Europe. Decided that the equivalent would be Paris->China (almost, I think the terminating point was eastern Kazakhstan).
I wonder if the answer changes if you consider actual geodesics instead of great circles (which are geodesics on a sphere, but the Earth is not a sphere). Even great ellipses (closer in length to actual geodesics) can deviate laterally from the geodesic by kilometers. See e.g. https://geographiclib.sourceforge.io/html/greatellipse.html#...
Weird, they talk about great circles yet insist on “straight line” rather than geodesic terminology. Still a good read though.
Looks like another cool app: Click somewhere on the coastline and it'll take you to what's directly on the other side of the sea. Results may be surprising:
https://ranlot.shinyapps.io/coastlinetrip/
This is cool. Though as you take into account more of the detail of the shape of the surface, your straight line can get a lot longer. I would be interested to see one that takes into account the large scale ocean surface topography, to see if that can change it by much.
I thought the solution set had to be convex or concave for branch and bound to work as an optimization algorithm, and this one clearly isn't?
EDIT: Nope. It's just the objective function that needs to be convex, not the constraints.
Technically, you need neither. Branch and bound works by solving a series of relaxations with fixed integer variables in order to better search the discrete space. Now, imagine we're solving a minimization problem. Finding a feasible solution to the problem may be difficult, but if we were to find one, discrete variables and all, we'd have an upper bound on how good the solution could be. Simply, it's feasible, but not optimal, so the objective value is higher. Now, since integer variables are hard to work with, we can relax them into continuous. For example, instead of a binary variable that's {0,1}, we could relax it into a continuous variable bounded between 0 and 1, [0,1]. If we do this relaxation and find the globally optimal solution, then we have a lower bound as to how good the solution is. In branch and bound, if we can show that the lower bound for one branch is not as good as the upper bound of another branch, we don't have to explore that branch.
Now, convexity comes into play because it allows us to correctly determine these lower bounds because we can guarantee a globally optimal solution to the relaxed problem. If we lack convexity, it's hard to find such solutions. Does that mean branch and bound requires convexity? No. You can still perform branch and bound using locally optimal solutions and even though you don't have a guaranteed global bound, there's relatively good information about where we should search next. This can lead to good, but not provably globally optimal solutions.
But the article was making a strong claim that this was the provably optimal solution. In this case it is, because the objective function (in the way I'm assuming they formulated the problem) is convex.
Its always surprised me that you cant get to the farthest eastern reaches of Russia by car.
Its such a massive amount of territory they hold in the east, you could easily start another country as large as China or the USA out there.
Further reading on the branch-and-bound technique: https://en.wikipedia.org/wiki/Branch_and_bound
Why not start at the most southern point of South America and go due east? Maybe even 0.01 degree South. Couldn't you do loops around the Earth before hitting Antarctica?
Going east is only straight at the equator.
I once flew from Chicago to Beijing and the thing that astonished me is that you are over land the entire flight except in the vicinity of the Bering Strait.
If your plane flies the shortest great-circle path, you'll fly over the arctic sea, which is definitely water, though sometimes solidified. IAH or DFW would be better examples :).
And what better way of showing it than a Mercator map!
The maps on the page are not Mercator, but what's the problem with Mercator? It is a perfectly reasonable projection, with some nice properties.
Ideally they'd show a map with a projection based on having the line drawn on it as a straight horizontal or vertical line in the middle of the map. So basically, a Mercator projection with the line being illustrated as its 'equator'.
That would require a custom rendering for each line though.
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A single map using that projection calls a lot of attention to the quirks of projecting a sphere onto a plane. A different projection or multiple maps centered on segments of the line would call attention to the straight path you'd take as you graze landmasses.
They need to account for currents as well otherwise a straight line wouldn't be straight at all
Why are these comments full of people trying to make what is an interesting and easy to understand problem into an insanely complicated problem? This is hacker news. We're not actually going to sail around the world. It's interesting because someone used a computer to find an answer.
Absolutely. Particularly under sail, where top speed is typically going to be under 15mph (often under 8mph), a 2mph side current can impact course over ground quite a bit over a day or two when you are only doing 200 miles a day.
The Earth is not a sphere.
https://en.m.wikipedia.org/wiki/Figure_of_the_Earth
From that page:
> the Earth deviates from spherical by only a third of a percent, sufficiently close to treat it as a sphere in many contexts
It's a sphere in human scale.
What do you mean by “human scale”? It may be a sphere in much larger scales (~1000km) but I’d say it’s far from being a sphere in human scale (~1m).
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