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."
I think it means that the writer doesn't really know what he is talking about.
And, by the way, there are five regular solids. The other is the dodecahedron. I imagine the writer would say that corresponds somehow to the "quintessence" some philosophers used to go on about. But that still doesn't mean this idea has anything to do with physical reality.
EDIT. By the way, in spite of all the mystical silliness that seems to surround it, personally, I think that the Soul of the World sculpture is a really cool idea.
the idea comes from ancient philosophy, when philosophers could make claims like "everything is made of fire" or "everything is made of water" or "everything is made of triangles" and argue it reasonably. the platonic solids, also, are five in number, where the fifth is a dodecahedron. i think it was plato, among others, who also believed in a fifth element, aether, but i cant remember the details.
how one would demonstrate any of these relationships mathematically is beyond 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.
>rivers don't fit in the conventional definition of "major body of water"
I suppose that's true. But it raises the question of what a river is. Lots of things that people would describe as lakes in fact have water flowing into them, through them, and out of them. So it's hard to distinguish them from a broad river.
Anyway, the distinction is moot since you can in fact squeeze just south of the Dead Sea and avoid hitting it at all.
Without looking up a formal definition, how I believe the common person would differentiate between a river and lake are the following:
* Is the body of fluid, within a set of bounds, mostly still (and not flowing)?
* Is it closer to a:
* > bowl (hemisphere with cut side facing up)
* > sliced cylinder (again flat slice facing up)
* > sheet* (a large thin expanse)
* Is the fluid at the bottom shifting location, how quickly?
Rivers tend to have high flow to volume ratios, faster moving flows at depth, are usually more shallow, and are defined more like a squashed tube in natural states.
Lakes tend to be deeper, more stable (slower flow, if discernible at all), and are generally placid. Lakes /usually/ have a large dimension in at least two surface directions while rivers usually have that in only one.
Exceptions to the above occur with canals/channels* (though that's an ocean thing) which might be closer to the fuzzy boundary.
River deltas also occur in high sediment deposit areas, such as the ends of rivers where they transition in to lakes / oceans; the extreme end of a river delta being bogs and other swamp like areas with shallow slow moving water. (I argue that such areas are neither river nor lake, but a third category.)
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)
What can I do to convince you that including an additional (2 PI)/N or N/(2 PI) factor into every single angle calculation you will ever do is a complete waste of time?
At the very least, you could take out the arbitrary N.
100 gradians is 0.25 rotations, or PI/2 radians. 30 degrees is 1/12 rotations, or PI/6 radians. There is no advantage whatsoever in multiplying everything by an arbitrary scaling factor that is not based on PI (or TAU).
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?
Oh I see what you mean. Yes, I suppose it is worded that way to impress the reader with the big numbers involved, when many of them can be trivially rejected.
Coastlines are close to infinite in length (they grow inversely to the length of the measure used; idealised mathematical 'coastlines' are fractal and infinite), we can limit them by limiting the resolution.
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?
Wind surfing has a surprisingly high fun/expertise ratio. You can start surfing on your own in as little as 2 hours of training. I was chasing dolphins on my second day of surfing.
Of course, like any sport/hobby, it can take years to master but even for complete beginners it can a lot of fun.
Correct. You can make it easy or tough simply by choosing smaller/bigger sail and applying less/more force. So it can be very relaxing in 20 knots, especially on a lake.
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.
Meh, this is the worst kind of pedantry imo. Like, first of all, doesn't matter. But if you must, you're going to have to go with 'as would be defined by mathematicians' - because they spend a lot of time thinking about these things, and thus are by far the most qualified to have an opinion....
And the mathematicians have thought long and hard about how the Euclidean concept of a straight line generalizes to other geometries... and came up with geodesics... aka great circles...
Why would mathematicians take precedence over people who navigate ships and airplanes for a living? Also, ellipsoidal geodesics aren't always circles.
"Thinking long and hard" isn't actually much of a qualification, if you think about it long enough and hard enough. Playing around with the definition of "straight line" is just an amusement, putting different theoretical constraints on the recreational problem. The whole thing is pedantry to begin with, so don't be surprised when someone pops their head in with something unexpected just to show off how clever they think they are.
Launching pointless academic arguments is almost the whole point. It shows the audience that everyone involved is very smart, and all possibilities have been duly considered, and therefore the agreed-upon answer must be very significant, reliable, and noteworthy.
If you navigate a ship while keeping the rudder straight, absent current or wind you will be following a geodesic. If you try to follow a parallel you will have to constantly be turning.
Planes follow geodesics too over oceans, not parallels.
Yes the geodesic on an ellipsoids aren't always great circle, but the earth's geodesics are commonly referred to as "great circles" because the earth is very nearly spherical.
Should we ignore wind and current? If you are in a medium that has winds and currents, and you do not touch your yaw controls, how will you know if your course remains on a straight line?
You either adjust your compass bearing (also correcting for magnetic variation), or you adjust course.
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.
It’s confusing because the latitude line 66° south looks like a straight line on a globe. But to sail in a “straight line”, which means keep your steering centered forever, you will travel in a great circle.
All great circles lie on a plane that intersects the sphere (earth) through the center of the sphere. You can see that the only latitude line that’s on a plane intersecting the center of the earth is the equator, and that 66° south doesn’t. This also means that all straight line paths on earth touch the equator at 2 opposing points. Or said another way, you can start with the equator, pick any one point on the equator and rotate it around that point to get a new great circle.
So in order to stay sailing along 66° south, you’d have to have your steering turned constantly just a little bit south.
It sure does, you’re right. So does any “straight line” great circle too, so that isn’t super helpful. The equator has the same circle projection that 66° does.
Look at it from the side and it looks straight. If you’re sitting in the plane of 66°, the projection is straight.
I was trying to be supportive of @dbatten while explaining. It’s easy to get confused about what straight means on a sphere, since nothing is actually straight.
May apologies if I sounded knee-jerky. The intention was to show that a picture was a better answer to @dbatten's very legitimate comment.
The idea is to demonstrate that a "straight" line on a surface needs to be viewed along a normal to that surface at the point of the line you are concerned with, assuming the definition of "straight" is "don't have to turn when travelling along line on the surface". That makes great circles look straight, and non-great circles not.
>> larkeith: 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.
> dbatten: Source? My gut tells me this is not true, but I'm willing to be convinced.
Consider any point on a sphere, pick a direction (which might be along a line of latitude) and visualise the plan defined by that vector and the centre of the sphere. The intersection of the plane with the surface of the sphere is a Great Circle, and that is where you would go if you didn't apply turn to the rudder.
The parallel of latitude defines a plane that does not go through the centre of the sphere (unless it's the equator) and so isn't a "straight line". If you want to stay on the parallel of latitude then you will deviant from the Great Circle, and that's why you need to apply rudder to stay on the parallel instead of the Great Circle.
Does that help?
This is all pretty obvious once you've done some spherical geometry, but can be completely opaque to anyone who hasn't.
The problem is that the proposed solution doesn't "count" for the purposes of this experiment, which uses a great circle definition of "straight". The easiest way to visualise a great circle, is to place a piece of string over a globe between your origin and destination and pull it tight, that will track a great circle route between the two (and how a flight flying straight between Europe and the US west coast will take off on an pretty northern bearing, and land on a ditto southern, despite not actually going over the north pole and Europe and the US being located east/west of each other).
If you tried to place the string around Antarctica and pull it tight, it'd slip off in the southern direction, which represents the parent's explanation that you'd be steering south (from a great circle course) to keep straight along a latitude.
Get a ball and a piece of string. Put the string on any two points of the ball and pull it tight. You'll observe that the path the string follows is part of an "equator" of the ball.
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.
Whoops, turns out I didn't actually understand what "constant bearing" meant. What I actually meant was "no rudder/steering" which is exactly what people would imagine as travelling in a straight line.
Obvious perhaps, in a mathematically idealised system that's idealised according to axioms that are unstated ... a non-equitorial latitudinal line on a sphere is straight.
Consider the degenerate case where you’re ten feet away from the North Pole. You’ll need to go in a tight circle. The situation at more moderate latitudes is the same, just less extreme. It only seems straight because we’re usually at moderate latitudes and the rate of turning is low.
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.
It's based on context. So something like a scale that makes sense when going about your average human business. For instance, the sun is 150 million miles from the sun. It's not, it's slightly less, but for "normal" human activity (eating, sleeping, sailing in straight lines, programming web apps, etc. etc.) it doesn't matter, it's close enough.
For most of what most of us do with our everyday lives it's good enough, to the extent that pointing out that "the earth is not actually a sphere" just sounds pedantic.
It did occur to me that sailing in a straight line could be interpreted as an actual straight line, but the surface of the ocean is spherical(fine, -ish) and so, modulo waves, we might actually be measuring an infinitely small distance on the earth that was sailable (Ah ha, got you all!) but realised that jokes about derivatives, despite the crowd, weren't going to be all that amusing :)
