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.)
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)) 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!
>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.
-- Dr. Gene Ray, Cubic and Wisest Human.
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.
Found a pic of the one in Nicaragua:
Does anyone have any idea what that even means?
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.
how one would demonstrate any of these relationships mathematically is beyond me.
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).
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  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."
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.
It's odd to say you "don't accept their defense" when you're really just operating from a different set of assumptions in the first place.
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.
* 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?
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.)
Why is that odd? Their set of assumptions is exactly what he is criticizing.
Put another way, if the problem is with the assumptions, the criticism goes far deeper than just that defense.
A global map with resolution
of 1.85 kilometers ...
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 ...
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...
"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.
The arbitrary factor of 400 (or 360), is simply not helpful for machine calculations.
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.
As natural as the number 360, and 60.
I would argue the meter seems more natural, since it is related to 10 fingers ...
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.
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!
If your parent comment is correct. 10 finger thing will merely be an afterthought.
Central America (?)
“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.
Speaking of big ships, you might also want to avoid icebergs.
I wouldn't volunteer for the water journey any time soon considering you're basically launching into a pirate haven.
For example: https://en.m.wikipedia.org/wiki/Wilfried_Erdmann
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?
Your point that you only need to fail once is fair - there is a lot of boundary conditions one can apply quite quickly
Of course, like any sport/hobby, it can take years to master but even for complete beginners it can a lot of fun.
Also could we challenge flat earthers to the same challenge and see who travels the furthest?
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.
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...
"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.
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.
You either adjust your compass bearing (also correcting for magnetic variation), or you adjust course.
And we'd need a frame of reference in any case.
The problem seems ill-defined.
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.
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.
On the contrary, latitude 66° on a globe looks to me like a fairly tight circle.
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.
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.
> 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.
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.
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.
A constant bearing actually gives a Rhumb line: https://en.wikipedia.org/wiki/Rhumb_line
None of these things is of course possible IRL.
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).
EDIT: Nope. It's just the objective function that needs to be convex, not the constraints.
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.
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.
That would require a custom rendering for each line though.
It should be possible to make one with this tool: http://mrgris.com/projects/merc-extreme/
> the Earth deviates from spherical by only a third of a percent, sufficiently close to treat it as a sphere in many contexts
So for the earth (as someone else pointed out):
> the Earth deviates from spherical by only a third of a percent, sufficiently close to treat it as a sphere in many contexts (https://en.m.wikipedia.org/wiki/Figure_of_the_Earth)
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 :)