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If I walk ten feet then take a 90' turn and walk ten more feet for a total of four times I end up where I started. If I start at 0'0' on earth then walk 6000 miles turn 90' walk 6000 miles I'm at the north pole. On my third turn of 90' I walk 6000 more miles and I'm back where I started. Where did the fourth turn go? Where do I end up at 3000 miles and turn four turns later? Back where I started?
That took me a second to figure out.
So in a gravity well do you make three turns like if you are out in space? -
https://en.wikipedia.org/wiki/Spherical_geometry
Not sure why you were thinking of gravity in this context?
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As the Sun's gravity can measuably curve the path of light, then a square formed by four points around the sun will have corners measurably more than 90 degrees... Slight, but there.
Maybe it is better to thing of a triangle, as all the points of a triangle in 3D space are in the same plane.... One less point, quite a few less problems to chew over :-)
Edit Replace 'less' with 'more'. Opps! -
It comes down to the fact a spheroid is not a a flat plane, equally as the worlds circumference is wider at the center than across the poles, you would not end up at the exact same point.
Your distances are also a little off, its a tad more than 6000 miles to the pole.
You begin at 0,0. you walk west 6000 miles, you will be at roughly 0.-86.7, you then walk north 6000 miles, and you will be at 86.7.-86.7, you turn to face longitude 0, and walk another 6000 miles, and you would be back where you began +- 80 miles depending on if your working on the assumption the earth is a smooth sphere or not.
Its the same at taking a toy plane, you pitch it 90 degrees up, yaw it 90 degrees, then pitch it down 90 degrees, your pointing the same way, but your orientation is different, and would take a 4th turn, a roll to end up the same way as you started. -
The parallel line postulate is a postulate for good reason; Euclid himself suspected it, but was unable to construct an argument against it. It took much later, as mathematicians were toying with complex, nonlinear spaces, for the postulate to finally be questioned, and for alternatives to apply.
Spherical geometry is such a case: the angles of a closed polygon always sum to less than the angles of the analogous figure on the flat plane, and parallel lines always intersect. The opposite case is hyperbolic geometry, where the angles sum greater than in the flat plane, and parallel lines always diverge.
The most extreme example, on a sphere, is to travel in a straight line for half the circumference, turn 360 degrees, and travel the same again. In the plane, this describes only a line segment, or at best, a triangle with one side zero, either way enclosing zero area.
Hyperbolic geometry isn't all that unfamiliar. Many naturally frilled and crumpled shapes exhibit this. Lettuce leaves and decorative frills are examples. The construction is simple: suppose you create a flat circular rug by sewing material onto the perimeter of an initial nucleus; but each course of material you add, is scrunched up so that it is longer than the actual perimeter it's being added to. It crumples up as you go, and the perimeter grows disproportionately to the distance from center (that is, you make the perimeter increase faster than pi times the diameter). Well, that sounds rather shoddily constructed, doesn't it -- it won't sit flat! But the benefit is packing a lot of surface area into a small volume, while making it easily accessible: that is, within a short distance from the center (depending on how much larger you made "pi" in this object).
On a frilly surface, because the circumference of a circle increases rapidly with distance from center -- that's literally how it's sewn together, eh? -- you can imagine moving from the origin out to some radial distance, then following a tangent and returning to the origin, and having subtended only a small angle at the origin. That is, if you draw triangles around the origin, you can pack many more than, say for example, six equilateral triangles around the origin!
Which leads to another construction method: suppose you join equilateral triangles by their edges. YOu can choose how many to join per vertex. Six gets you the flat plane (with a triangular / isometric grid), five or fewer gets you a somewhat spherical section (the icosahedron; four gets you an octahedron and so on!), seven or more gets you a hyperbolic frill. If you count the grid distance (number of triangles) traveled between points and along paths, you get the same results as before.
Now, on Earth, or on the surface of any round-ish object, really: the curvature is a consequence of that object's surface, and the confinement of activity to its surface (because we can't really swim through the Earth's crust, and it's rather hard to fly arbitrary heights above its surface). This is not at all due to relativity.
Relativity does have an effect on geometry, but it's only significant for extremely precise calculations (e.g. GPS), extremely massive bodies (solar masses and up), or extremely high speeds (near the speed of light). In this case -- now, let me see if I have this right? -- hyperbolic space is present locally around a gravity well, or a property of the universe itself if the matter density is a little too low (which doesn't quite seem to be the case), while spherical geometry is present inside a gravity well of sufficient size: a black hole's event horizon, or a universe with slightly too much matter (which also doesn't quite seem to be the case). You also can't move or see beyond your causal horizon, which is also the extent of the visible universe if the universe happens to be flat (which seems to be very nearly the case).
If one constructs an experiment to measure the distances and angles in a triangle, say -- then the geometry of that experiment is affected by fluctuations in space itself: gravity waves. A number of active and proposed experiments are working on this very problem, and they've shown exciting (if extraordinarily weak) signals already!
Tim -
If I walk ten feet then take a 90' turn and walk ten more feet for a total of four times I end up where I started. If I start at 0'0' on earth then walk 6000 miles turn 90' walk 6000 miles I'm at the north pole. On my third turn of 90' I walk 6000 more miles and I'm back where I started. Where did the fourth turn go? Where do I end up at 3000 miles and turn four turns later? Back where I started?
That took me a second to figure out.
So in a gravity well do you make three turns like if you are out in space?
You missed the 90º downward pitch as you walked each of those 6000 mile legs.
Also, gravity plays no part in this, other than keeping you walking on the surface. -
In the original "10 foot" case, there are only three turns, so your basic assumption is incorrect.
Run your finger around your monitor screen in a clockwise direction, starting at the bottom LH corner.
First side :- LH bottom corner up to the LH top corner.
First 90 degree turn
Second side :-LH top corner to RH top corner.
Second 90 degree turn
Third side:- RH top corner down to RH bottom corner.
Third 90 degree turn
Fourth side:- RH bottom corner back to LH bottom corner.
Four sides, but only three 90 degree turns. -
As the Sun's gravity can measuably curve the path of light, then a square formed by four points around the sun will have corners measurably less than 90 degrees... Slight, but there.
Given that light bends towards the sun, wouldn't the angles appear to be more than 90 degrees?
Opps, yes. Went back and corrected. Glad I don't drive a intergalactic spaceship for a job! -
Four sides, but only three 90 degree turns.
True - but you are not looking in the same direction.
To make the trip repeatable over the same course, you would need to make that final turn. -
Four sides, but only three 90 degree turns.
True - but you are not looking in the same direction.
To make the trip repeatable over the same course, you would need to make that final turn.
The OP only said "get back where I started"----nothing about orientation or repeatability.
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I realize that this doesn't work if you are in a room because the room's floor is plat and not the curve of the earth. But what if you went somewhere totally flat like the salt flats in western USA. Cody's lab did a video where he showed the curve of the earth out there using a tele scope. So what if you drove 100 miles and started making right angle turns. Or better yet you did this out on the ocean. If you take a globe and map out a tiny triangle on the ocean you will see it ends at the origin. How small do you have to make your triangle before it becomes a square? Four bouys 100' away would be 90' apart. But what about 100 miles away does it become three?
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How small do you have to make your triangle before it becomes a square?
Well, aside from the fact that the "triangle" ONLY works properly when you traverse exactly 90º of a spheroid surface, it depends on your tolerances.
If you stand on a truly flat surface, then the 4 sided square movement will always work ... but that is for a true geometrically flat surface. The floor of a dried out salt flat is not geometrically flat - it follows the curve of the earth. On any such surface, the 4 sided movement can only get you back to your starting point when the distance you move in each direction approaches zero. For any larger distances, you must specify your "tolerance" for getting back to your original starting point.
This logic also applies in reverse for your triangle traverse - except the limiting case for perfect alignment is a much more observable 90º per leg.
You could calculate the error for the square traverse as distance increases - and you could also calculate the error for the triangular traverse as the angular distance decreases.
There is only ever "neatness" at the limiting conditions of each. Everywhere in between is messy ... unless you define a range where you consider it's not "messy" enough to bother you. -
In your example, you walk towards the north pole on a line of longitude to the exact north pole and when you turn 90 degrees you follow a second line back to the same latitude as you started. Then you turn 90 degrees and follow a line around the globe to your original starting point. This makes sense (assuming perfect sphere etc).
When compared to flat plane you are not comparing apples to apples. The earth is not a flat plane and map makers have struggled with this for years. A flat map can never represent places near the polar correctly on a flat map due to stretching out decreasing distances per lines of longitude at the poles. Greenland is too large on a flat map.
Also, the shortest distance between two places is not a straight line on a globe. Take a ball (larger the better) and mark two random places on the ball about 90 to 120 degrees apart (just for ease of do this) and then take a piece of string and pull the string tight at the two marked points and you will see that the string does not flow in a straight line, but is curved. Airplanes flying long distances follow that curve to reduce distance traveled.
Also, there is no north, east or west when at the north pole, only south. This was a major concern for nuke submarines when they first attempted to traverse the north pole - they weren't sure they could use their navigation equipment.