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It is obvious that the Earth is not flat. We know this from looking at NASA pictures other experiments. But, mathematically speaking, the Earth, along with other celestial bodies, are measured with mathematics and graphs that display information using a 2 dimensional point of reference.

For example, suppose I want to show the different temperatures at the earths inner core compared to its outer-shells. I would display the information on a graph, placing temperature on the $y$-axis and the distance from the surface to the core on the $x$-axis. This shows the proper data but it will also means that I'm using a 2 dimensional model of the earth to display the data, in other words, flat.

Genesis Productions okc
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  • Hi and welcome to the Physics SE! You can answer your own questions, but please post the answer as such and not in the body of the question. – stafusa Sep 27 '17 at 20:43
  • You're actually only plotting one spatial dimensions in that example of temperature vs. radius. When you assume spherical symmetry, you can always change $d^3 r \to 4 \pi r^2 dr$ – Señor O Sep 27 '17 at 20:44
  • can this rule also be apply in a quantum level. – Genesis Productions okc Sep 27 '17 at 20:50

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No. The Earth is not defined by a flat geometry. To see why this is the case, just look at what happens to two parallel geodesics. A geodesic is the general version of what we call a "straight line" in flat geometries. On a spherical surface, such geodesics are called "great circles"; common examples of great circles: the Equator, lines of longitude. Take any two lines of longitude - they're parallel at the equator, and they cross at the poles, therefore the surface of the Earth is not flat.

In order to be flat, two parallel lines have to not only not cross, but maintain a constant distance from each-other. If the distance between the parallel lines grows from a point of closest approach, then the geometry is hyperbolic.

When you talk about drawing something on a flat graph, you're going to inevitably have to distort something badly to do it, or just not draw the whole thing. In most cases, we cut along a line to make it two of the different edges of the map, and then we take single points (usually the North and South poles) and stretch them out to be entire lines. If we don't do the second, we get something like the sinusoidal projection where straight lines of longitude are badly bent. You can go through Wikipedia's list of map projections to get an idea of some of the trade-offs among the various distortions to shape and area that have to be made to produce a flat projection of the whole world.

Sean E. Lake
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It sounds like you're curious about data representation. In the example you've given (temperature as a function of depth from the Earth's surface), you've actually compressed three spatial dimensions into one dimension and then plotted the temperature as a function of depth. Because of the rough spherical symmetry of the earth, this is a pretty good representation (though it'd also make sense to start that axis at the center of the earth and run it outwards), but it ignores the temperature variation at different places on Earth (at a specific depth). Since we don't know local temperatures underground at every longitude and latitude, we don't lose much by displaying only the depth, but to show this data completely, you'd need a 3 dimensional display which could tell you temperatures at every point.

You're right to recognize that our computer screens (and most of the charts we put on them) are two dimensional, but that doesn't have any bearing on how many numbers we need to describe a physical system. In fact, computers store data in one dimensional arrays - everything else is data representation.

user121330
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