Draw a triangle on an ideal sphere using the north pole and two points on the equator separated by a quarter of the equator. Connect the pole to each equatorial point along a meridian, then connect those points along the shorter equatorial arc. Each corner is a right angle. The angles add to 270 degrees.
Nothing has gone wrong with the arithmetic. The triangle lives on a surface with a different geometry from a plane. Its sides follow the sphere's straightest available paths, and its angles are measured between those paths at their intersections.
This original construction gives us a concrete route into curvature. We can discover the difference by measuring on the surface, without standing outside the sphere and admiring its roundness. That distinction becomes especially useful when geometry describes spacetime, where there is no requirement for an outside room in which the universe is bent.
Why the three corners are right
A meridian meets the equator at a right angle. Our two meridians therefore supply right angles at the two lower corners. At the pole they meet at the same angle as the separation of their longitudes: one quarter-turn, or 90 degrees. Adding the three corners gives 90 + 90 + 90 = 270.
The sides are portions of great circles: circles made by cutting the sphere through its center. The equator is one; a full meridian belongs to another. Great-circle paths are geodesics on a sphere. A geodesic continues without an intrinsic turn along the surface; locally it supplies the straightest available path. On a plane it becomes an ordinary straight line. Einstein Online, spherical and flat geometry.
“Shortest path” needs a little care. Between ordinary, non-antipodal points on an ideal sphere, the shorter great-circle arc gives the shortest surface route. Continue along that same great circle the long way around and the whole journey is no longer shortest. A geodesic's local property does not guarantee that any arbitrarily long segment wins a global distance contest.
Our triangle covers one eighth of the sphere. One way to see this is to divide the northern hemisphere into four equal wedges between meridians a quarter-turn apart. The northern hemisphere is half the surface, so each wedge occupies an eighth of the whole. For a sphere of radius R, whose surface area is 4πR², the triangle's area is therefore πR²/2.
On a sphere, a geodesic triangle's angle excess above π radians equals its area divided by R². Here the excess is π/2, or 90 degrees, matching the extra right angle. This is an exact mathematical example. It does not assert that the physical Earth is a perfect sphere or that an actual route has been measured.
A curved drawing can describe a flat surface
Roll an ideal sheet into a cylinder without stretching it. Its appearance in three dimensions changes, but its internal distances do not. Unroll it again and a small triangle drawn on the sheet still has the same side lengths and angles. The cylinder demonstrates why bending seen from outside and intrinsic curvature are different concepts.
The sphere cannot be flattened in that fashion while preserving its entire surface geometry. Its large triangle already gives the problem away: a planar triangle cannot retain three right angles. A map must give up some exact relationship—perhaps distance, angle or area—to put the sphere on a flat page.
This also explains why a bent line on a diagram is insufficient evidence of curved space. Coordinate choices and projections can change a picture's appearance. Intrinsic geometry concerns relationships measured within the space. Distinguishing the labels from the measurements was important enough that Einstein discussed it through measuring rods and coordinate systems in his popular account of relativity. Einstein, sections XXIV and XXVII.
How a small region can look flat
The big polar triangle makes curvature obvious. A sufficiently small region can fit a flat approximation closely enough for a particular purpose. “Sufficiently” depends on the size of the region and the required precision.
For a transparent mathematical example, compare a circular arc of radius R with the straight chord joining its ends. Let s be the arc length. The chord length is 2R sin(s/(2R)). The arc is longer, though the difference can become very small when s is small compared with R.
Take an ideal radius of 1,000 units. An arc of length one unit has a chord shorter by roughly 0.0000000417 unit. For an arc of length 100 units, the difference grows to approximately 0.0417 unit. The radius is unchanged; the size of the region being approximated is what changed. These calculated examples are not geographic measurements.
A flat approximation can therefore be excellent within one error allowance and inadequate within a tighter one. Calling a region “locally flat” is not permission to ignore curvature at every scale. It tells us where to start comparing an approximation with the relationships it leaves out.
Free fall asks about spacetime paths
General relativity uses geometry to describe gravity. An ideal test particle subject to no nongravitational force follows a geodesic in spacetime. Its path joins events at different times. Calling it a geodesic does not make it a shortest route across a spatial map. Einstein Online, geometry of gravity.
Nearby freely falling particles can change their separation. These tidal effects express a gravitational difference that choosing a freely falling reference frame cannot erase across an extended region. A small enough freely falling laboratory approximates the setting of special relativity, but extending the region or demanding more precision can reveal the remaining curvature. Einstein Online, tidal effects and local frames.
The sphere helps explain intrinsic measurement and local approximation. It does not reproduce spacetime's geometry: time and space enter relativistic intervals differently. Nor does a rubber sheet's sag explain gravity by itself. A sheet demonstration relies on ordinary gravity to pull objects down, which makes it unsuitable as a literal mechanism for gravity's origin.
The payoff of the geometric description is a way to relate measured intervals and paths under a physical theory. Curvature acquires its significance through those relationships. A three-right-angle triangle gives us a simple example of how an apparently broken rule can instead reveal which geometry we are using.
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