Flat

Flat

zero curvature everywhere

The three curvatures: a surface may bend one way (positive, K = 1/R²), not at all (zero, K = 0), or two opposite ways at once (negative, K < 0). Flat is the middle case; the Earth, measured, is the first.

Exact sense	A plane; no curvature anywhere
Curvature	K = 0 everywhere (a sphere: K = 1/R²)
In the small	Every smooth surface is nearly flat
Flat but not level	A tilted plane
Level but not flat	The geoid; still water
As a claim about the Earth	Zero curvature everywhere
Status of that claim	Refuted by any measured curve

Flat describes a surface — or a space — with no curvature anywhere: a plane.^[1] In ordinary speech it is treated as a synonym for level and horizontal; it is neither. The horizontal is flat but local; the level surface is perpendicular to gravity but curved; flat is the property of bending nowhere at all, and is the strongest of the three.

That it is the strongest is most clearly seen by how easily the others fail it. A floor tilted to drain is flat and not level. The still surface of the sea is level and not flat. The one surface that everybody points to as the very type of flatness — calm water — is, being level, curved.^[2]

Flat, level, and horizontal

Three words, three things. Flat is zero curvature — a plane. Level is everywhere perpendicular to gravity — an equipotential, which over any distance curves. Horizontal is the plane tangent to a level surface at one point — flat, but true only there. The three are independent: a surface may have any one without the others, and the careless habit of treating them as one word is the root from which a great deal of confident error grows.^[3]

In the small and in the large

Any smooth surface, examined over a small enough patch, is indistinguishable from its tangent plane: flat to first order, which is why a field looks flat and a builder may treat it so. Flatness proper is the absence of curvature over the whole, not the patch. The measure is the Gaussian curvature $K$: a plane has $K = 0$ at every point; a sphere of radius $R$ has $K = 1/R^{2}$ at every point — small for a large sphere, but nowhere zero.^[4] Curvature takes three signs: positive where a surface bends the same way in every direction (a dome), negative where it bends opposite ways at once (a saddle), and zero where it does not bend — the plane, and the flat. Flatness is the vanishing of all of it. The gap between a sphere and its tangent plane is exactly the drop kept under eight inches per mile, squared: not flat, and measurably so, at the first mile.

The global claim

To call the Earth flat is to assert $K = 0$ everywhere — the strongest of the three claims and, for that reason, the easiest to disprove, since a single curved mile is enough.^[5] The curve has been measured times beyond counting, and most reliably along long, still water, where the surface is at its most obligingly level; which is the one place flat-Earth demonstrations are most fond of being held, and the worst they could choose. The shape that remains, when the flat claim is set down, is the business of the figure of the Earth.

See also

• Level — perpendicular to gravity, and curved; flat's most-confused neighbour
• Horizontal — the local tangent plane; flat, but only at a point
• Eight inches per mile, squared — the measured departure from flat
• Flat Earth — the global claim of zero curvature
• The figure of the Earth — the shape left when the claim is withdrawn

References

1. ^ A plane is the model of a flat surface: through any point, straight lines run in it without ever leaving it, and parallel lines stay parallel. The flat condition is intrinsic — it can be detected by measurements made within the surface, without reference to any space outside it.
2. ^ On why still water is a curved surface, see level: water at rest settles to an equipotential of gravity, which closes round the Earth.
3. ^ A surface can hold any one of the three properties without the others: a tilted plane (flat, not level); the geoid (level, not flat); a small patch of pavement (effectively flat and horizontal, yet part of a curved whole).
4. ^ A cylinder, curiously, has $K = 0$ and is not a plane — it can be unrolled flat without stretching, being curved in only one direction. A sphere cannot be unrolled flat at all, which is the standing difficulty of every flat map; see cartography. The Earth is curved in the manner of the sphere, not the cylinder.
5. ^ The claim is unusually falsifiable for a cosmography: it forbids curvature everywhere, so any curvature anywhere refutes it. The drop under eight inches per mile, squared is that curvature, stated as a number.

External links

• Carl Friedrich Gauss, General Investigations of Curved Surfaces (1827; English trans. 1902) — the memoir that defined the curvature by which the flat is told from the curved
• Gaussian curvature — the measure of how a surface departs from flat