elliptic geometry
If hyperbolic geometry is what you get by allowing TOO MANY parallels, elliptic geometry is the opposite extreme — a geometry with NO parallels at all. Through a point off a given line, there is not one parallel (Euclid) nor infinitely many (hyperbolic) but zero: every pair of lines meets. Bernhard Riemann opened this third door in 1854. To make it work, two of Euclid's assumptions have to be loosened — lines must be finite in total length (they close up on themselves), and the assumption that a line has two distinct endpoints in each direction is dropped.
The model to keep in your head is the sphere with one crucial adjustment. On a sphere the 'lines' are great circles (circles that cut the sphere into equal halves, like the equator), since they are the shortest paths and the straightest available routes. Any two great circles always cross — at two diametrically opposite points. To get genuine ELLIPTIC geometry (where two lines meet in exactly ONE point, as proper geometry demands), you identify each pair of antipodal points as a single point; the result is the real projective plane. Now any two lines meet exactly once, there are no parallels, and triangles bulge: their angles add up to MORE than 180 degrees, the excess growing with area.
Elliptic geometry has its own elegant signature, a perfect duality between points and lines: every true statement stays true if you swap the words 'point' and 'line' throughout, because in the projective plane two points determine a line and two lines determine a point symmetrically. As with the other geometries, the honest framing is that elliptic geometry does not refute Euclid — it is a third consistent option, obtained by replacing the parallel postulate (and relaxing the line-length axiom) rather than by finding any flaw in the original.
On a globe, two meridians (lines of longitude) are both great circles, so both are 'lines'. They run parallel at the equator yet inevitably cross at the North and South Poles — proof that there are no parallels. A triangle joining the North Pole to two equator points 90 degrees apart has three right angles, summing to 270 degrees: an angle excess of 90 degrees, impossible in Euclid.
No parallels, every two lines meet, and triangle angles exceed 180 degrees — the third consistent geometry, dual in points and lines.
Spherical and elliptic geometry are close cousins but not identical: on the sphere two great circles meet TWICE (at antipodes), violating 'two points, one line'. True elliptic geometry glues antipodal points together so any two lines meet exactly once.