Non-Euclidean geometry.

Step 1 of 4

A triangle on a ball

On a flat plane every triangle’s angles add up to $180^\circ$⁠. The proof rests on Euclid’s parallel postulate: through a point off a line passes just one line that never meets it. A curved surface has no straight lines; instead it has geodesics: the shortest paths along the surface. On a ball these are arcs of a great circle.

Curvature K is fixed now on a ball, $K = 0.25$⁠.

Step 2 of 4

The Gauss–Bonnet theorem

On a surface of constant curvature, the difference between the angle sum and $180^\circ$⁠ equals the curvature times the area:

$\alpha + \beta + \gamma - \pi = K \cdot A$$\alpha + \beta + \gamma - \pi =$$K \cdot A$⁠

The angles here are in radians, which is why it has $\pi$⁠ and not $180^\circ$⁠. On a ball $K > 0$⁠ and the excess is positive; on a saddle $K < 0$⁠, and the sum is less than $180^\circ$⁠. At the Angles stage the board shows each angle and the sum.

Step 3 of 4

A small triangle

Triangle size is free now, and the curvature stays $0.25$⁠.

Step 4 of 4

Parallels

At the Parallels stage two lines leave the same base line at right angles. On a Flat surface they stay a fixed distance apart, as Euclid required. On a ball they close in and meet, like lines of longitude at the pole.

Try it: after checking, switch between Saddle, Flat and Ball, and compare the gap at the lines’ ends.

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