Flagbound

You are standing on the surface of a closed world. You can't see it from outside. Work out its shape from the inside, using flags and an odometer.

Sphere, torus, double torus, triple torus or Klein bottle, chosen at random.

Or practice on a known world (outside view unlocked)

How to figure it out

Plant a flag, then walk straight (don't turn). Flags that you come back to announce the distance you walked.
On a sphere every straight line comes home after the same distance. Does that hold here, in every direction?
Some straight lines come back, some never do, and the distances differ. What does that tell you about holes?
You get a guess at the end. Mystery worlds reveal an outside view afterwards.

Controls

Click the view to capture the mouse. If capture is blocked, drag to look instead. Esc frees the cursor.

WASD walk and strafe. Shift runs.
Mouse or arrow keys look around. Looking left or right turns your heading; holding only W walks a straight line (a geodesic).
F plants a flag a few steps ahead. X removes the last one. Up to 24 flags.
R resets the trip meter. The odometer never resets.
The world turns once every 5 minutes: the sun rises and sets, and the stars wheel around one fixed pole star. C toggles faint constellation lines.
T toggles a bright red string that runs from your first flag, along your path through every later flag, to you. Off by default.
G opens the guess panel. O toggles the outside view when it is unlocked.
[ / ] change render resolution.

Flags light up in the list when you walk back to them: you see how far you walked since you last stood there.

One guess. You will see the world from outside afterwards.