Room 10 of 11 · Attic
The Turn
“Can you come home changed?”
Lamp waiting
▶ Hear the storyteller tell this roomLook
A tiny doll holds an arrow. A round hill rises under her feet. A loop is marked on the hill.
Try
Walk her around the loop. Watch the arrow when she gets back.
Notice
Try the experiment first. Then this will fill in.
What keeps this room working?
A closed journey on the hill turns the arrow by a real amount.
What would break this room?
If every closed journey brought everything back exactly as it began.
Child-sized version
“You can come home changed.”
Try to break the room
For grown-upsShow me the math
Carry an arrow around a loop on a curved surface without ever turning it on purpose. It comes back turned anyway. On a sphere, the turn equals the area the loop encloses.
- Parallel transport
- Carrying an arrow along a path without turning it on purpose.
- Holonomy
- The turn you come home with.
- Curvature
- What makes the turn nonzero. A flat floor gives no turn at all.