Foundation A · beneath the house
The Hidden Circles
“Can one dot carry a whole circle?”
Lamp waiting
▶ Hear the storyteller tell this roomLook
A round world. A gold dot sits on its surface. Above the dot floats a circular track with a bead on it.
Try
Move the bead.
Notice
Try the experiment first. Then this will fill in.
What keeps this room working?
Every dot has its own hidden circle, and the dot never reveals the bead's place on it.
What would break this room?
If one visible dot could uniquely reveal its entire hidden circle, everywhere.
Child-sized version
“One little dot cannot carry the whole circle.”
Try to break the room
For grown-upsShow me the math
The round world is the 2-sphere. Each dot's hidden circle is its fibre in the Hopf fibration. Two fibres over different dots are linked circles that never meet. This is used here as a rigorous structural example, not a claim that reality is a Hopf fibration.
- Base space
- The round world where the dots live.
- Fibre
- The hidden circle above one dot.
- Total space
- All the circles together. A 3-sphere, bigger than the world below.
- Linked fibres
- Any two hidden circles thread through each other once and never touch.