Room 6 of 11 · Middle floor

The Bead

“What can the dot tell us?”

Lamp waiting

▶ Hear the storyteller tell this room

Look

A circular rail floats above a floor tile. A bead rides the rail. A gold dot glows on the floor beneath it.

Try

Move the bead. Then draw the curtain and guess where it is.

Notice

Try the experiment first. Then this will fill in.

What keeps this room working?

Many bead positions on the hidden circle sit above the same visible dot.

What would break this room?

If the dot always told you exactly where the bead was.

Child-sized version

“The dot cannot tell the whole story.”

Try to break the room

For grown-upsShow me the math

The dot is a point of the base. The rail is its fibre. In the Hopf fibration every point of the 2-sphere has a circle above it, and there is no continuous way to pick one bead on every circle at once.

Fibre: The hidden circle above a point. Every point of the base has one.
Hopf fibration: The map from the 3-sphere to the 2-sphere whose fibres are circles.
No global section: No continuous choice of one bead over every dot exists.
Chern class: The number that measures why no global section can exist here. It is not zero.

The full statement, through the looking glass →