A Line Is a Square

Same infinity, different dimensions

An interactive lesson with 7 steps, about 6 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• A segment, a line, a square and even a cube all have the same number of points.
• Interleaving digits pairs points of a square with points of a segment.
• Dimension is about arrangement (continuity), not about how many points there are.

Steps

1. Stretching a segment
2. Squash a square
3. “I see it, but I don’t believe it”
4. Where does 0.75 go?
5. Hilbert’s squares
6. A curve that fills the square
7. Back to the square