Why does the number of regions in a circle cut by chords joining $n+1$ points equal the number of regions in $\mathbb{R}^4$ cut by $n$ hyperplanes?

Question

It is well-known that if $n+1$ points are placed on a circle ($n$ a nonnegative integer), the $\binom{n+1}{2}$ chords joining them cut the interior into $$1 + \binom{n+1}{2} + \binom{n+1}{4} = \sum_{k=0}^4 \binom{n}{k}$$ regions (in the general case where no three chords have a common intersection). This is also equal to the number of regions that $4$-dimensional Euclidean space $\mathbb{R}^4$ is cut by $n$ general hyperplanes. (This is sequence A000127.) (Equivalently, it is the number of regions cut by $n+1$ hyperplanes in $4$-dimensional projective space $\mathbb{RP}^4$.)

The question: Is there a direct proof that these two numbers are the same, without explicitly counting them? For example, is there a natural bijection between the regions in the circle and the regions in $4$-space?

Answer 1

Put one marble inside each region and allow the marbles to fall and roll downwards, where we pick “downwards” as some direction that’s not parallel to anything of interest.

In the case of the circle:

• One marble rolls to the bottom of the circle.
• For every set of two points, one marble rolls to the lower of the two, resting on the chord between them. (Each such marble actually rests between two chords, or between one chord and the upper part of circle; we assign it to the chord whose angle is farther from the upper part of the circle.)
• For every set of four points, one marble rolls to the intersection of the diagonals of the quadrilateral formed by them.

This gives a bijection between the regions and the sets of 0, 2, or 4 of the $n + 1$ points.

In the case of 4-dimensional space, for convenience, draw an extra slightly slanted “ground hyperplane” below all of the existing intersection points that catches all the falling marbles (but does not create additional regions).

• One marble rolls forever down the ground hyperplane without hitting any other hyperplanes.
• For every set of two hyperplanes (possibly including the ground hyperplane), one marble rolls forever down the line or plane defined by the intersection of those hyperplanes with the ground hyperplane.
• For every set of four hyperplanes (possibly including the ground hyperplane), one marble rolls to their intersection.

This gives a bijection between the regions and the sets of 0, 2, or 4 of the $n + 1$ hyperplanes (possibly including the ground hyperplane).

Composing these bijections gives a bijection between the regions of the circle and the regions of 4-dimensional space.