In the realm of convex polyhedra, a pivot point is a special point within the interior of the polyhedron. It holds the property that every line passing through it contains either zero or twovertices of the polyhedron. Now, let’s explore the intriguing question: What is the maximum number of pivot points that a polyhedron can contain?
To unravel this mystery, let’s embark on a geometric journey:
- Suppose we have a pivot point P with two other pivot points x and y. We’ll pick a vertex v₁ of the polyhedron that is not on the line connecting x and y. The line from v₁ to xintersects another vertex v₂ of the polyhedron. Continuing this process, we find vertices v₃, v₄, and so on. These vertices must lie in the same plane as x, y, and P due to the convexity of the polyhedron.
- By construction, these vertices form a convex polygon with x and y as its pivot points. This polygon lies in the same plane as x, y, and P.
- Now, let’s focus on the convex polygon. For any convex polygon, there exists a unique pairing of its vertices for the pivot point. Why? Because there must be an equal number of vertices on each side of any line segment connecting two vertices if that line segment is to pass through the pivot point. This unique pairing ensures that at most one pivot point exists for the convex polygon.
- Therefore, we conclude that a polyhedron can have at most one pivot point.
In summary, the maximum number of pivot points in a polyhedron is one. Beyond that, the geometry of convex polygons restricts any additional pivot points.
