Question

Prove that faces, cross sections, and projections of convex polyhedra are convex polygons.

Short answer

Faces, cross sections, and projections of convex polyhedra are all convex polygons.

Step-by-step solution

Understanding Convex Polyhedra

Convex polyhedra are 3-dimensional shapes where a line segment connecting any two points within the polyhedron lies entirely inside it. This property must be considered when proving the properties of their faces, cross sections, and projections.

Proving Faces Are Convex Polygons

Each face of a convex polyhedron is a polygon made by the intersection of a plane with the polyhedron. Since the polygon is formed by the intersection of half-spaces (which represent the volume of the polyhedron), the resulting face is a convex polygon as it retains the convex property of the whole shape.

Demonstrating Cross Sections As Convex Polygons

A cross section of a polyhedron is the shape formed by the intersection of a plane with the polyhedron. Since the polyhedron is convex, any cross-sectional plane will intersect in a convex shape due to the flatness and continuity of the polyhedron's surface, resulting in a convex polygon.

Showing Projections Are Convex Polygons

A projection of a polyhedron onto a plane involves mapping each vertex to the nearest point on the plane. Given that convexity ensures line segments within the polyhedron's boundary connect all vertices, their projections similarly connect, thus forming a convex polygon in the plane.

Key concepts

Convex Polygons

Convex polygons are shapes where a line segment joining any two points within the polygon lies entirely inside the shape. These are notably different from non-convex polygons, where such a line might extend outside the polygon. Convex polygons have simple and useful properties, such as:

• Their interior angles always total less than 360 degrees.
• All internal angles are less than or equal to 180 degrees.

In the context of a convex polyhedron, understanding the nature of these polygons is essential. Every face of a polyhedron is a polygon. Due to the polyhedron’s convex nature, these polygons are also convex. This occurs because the plane intersection that forms each face cannot extend beyond the outline of the polyhedron, ensuring all parts of the face are contained within its boundary.

Cross Sections

Cross sections occur when a plane intersects a polyhedron. Imagine slicing through a loaf of bread with a knife; the cut surface is the cross section. For a convex polyhedron, this cross-sectional area will be a convex polygon. So why is that?

• The intersection plane doesn't cut across itself.
• The nature of the surface it crosses is only unbroken and continuous due to the convexity.

As a result, the shape formed is a convex polygon. This is because no part of the section is pinched inward. The smooth, non-intersecting perimeter guarantees a convex shape. This is true no matter the angle or position of the intersecting plane.

Projections

When projecting a polyhedron, imagine shining a flashlight on it and observing the shadow on a wall. This shadow is the projection. The process involves mapping each vertex of the polyhedron to its nearest point on a chosen plane, and this creates a new 2D shape. For convex polyhedra, projections will always be convex polygons. Here’s why:

• Convexity means any linear path on the polyhedron stays inside its limits.
• Similarly, projections of these connected vertices onto a plane will not disrupt the line segments’ connections.

This ensures that the resulting shape preserves the original polyhedron's convex property's integrity, yielding a connected and continuous convex polygon on the plane.

Geometry

Geometry is the branch of mathematics that deals with sizes, shapes, and the relative positions of figures. It plays a crucial role in understanding concepts like convex polyhedra, convex polygons, cross sections, and projections. Why is geometry important here?

• It provides the vocabulary and methods needed to discuss and prove properties of shapes and figures.
• Understanding geometric principles underpins the ability to explain why faces of convex polyhedra, their cross sections, and projections all result in convex polygons.

From the basic principles of points, lines, and planes to the intricate discussions of 3D modeling, geometry allows us to describe with clarity why convex constraints exist and how they manifest in different contexts like those of polyhedra.