3. The 3-dimensional cube
Here is a brief review of the relevant ideas. A regular polyhedron is one which consists of congruent convex regular polygons, and where all of the vertices of the polyhedra are congruent (alike) as well. Recall that a set or figure X is convex if, whenever p and q are points of X, then all the points of a line segment joining p and q are also in X. The set X below (in red) is non-convex because the blue line from point p to point q is not completely within the set.
Loosely speaking, plane sets fail to be convex when they have notches or holes. It is sometimes convenient to be able to "convexify" a non-convex set in a non-ambiguous way. This is achieved by the concept of the convex hull of a set X: The intersection of all convex sets that contain X (also characterized as the smallest convex set that contains X). For the non-convex polygon above, its convex hull is shown below:
Welcome to the
These web essays are designed for those who have already discovered the joys of mathematics as well as for those who may be uncomfortable with mathematics.
Search Feature Column
Feature Column at a glance