Graph Theory Il The theorem follows by the principle of induction In this argument, we implicitly assumed two geometric facts: a drawing of a tree can not have multiple faces and removing an edge on a cycle merges two faces into one 3. 2 Classifying Polyhedra The Pythagoreans had two great mathematical secrets, the irrationality of 2 and a geo- metric construct that were about to rediscover A polyhedron is a convex, three-dimensional region bounded by a finite number of polygonal faces. If the faces are identical regular polygons and ual number of ol poly- gons meet at each corner, then the polyhedron is regular. Three examples of regular polyhedra are shown below: the tetraheron, the cube, and the octahedron How many more polyhedra are there? Imagine putting your eye very close to one face of a translucent polyhedron. The edges of that face would ring the periphery of your vision and all other edges would be visible within. For example, the three polyhedra above would look something like this Ox Thus, we can regard the corners and edges of these polyhedra as the vertices and edges nother logical leap based on geometric intuition. ) This mea Euler's formula for planar graphs can help guide our search for regular polyhedra8 Graph Theory II The theorem follows by the principle of induction. In this argument, we implicitly assumed two geometric facts: a drawing of a tree can not have multiple faces and removing an edge on a cycle merges two faces into one. 3.2 Classifying Polyhedra The Pythagoreans had two great mathematical secrets, the irrationality of 2 and a geo￾metric construct that we’re about to rediscover! A polyhedron is a convex, three­dimensional region bounded by a finite number of polygonal faces. If the faces are identical regular polygons and an equal number of poly￾gons meet at each corner, then the polyhedron is regular. Three examples of regular polyhedra are shown below: the tetraheron, the cube, and the octahedron. How many more polyhedra are there? Imagine putting your eye very close to one face of a translucent polyhedron. The edges of that face would ring the periphery of your vision and all other edges would be visible within. For example, the three polyhedra above would look something like this: Thus, we can regard the corners and edges of these polyhedra as the vertices and edges of a planar graph. (This is another logical leap based on geometric intuition.) This means Euler’s formula for planar graphs can help guide our search for regular polyhedra