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Going 3D

Investigating polyhedra and Euler's facet counting formula.

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A polyhedra is the three-dimensional version of a polygon: a 3D solid that has 2d, polygonal faces. Most people are only familiar with the most basic polyhedra: cubes, prisms, and pyramids. However, there are many fascinating types of polyhedra and much that we can study about them!

For example, the sequence of polyhedra below demonstrates the process of rectifying a cube.

In its final, fully rectified state, the cube is transformed into a strange polyhedra called a cuboctahedron.

Additionally, instead of the classic approach of introducing 3D geometry with calculations of volume and surface area, this chapter takes a completely different tract and introduces 3D geometry with an in-depth exploration of one single, tremendous result known as Euler’s Facet Counting Formula: \[V + F = E + 2\]

As an example, for a cube, the number of vertices, V = 8, the number of faces, F = 6, and the number of edges, E=12 fit this formula as follows:

\[8 + 6 = 12 + 2.\]

With many examples, we can test that \(V + F = E + 2\) always holds.

However, why is this relationship between vertices, edges, and faces of polyhedra always true and how can it be applied to problem solving? These are the two fundamental questions that this chapter will explore in depth.

Master the problem solving skills of Outside the Box Geometry.

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