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12.7: Classifying Polyhedra

  • Page ID
    48383
    • Eric Lehman, F. Thomson Leighton, & Alberty R. Meyer
    • Google and Massachusetts Institute of Technology via MIT OpenCourseWare
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    The Pythagoreans had two great mathematical secrets, the irrationality of \(\sqrt{2}\) and a geometric construct that we’re 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 an equal number of polygons meet at each corner, then the polyhedron is regular. Three examples of regular polyhedra are shown in Figure 12.13: the tetrahedron, the cube, and the octahedron.

    We can determine how many more regular polyhedra there are by thinking about planarity. Suppose we took any polyhedron and placed a sphere inside it. Then we could project the polyhedron face boundaries onto the sphere, which would give an image that was a planar graph embedded on the sphere, with the images of the corners of the polyhedron corresponding to vertices of the graph. We’ve already observed that embeddings on a sphere are the same as embeddings on the plane, so Euler’s formula for planar graphs can help guide our search for regular polyhedra.

    For example, planar embeddings of the three polyhedra in Figure 12.1 are shown in Figure 12.14.

    clipboard_e851b2f7d8c0e531770a4013b66803200.png
    Figure 12.13 The tetrahedron (a), cube (b), and octahedron (c).
    clipboard_ef8833a52898e22d3027e91f65728b1cc.png
    Figure 12.14 Planar embeddings of the tetrahedron (a), cube (b), and octahedron (c).

    Let \(m\) be the number of faces that meet at each corner of a polyhedron, and let \(n\) be the number of edges on each face. In the corresponding planar graph, there are \(m\) edges incident to each of the \(v\) vertices. By the Handshake Lemma 11.2.1, we know:

    \[\nonumber mv = 2e.\]

    Also, each face is bounded by \(n\) edges. Since each edge is on the boundary of two faces, we have:

    \[\nonumber nf = 2e\]

    Solving for \(v\) and \(f\) in these equations and then substituting into Euler’s formula gives:

    \[\nonumber \dfrac{2e}{m} - e + \dfrac{2e}{n} = 2\]

    which simplifies to

    \[\label{12.7.1.}\dfrac{1}{m} + \dfrac{1}{n} = \dfrac{1}{e} + \dfrac{1}{2}\]

    Equation \(\ref{12.7.1.}\) places strong restrictions on the structure of a polyhedron. Every nondegenerate polygon has at least 3 sides, so \(n \geq 3\). And at least 3 polygons must meet to form a corner, so \(m \geq 3\). On the other hand, if either \(n\) or \(m\) were 6 or more, then the left side of the equation could be at most \(\frac{1}{3} + \frac{1}{6} = \frac{1}{2}\), which is less than the right side. Checking the finitely-many cases that remain turns up only five solutions, as shown in Figure 12.15. For each valid combination of \(n\) and \(m\), we can compute the associated number of vertices \(v\), edges \(e\), and faces \(f\). And polyhedra with these properties do actually exist. The largest polyhedron, the dodecahedron, was the other great mathematical secret of the Pythagorean sect.

    clipboard_e4bd40211e2d18d1061828c34c81a0dcd.png
    Figure 12.15 The only possible regular polyhedra.

    The 5 polyhedra in Figure 12.15 are the only possible regular polyhedra. So if you want to put more than 20 geocentric satellites in orbit so that they uniformly blanket the globe—tough luck!


    This page titled 12.7: Classifying Polyhedra is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Eric Lehman, F. Thomson Leighton, & Alberty R. Meyer (MIT OpenCourseWare) .

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