Abstract
We begin this chapter by defining and computing the area of a sphere and establishing a famous result of Girard on the area of a spherical triangle. Then, we present the important concept of convex polyhedron, which encompasses prisms and pyramids, and apply Girard’s theorem to prove the celebrated theorem of Euler, which asserts that the Euler characteristic of every convex polyhedron is equal to 2. The chapter finishes with using Euler’s theorem to obtain the classification of all regular polyhedra, and showing that all found possibilities do exist.
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- 1.
By the definitions just presented, we are forced to conclude that the interior of the empty set is itself.
- 2.
At this point, if we had at our disposal the concept of continuity for functions \(f:\Sigma \rightarrow \partial \mathcal P\), we could easily prove that this function f is a homeomorphism , i.e., a continuous bijection with continuous inverse. Thanks to this result, the boundary of a convex polyhedron is homeomorphic to (i.e., has essentially the same shape as) a sphere. Intuitively, such a statement means that one can continuously deform a sphere until transform it into the boundary of \(\mathcal P\).
- 3.
The maximum value of 30 edges is attained in regular icosahedra (see next section).
- 4.
In honor of Plato , one of the great philosophers of Classical Greek Antiquity, who lived in the fifth century bc.
References
T. Apostol, Calculus, vol. 1 (Wiley, New York, 1967)
A. Caminha, An Excursion Through Elementary Mathematics I - Real Numbers and Functions (Springer, New York, 2017)
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Caminha Muniz Neto, A. (2018). Convex Polyhedra. In: An Excursion through Elementary Mathematics, Volume II. Problem Books in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-77974-4_12
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DOI: https://doi.org/10.1007/978-3-319-77974-4_12
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