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An Elementary Proof of Euler's Theorem for Polytopes

An Elementary Proof of Euler's Theorem for Polytopes AN ELEMENTARY PROOF OF EULER'S THEOREM FOR POLYTOPES GEORGE MAXWELL Suppose that P is a d-dimensional convex polytope in W and let f be the number of /c-faces of P. We count the empty face as a ( —l)-face and P itself as a d-face. Euler's theorem asserts that, if P ± 0, an alternating sum of the f is equal to zero. This is of course equivalent to the more familiar equation The purpose of this note is to give a brief elementary proof of Euler's theorem by induction on the number of vertices of P. However, we shall first make some preliminary remarks. If F is a j-face of P, let f {F) denote the number of /c-faces of P containing F. There exists a "quotient" polytope P/F whose (/c-/-l)-faces correspond to the /c-faces of P containing F. Euler's theorem applied to P/F would state that an alternating sum of the numbers f {F) is also equal to zero. To define quotients, it is n+1 +1 simpler to use the corresponding cones in U . Namely, if C is a cone in U" and D is a face of C which spans a subspace U, the quotient http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

An Elementary Proof of Euler's Theorem for Polytopes

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/16.6.597
Publisher site
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Abstract

AN ELEMENTARY PROOF OF EULER'S THEOREM FOR POLYTOPES GEORGE MAXWELL Suppose that P is a d-dimensional convex polytope in W and let f be the number of /c-faces of P. We count the empty face as a ( —l)-face and P itself as a d-face. Euler's theorem asserts that, if P ± 0, an alternating sum of the f is equal to zero. This is of course equivalent to the more familiar equation The purpose of this note is to give a brief elementary proof of Euler's theorem by induction on the number of vertices of P. However, we shall first make some preliminary remarks. If F is a j-face of P, let f {F) denote the number of /c-faces of P containing F. There exists a "quotient" polytope P/F whose (/c-/-l)-faces correspond to the /c-faces of P containing F. Euler's theorem applied to P/F would state that an alternating sum of the numbers f {F) is also equal to zero. To define quotients, it is n+1 +1 simpler to use the corresponding cones in U . Namely, if C is a cone in U" and D is a face of C which spans a subspace U, the quotient

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Nov 1, 1984

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