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Illustration of a Property of Polyhedra: That the Sum of the Angular Deficits at All Vertices is equal to 4π radians |
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Mathematicians worked with polyhedra for two thousand years before the Swiss mathematician, Leonard Euler, noticed that for all of them the number of faces less the number of edges plus the number of vertices is equal to 2. This became known as their Euler characteristic. This applied only to polyhedra without any holes. For a polyhedral figure shaped generally like a doughnut the Euler characteristic is zero.
What is illustrated below is that for polyhedra without holes the sum of the angular deficits at the vertices is equal to 4π radians (720°). The angular deficit at a vertex is the amount by which the sum of the interior angulars of the faces that meet at that vertice fall short of a full circle, 2π radians. The angular deficit at a vertex can also be visualized in terms of cutting off the polyhedron near the vertex and then opening up the figure along one edge and flattening it out. The angular deficit is the amount the flattened figure falls short of a full circle.
In the above illustration the blue sector represents the sum of the angles of the polygons meeting at a vertex and the white sector is the angular deficit.
This property of polyhedra is a special case of an extension of the very beautiful Gauss-Bonnet Theorem. The Gauss-Bonnet Theorem says that the integral of the Gaussian curvature of a closed smooth surface is equal to 2π times the Euler characteristic for the figure. For a surface that is topologically equivalent to a sphere that integral is 4π. The extension of the Gauss-Bonnet Theorem for surfaces like spheres but which have isolated singularities like conical points or vertices is that the sum of the angular deficits at those points plus the integral of the curvature over the smooth part of the surfaces is equal to 4π. For polyhedra the curvature on the faces is zero so the theorem reduces to the sum of the angular deficits at the vertices being equal to 4π radians. This is illustrated below for the regular polyhedra. (These are also known as the Platonic Solids.)
The Angular Deficits at the Vertices of the Regular Polyhedra |
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Regular Polyhedron | Face | Interior angle | Number of Faces at Vertex | Angular Deficit |
Tetrahedron | triangle | π/3 (60°) | 3 | π (180°) |
Cube | square | π/2 (90°) | 3 | π/2 (90°) |
Rhomboid (Octahedron) | triangle | π/3 (60°) | 4 | 2π/3 (120°) |
Dodecahedron | pentagon | 3π/5 (108°) | 3 | π/5 (36°) |
Icosahedron | triangle | π/3 (60°) | 5 | π/3 (60°) |
The Total Angular Deficits of the Regular Polyhedra | |||
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Polyhedron | Angular Deficit at each Vertex | Number of Vertices | Total Angular Deficit |
Tetrahedron | π (180°) | 4 | 4π (720°) |
Cube | π/2 (90°) | 8 | 4π (720°) |
Rhomboid | 2π/3 (120°) | 6 | 4π (720°) |
Dodecahedron | π/5 (36°) | 20 | 4π (720°) |
Icosahedron | π/3 (60°) | 12 | 4π (720°) |
Although the proposition is illustrated only for the regular polyhedra it is true for all polyhedra without holes, and a corresponding proposition is true for polyhedra with holes.
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