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897:
858:
of vertices of a polyhedron by totaling the angles of all the faces, and adding the total defect. This total will have one complete circle for every vertex in the polyhedron. Care has to be taken to use the correct Euler characteristic for the polyhedron.
215:, the defect at a vertex equals 2π minus the sum of all the angles at the vertex (all the faces at the vertex are included). If a polyhedron is convex, then the defect of each vertex is always positive. If the sum of the angles exceeds a full
830:
to a sphere (i.e. topologically equivalent to a sphere, so that it may be deformed into a sphere by stretching without tearing), the "total defect", i.e. the sum of the defects of all of the vertices, is two full circles (or 720° or
850:
to the Euler characteristic. Here the
Gaussian curvature is concentrated at the vertices: on the faces and edges the Gaussian curvature is zero and the integral of Gaussian curvature at a vertex is equal to the defect there.
104:
angles add up to 360°). However, on a convex polyhedron the angles at a vertex add up to less than 360°, on a spherical triangle the interior angles always add up to more than 180° (the exterior angles add up to
924:
is convex and the defects at each vertex are each positive. Now consider the same cube where the square pyramid goes into the cube: this is concave, but the defects remain the same and so are all positive.
768:
659:
550:
441:
250:
meet at each vertex) is 36°, or π/5 radians, or 1/10 of a circle. Each of the angles measures 108°; three of these meet at each vertex, so the defect is 360° − (108° + 108° + 108°) = 36°.
814:
705:
596:
487:
376:
330:
878:
It is tempting to think that every non-convex polyhedron must have some vertices whose defect is negative, but this need not be the case. Two counterexamples to this are the
178:
201:
149:
866:, according to which a metric space that is locally Euclidean except for a finite number of points of positive angular defect, adding to 4
863:
1032:
883:
879:
97:
724:
615:
506:
395:
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than 360°), and the angles in a hyperbolic triangle always add up to less than 180° (the exterior angles add up to
843:
125:
921:
17:
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556:
447:
336:
903:
293:
896:
839:
152:
121:
82:
70:
1027:
847:
231:
117:
89:
1000:
956:
157:
826:
Descartes's theorem on the "total defect" of a polyhedron states that if the polyhedron is
183:
131:
227:
216:
52:
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The concept of defect extends to higher dimensions as the amount by which the sum of the
219:, as occurs in some vertices of many non-convex polyhedra, then the defect is negative.
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917:
254:
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933:
827:
932:(negative curvature), whereas positive defect indicates that the vertex resembles a
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929:
711:
243:
56:
602:
280:
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212:
1008:
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A generalization says the number of circles in the total defect equals the
51:
to add up to the expected amount of 360° or 180°, when such angles in the
870:, can be realized in a unique way as the surface of a convex polyhedron.
247:
28:
116:
In modern terms, the defect at a vertex is a discrete version of the
96:
In the
Euclidean plane, angles about a point add up to 360°, while
912:
A counterexample which does not intersect itself is provided by a
48:
913:
493:
886:, which have twelve convex points each with positive defects.
985:
Euler's Gem: The
Polyhedron Formula and the Birth of Topology
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Negative defect indicates that the vertex resembles a
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842:of the polyhedron. This is a special case of the
253:The same procedure can be followed for the other
763:{\displaystyle {\pi \over 5}\ \ (36^{\circ })}
654:{\displaystyle {\pi \over 3}\ \ (60^{\circ })}
545:{\displaystyle {\pi \over 2}\ \ (90^{\circ })}
436:{\displaystyle {2\pi \over 3}\ (120^{\circ })}
242:The defect of any of the vertices of a regular
18:Descartes' theorem on total angular defect
835:radians). The polyhedron need not be convex.
8:
100:in a triangle add up to 180° (equivalently,
62:Classically the defect arises in two ways:
854:This can be used to calculate the number
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133:
888:
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77:and the excess also arises in two ways:
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862:A converse to this theorem is given by
809:{\displaystyle 4\pi \ \ (720^{\circ })}
700:{\displaystyle 4\pi \ \ (720^{\circ })}
591:{\displaystyle 4\pi \ \ (720^{\circ })}
482:{\displaystyle 4\pi \ \ (720^{\circ })}
371:{\displaystyle 4\pi \ \ (720^{\circ })}
66:the defect of a vertex of a polyhedron;
874:Positive defects on non-convex figures
325:{\displaystyle \pi \ \ (180^{\circ })}
7:
961:Progymnasmata de solidorum elementis
988:, Princeton (2008), Pages 220–225.
846:which relates the integral of the
55:would. The opposite notion is the
25:
936:or minimum (positive curvature).
916:where one face is replaced by a
902:
895:
890:Polyhedra with positive defects
269:Polygons meeting at each vertex
864:Alexandrov's uniqueness theorem
180:, so the sum of the defects is
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234:falls short of a full circle.
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128:gives the total curvature as
884:great stellated dodecahedron
880:small stellated dodecahedron
47:) means the failure of some
288:Three equilateral triangles
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610:Five equilateral triangles
390:Four equilateral triangles
122:concentrated at that point
120:of the polyhedral surface
922:elongated square pyramid
246:(in which three regular
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719:Three regular pentagons
173:{\displaystyle \chi =2}
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272:Defect at each vertex
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967:, vol. X, pp. 265–276
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196:{\displaystyle 4\pi }
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144:{\displaystyle 2\pi }
965:Oeuvres de Descartes
844:Gauss–Bonnet theorem
840:Euler characteristic
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153:Euler characteristic
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126:Gauss–Bonnet theorem
1033:Hyperbolic geometry
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822:Descartes's theorem
266:Number of vertices
83:toroidal polyhedron
71:hyperbolic triangle
1001:Weisstein, Eric W.
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848:Gaussian curvature
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16:(Redirected from
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255:Platonic solids
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53:Euclidean plane
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918:square pyramid
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934:local maximum
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275:Total defect
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980:Richeson, D.
974:Bibliography
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960:
952:
930:saddle point
927:
911:
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855:
853:
837:
828:homeomorphic
825:
712:dodecahedron
252:
244:dodecahedron
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115:
113:than 360°).
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603:icosahedron
281:tetrahedron
1022:Categories
940:References
383:octahedron
213:polyhedron
151:times the
124:, and the
45:deficiency
1028:Polyhedra
1009:MathWorld
799:∘
782:π
753:∘
731:π
690:∘
673:π
644:∘
622:π
581:∘
564:π
535:∘
513:π
472:∘
455:π
426:∘
406:π
361:∘
344:π
315:∘
298:π
248:pentagons
191:π
162:χ
139:π
118:curvature
882:and the
238:Examples
102:exterior
29:geometry
920:: this
226:of the
41:deficit
33:angular
31:, the (
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263:Shape
211:For a
57:excess
49:angles
37:defect
963:, in
945:Notes
230:at a
228:cells
914:cube
494:cube
232:peak
217:turn
111:more
107:less
39:(or
795:720
686:720
577:720
468:720
422:120
357:720
311:180
43:or
27:In
1024::
1006:.
982:;
959:,
749:36
716:20
640:60
607:12
531:90
257::
203:.
59:.
35:)
1012:.
868:π
856:V
833:π
831:4
804:)
791:(
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758:)
745:(
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695:)
682:(
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636:(
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573:(
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527:(
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477:)
464:(
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431:)
418:(
410:3
403:2
387:6
366:)
353:(
341:4
320:)
307:(
285:4
188:4
168:2
165:=
136:2
92:;
85:.
73:;
20:)
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