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In `F33f`_`[geometry`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Geometry]`_`f, the `!angular defect`! or simply `!defect`! (also called `!deficit`! or `!deficiency`!) is the failure of some `F33f`_`[angles`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Angle]`_`f to add up to the expected amount of 360° or 180°, when such angles in the `F33f`_`[Euclidean plane`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Euclidean_plane]`_`f would. The opposite notion is the `F33f`_`[excess`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Angle_excess]`_`f.

Classically the defect arises in two contexts: in the Euclidean plane, angles about a point add up to 360°, while `F33f`_`[interior angles`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Internal_and_external_angle]`_`f in a triangle add up to 180°. However, on a `F33f`_`[convex polyhedron`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Polyhedron]`_`f, the angles of the faces meeting at a vertex add up to `*less`* than 360° (a defect), while the angles at some vertices of a `F33f`_`[nonconvex polyhedron`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Nonconvex_polyhedron]`_`f may add up to `*more`* than 360° (an excess). Also the angles in a `F33f`_`[hyperbolic triangle`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Hyperbolic_triangle]`_`f add up to `*less`* than 180° (a defect), while those on a `F33f`_`[spherical triangle`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Spherical_triangle]`_`f add up to `*more`* than 180° (an excess).

In modern terms, the defect at a vertex is a discrete version of the `F33f`_`[curvature`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Gaussian_curvature]`_`f of the polyhedral surface `F33f`_`[concentrated at that point`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Dirac_delta_function]`_`f. Negative defect indicates that the vertex resembles a `F33f`_`[saddle point`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Saddle_point]`_`f (negative curvature), whereas positive defect indicates that the vertex resembles a `F33f`_`[local maximum`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Local_maximum]`_`f or minimum (positive curvature). The `F33f`_`[Gauss–Bonnet theorem`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Gauss–Bonnet_theorem]`_`f gives the total curvature as 2 π π {\\displaystyle 2\\pi } times the `F33f`_`[Euler characteristic`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Euler_characteristic]`_`f χ χ = 2 {\\displaystyle \\chi =2} , so for a convex polyhedron the sum of the defects is 4 π π {\\displaystyle 4\\pi } , while a `F33f`_`[toroidal polyhedron`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Toroidal_polyhedron]`_`f has χ χ = 0 {\\displaystyle \\chi =0} and total defect zero.

>>Contents

• `F0af`_`[Defect of a vertex`#defect-of-a-vertex]`_`f

• `F0af`_`[Examples`#examples]`_`f

• `F0af`_`[Descartes's theorem`#descartes-s-theorem]`_`f

• `F0af`_`[Positive defects on non-convex figures`#positive-defects-on-non-convex-figures]`_`f

• `F0af`_`[References`#references]`_`f

• `F0af`_`[Notes`#notes]`_`f

• `F0af`_`[Bibliography`#bibliography]`_`f

• `F0af`_`[External links`#external-links]`_`f

-─

>>Defect of a vertex

For a `F33f`_`[polyhedron`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Polyhedron]`_`f, 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 `F33f`_`[turn`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Turn_(geometry)]`_`f, as occurs in some vertices of many non-convex polyhedra, then the defect is negative.

The concept of defect extends to higher dimensions as the amount by which the sum of the `F33f`_`[dihedral angles`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Dihedral_angle]`_`f of the `F33f`_`[cells`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Cell_(geometry)]`_`f at a `F33f`_`[peak`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Peak_(mathematics)]`_`f falls short of a full circle.

>>Examples

The defect of any of the vertices of a regular `F33f`_`[dodecahedron`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Dodecahedron]`_`f (in which three regular `F33f`_`[pentagons`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Pentagon]`_`f 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°.

The same procedure can be followed for the other `F33f`_`[Platonic solids`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Platonic_solid]`_`f:

`t

| Shape | Number of vertices | Polygons meeting at each vertex | Defect at each vertex | Total defect |

|---|---|---|---|---|

| tetrahedron | 4 | Three equilateral triangles | π ( 180 ∘ ) {\\displaystyle \\pi \\ \\ (180^{\\circ })} | 4 π ( 720 ∘ ) {\\displaystyle 4\\pi \\ \\ (720^{\\circ })} |

| octahedron | 6 | Four equilateral triangles | 2 π 3 ( 120 ∘ ) {\\displaystyle {2\\pi \\over 3}\\ (120^{\\circ })} | 4 π ( 720 ∘ ) {\\displaystyle 4\\pi \\ \\ (720^{\\circ })} |

| cube | 8 | Three squares | π 2 ( 90 ∘ ) {\\displaystyle {\\pi \\over 2}\\ \\ (90^{\\circ })} | 4 π ( 720 ∘ ) {\\displaystyle 4\\pi \\ \\ (720^{\\circ })} |

| icosahedron | 12 | Five equilateral triangles | π 3 ( 60 ∘ ) {\\displaystyle {\\pi \\over 3}\\ \\ (60^{\\circ })} | 4 π ( 720 ∘ ) {\\displaystyle 4\\pi \\ \\ (720^{\\circ })} |

| dodecahedron | 20 | Three regular pentagons | π 5 ( 36 ∘ ) {\\displaystyle {\\pi \\over 5}\\ \\ (36^{\\circ })} | 4 π ( 720 ∘ ) {\\displaystyle 4\\pi \\ \\ (720^{\\circ })} |

`t

>>Descartes's theorem

Descartes's theorem on the "total defect" of a polyhedron states that if the polyhedron is `F33f`_`[homeomorphic`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Homeomorphism]`_`f 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 4π radians). The polyhedron need not be convex.`:cite-ref-1[`F5bf`_`[1`#cite-note-1]`_`f]

A generalization says the number of circles in the total defect equals the `F33f`_`[Euler characteristic`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Euler_characteristic]`_`f of the polyhedron. This is a special case of the `F33f`_`[Gauss–Bonnet theorem`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Gauss–Bonnet_theorem]`_`f which relates the integral of the `F33f`_`[Gaussian curvature`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Gaussian_curvature]`_`f to the Euler characteristic. Here the Gaussian curvature is concentrated at the vertices: on the faces and edges the curvature is zero (the surface is locally `F33f`_`[isometric`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Isometry]`_`f to a Euclidean plane) and the integral of curvature at a vertex is equal to the defect there (by definition).

This can be used to calculate the number `*V`* of vertices of a polyhedron by totaling the angles of all the faces, and adding the total defect (which is 2 π π {\\displaystyle 2\\pi } times the Euler characteristic). This total will have one complete circle for every vertex in the polyhedron.

A converse to Descartes' theorem is given by `F33f`_`[Alexandrov's uniqueness theorem`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Alexandrov's_uniqueness_theorem]`_`f, according to which a metric space that is locally Euclidean (hence zero curvature) except for a finite number of points of positive angular defect, adding to 4 π π {\\displaystyle 4\\pi } , can be realized in a unique way as the surface of a convex polyhedron.

>>Positive defects on non-convex figures

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 if the Euler characteristic is positive (a topological sphere).

A counterexample is provided by a `F33f`_`[cube`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Cube]`_`f where one face is replaced by a `F33f`_`[square pyramid`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Square_pyramid]`_`f: this `F33f`_`[elongated square pyramid`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Elongated_square_pyramid]`_`f 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.

Two counterexamples which are self-intersecting polyhedra are the `F33f`_`[small stellated dodecahedron`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Small_stellated_dodecahedron]`_`f and the `F33f`_`[great stellated dodecahedron`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Great_stellated_dodecahedron]`_`f, with twelve and twenty convex points respectively, all with positive defects.

>>References

>>>Notes

`:cite-note-1`!1.`! `F0af`_`[↑`#cite-ref-1]`_`f `F33f`_`[Descartes, René`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=René_Descartes]`_`f, `*Progymnasmata de solidorum elementis`*, in `*Oeuvres de Descartes`*, vol. X, pp. 265–276

>>>Bibliography

• `F33f`_`[Richeson, D.`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=David_Richeson]`_`f; `*`F33f`_`[Euler's Gem: The Polyhedron Formula and the Birth of Topology`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Euler's_Gem]`_`f`*, Princeton (2008), Pages 220–225.

>>External links

Look up

defect

in Wiktionary, the free dictionary.

• `:reference-mathworld-angular-defect`a`:citerefweisstein`a`F33f`_`[Weisstein, Eric W.`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Eric_W._Weisstein]`_`f "Angular defect". `*`F33f`_`[MathWorld`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=MathWorld]`_`f`*.

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