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In [[geometry]], the '''exsphere''' of a face of a regular polyhedron is the sphere outside the polyhedron which touches the face and the planes defined by extending the adjacent faces outwards. It is tangent to the face externally and tangent to the adjacent faces internally.
 
It is the 3-dimensional equivalent of the [[excircle]].
 
The sphere is more generally well-defined for any face which is a regular
polygon and delimited by faces with the same dihedral angles
at the shared edges. Faces of semi-regular polyhedra often
have different types of faces, which define exspheres of different size with each type of face.
 
 
== Parameters ==
The exsphere touches the face of the regular polyedron at the center
of the incircle of that face. If the exsphere radius is denoted {{math|r<sub>ex</sub>}}, the radius of this incircle {{math|r<sub>in</sub>}}
and the dihedral angle between the face and the extension of the  
adjacent face {{math|&delta;}}, the center of the exsphere
is located from the viewpoint at the middle of one edge of the
face by bisecting the dihedral angle. Therefore
 
:<math>\tan\frac{\delta}{2} = \frac{r_{ex}}{r_{in}}.</math>
 
{{math|&delta;}} is the 180-degree complement of the
internal face-to-face angle.
 
=== Tetrahedron ===
Applied to the geometry of the [[Tetrahedron]] of edge length {{math|a}},
we have an [[Incircle|incircle radius]] {{math|r<sub>in</sub>{{=}}a/(2&radic;3)}} (derived by
dividing twice the face area {{math|(a<sup>2</sup>&radic; 3)/4}} through the
perimeter {{math|3a}}), a dihedral angle {{math|&delta;{{=}}&pi;-arccos(1/3)}}, and in consequence {{math|r<sub>ex</sub>{{=}}a/&radic; 6}}.
 
=== Cube ===
The radius of the exspheres of the 6 faces of the [[Cube]]
is the same as the radius of the inscribed
sphere, since {{math|&delta;}} and its complement are the same, 90 degrees.
 
=== Icosahedron ===
The dihedral angle applicable to the [[Icosahedron]] is derived by
considering the coordinates of two triangles with a common edge,
for [[Icosahedron#Cartesian coordinates|example]] one face with vertices
at
 
:<math>(0,-1,g), (g,0,1), (0,1,g),</math>
 
the other at
 
:<math>(1,-g,0), (g,0,1), (0,-1,g),</math>
 
where {{math|g}} is the [[golden ratio]]. Subtracting vertex coordinates
defines edge vectors,
 
:<math>(g,1,1-g), (-g,1,g-1)</math>
 
of the first face and
 
:<math>(g-1,g,1), (-g,-1,g-1)</math>
 
of the other. [[Cross product]]s of the edges of the first face and second
face yield (not normalized) face [[Normal (geometry)|normal vectors]]
 
:<math>(2g-2,0,2g) \sim (g-1,0,g)</math>
of the first and
:<math>(g^2-g+1,-g-(g-1)^2,1-g+g^2) = (2,-2,2)\sim (1,-1,1)</math>
of the second face, using {{math|g<sup>2</sub>{{=}}1+g}}.
The [[dot product]] between these two face normals yields the cosine
of the dihedral angle,
 
:<math>\cos\delta = \frac{(g-1)\cdot 1+g\cdot 1}{\sqrt{(g-1)^2+g^2} \sqrt{3}} =\frac{2g-1}{3} =\frac{\surd 5}{3}\approx 0.74535599.</math> {{OEIS2C|A208899}}
:<math>\therefore \delta \approx 0.72973 \,\mathrm{rad} \approx 41.81^\circ</math>
:<math>\therefore \tan\frac{\delta}{2} = \frac{\sin\delta}{1+\cos\delta}
=\frac{2}{3+\surd 5} \approx 0.3819660</math> {{OEIS2C|A132338}}
 
For an icosahedron of edge length {{math|a}}, the incircle radius
of the triangular faces is {{math|r<sub>in</sub>{{=}}a/(2&radic; 3)}}, and finally the
radius of the 20 exspheres
:<math>r_{ex} = \frac{a}{(3+\sqrt{5})\sqrt 3} \approx 0.1102641 a.</math>
 
== See also ==
[[Inscribed sphere|Insphere]]
 
== External links ==
*{{cite journal
|first1=Leon
|last1=Gerber
|title=Associated and skew-orthologic simplexes
|year=1977
|journal=Trans. Am. Math. Soc.
|volume=231
|number=1
|pages=47–63
|jstor=1997867
|mr=0445393
|doi=10.1090/S0002-9947-1977-0445393-6
}}
*{{cite journal
|first1=Mowaffaq
|last1=Hajja
|title=The Gergonne and Nagel centers of an n-dimensional simplex
|journal=J. Geom.
|doi=10.1007/s00022-005-0011-3
|volume=83
|number=1-2
|pages=46–56
|year=2005
}}
 
[[Category:Geometry]]

Revision as of 02:42, 24 February 2014

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