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 regularpolygon and delimited by faces with the same dihedral anglesat 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.
The exsphere touches the face of the regular polyedron at the centerof the incircle of that face. If the exsphere radius is denoted, the radius of this incircle and the dihedral angle between the face and the extension of the adjacent face, the center of the exsphereis located from the viewpoint at the middle of one edge of theface by bisecting the dihedral angle. Therefore
| \tan | \delta |
| 2 |
=
| rex | |
| rin |
.
is the 180-degree complement of the internal face-to-face angle.
Applied to the geometry of the Tetrahedron of edge length,we have an incircle radius (derived by dividing twice the face area through theperimeter), a dihedral angle, and in consequence .
The radius of the exspheres of the 6 faces of the Cubeis the same as the radius of the inscribedsphere, since and its complement are the same, 90 degrees.
The dihedral angle applicable to the Icosahedron is derived byconsidering the coordinates of two triangles with a common edge,for example one face with verticesat
(0,-1,g),(g,0,1),(0,1,g),
the other at
(1,-g,0),(g,0,1),(0,-1,g),
where is the golden ratio. Subtracting vertex coordinatesdefines edge vectors,
(g,1,1-g),(-g,1,g-1)
of the first face and
(g-1,g,1),(-g,-1,g-1)
of the other. Cross products of the edges of the first face and secondface yield (not normalized) face normal vectors
(2g-2,0,2g)\sim(g-1,0,g)
(g2-g+1,-g-(g-1)2,1-g+g2)=(2,-2,2)\sim(1,-1,1)
\cos\delta=
| (g-1) ⋅ 1+g ⋅ 1 | |
| \sqrt{(g-1)2+g2 |
\sqrt{3}}=
| 2g-1 | = | |
| 3 |
| \surd5 | |
| 3 |
≈ 0.74535599.
\therefore\delta ≈ 0.72973rad ≈ 41.81\circ
\therefore\tan
| \delta | |
| 2 |
=
| \sin\delta | = | |
| 1+\cos\delta |
| 2 | |
| 3+\surd5 |
≈ 0.3819660
For an icosahedron of edge length, the incircle radius of the triangular faces is, and finally the radius of the 20 exspheres
rex=
| a | |
| (3+\sqrt{5 |
)\sqrt3} ≈ 0.1102641a.