There are a number of algebraic equations known as the icosahedral equation, all of which derive from the projective geometry of the icosahedron.
Consider an icosahedron centered 
, oriented with 
-axis along a fivefold (
) rotational symmetry axis, and with one of the top five
edges lying in the 
-plane
(left figure). In this figure, vertices are shown in black, face centers in red,
and edge midpoints in blue.
The simplest icosahedral equation is defined by projecting the vertices of the icosahedron with unit circumradius using a stereographic
projection from the south pole of its circumsphere
onto the plane 
,
and expressing these vertex locations (interpreted as complex quantities in the complex
-plane)
as roots of an algebraic equation. The resulting projection is shown as the left
figure above, with black dots being the vertex positions. The resulting equation
is
 |
(1)
|
where 
here refers to the coordinate in the complex plane (not the height above the
projection plane), and the equation is of order 11 instead of 12 since the vertex
at 
is transformed to infinity and has been omitted. Writing the above equation in symmetric
form gives
 |
(2)
|
If the icosahedron with unit inradius is instead projected (second figure above), the equation expressing the positions of the face centers (red dots) is given by
 |
(3)
|
or in symmetric form,
 |
(4)
|
Finally, if the icosahedron with unit midradius is projected (right figure above), the equation expressing the positions of the edge midpoints (blue dots) is given by
 |
(5)
|
or in symmetric form,
 |
(6)
|
Note that because these equations involve variables to multiples of the power 5, rotating the solid by 
radians changes transforms the quantities from 
to 
, producing the same equations modulo minus
signs in odd powers of 
, corresponding to flipping the positions of the roots about
the imaginary axis.
Combining 
and 
gives a general equation commonly known as "the" icosahedral equation,
 |
(7)
|
Hunt (1996) considers a "dehomogenized" icosahedral equation given by
 |
(8)
|
If the icosahedron is instead oriented so that the top and bottom faces are parallel to the 
-plane,
the corresponding equation giving its projected vertices is
 |
(9)
|
