Pentagramma mirificum


Pentagramma mirificum is a star polygon on a sphere, composed of five great circle arcs, all of whose internal angles are right angles. This shape was described by John Napier in his 1614 book Mirifici logarithmorum canonis descriptio along with rules that link the values of trigonometric functions of five parts of a right spherical triangle. The properties of pentagramma mirificum were studied, among others, by Carl Friedrich Gauss.

Geometric properties

On a sphere, both the angles and the sides of a triangle are measured as angles.
There are five right angles, each measuring at,,,, and
There are ten arcs, each measuring ,,,,,,,,, and
In the spherical pentagon, every vertex is the pole of the opposite side. For instance, point is the pole of equator, point — the pole of equator, etc.
At each vertex of pentagon, the external angle is equal in measure to the opposite side. For instance, etc.
Napier's circles of spherical triangles,,,, and are rotations of one another.

Gauss's formulas

Gauss introduced the notation
The following identities hold, allowing the determination of any three of the above quantities from the two remaining ones:
Gauss proved the following "beautiful equality" :
It is satisfied, for instance, by numbers, whose product is equal to.
Proof of the first part of the equality:
Proof of the second part of the equality:
From Gauss comes also the formula
where is the area of pentagon.

Gnomonic projection

The image of spherical pentagon in the gnomonic projection onto any plane tangent to the sphere is a rectilinear pentagon. Its five vertices unambiguously determine a conic section; in this case — an ellipse. Gauss showed that the altitudes of pentagram cross in one point, which is the image of the point of tangency of the plane to sphere.
Arthur Cayley observed that, if we set the origin of a Cartesian coordinate system in point, then the coordinates of vertices : satisfy the equalities , where is the length of the radius of the sphere.