On a sphere, both the angles and the sides of a triangle (arcs of great circles) 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.
The following identities hold, allowing the determination of any three of the above quantities from the two remaining ones:[2]
Gauss proved the following "beautiful equality" (schöne Gleichung):[2]
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[2]
where is the area of pentagon .
Gnomonic projection
The image of spherical pentagon in the gnomonic projection (a projection from the centre of the sphere) 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 (lines passing through vertices and perpendicular to opposite sides) 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.[3]
References
^Gauss, Carl Friedrich (1866). "Pentagramma mirificum". Werke, Band III: Analysis. Göttingen: Königliche Gesellschaft der Wissenschaften. pp. 481–490.
^ a b cCoxeter, H. S. M. (1971). "Frieze patterns" (PDF). Acta Arithmetica. 18: 297–310. doi:10.4064/aa-18-1-297-310.
^Cayley, Arthur (1871). "On Gauss's pentagramma mirificum". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 42 (280): 311–312. doi:10.1080/14786447108640572.
External links
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