MCiura: minor
'''Pentagramma mirificum''' (Latin for ''miraculous pentagram'') is a [[star polygon]] on a [[sphere]], composed of five [[great circle]] [[arc (geometry)|arcs]], whose all [[internal and external angles|internal angles]] are [[right angle]]s. This shape was described by [[John Napier]] in his 1614 book ''Mirifici logarithmorum canonis descriptio'' (''Description of the wonderful rule of logarithms'') along with [[spherical trigonometry#Napier's rules for right spherical triangles|rules]] that link the values of [[trigonometric functions]] of five parts of a [[right triangle|right]] [[spherical trigonometry|spherical triangle]] (two angles and three sides). The properties of ''pentagramma mirificum'' were studied, among others, by [[Carl Friedrich Gauss]].<ref></ref>
[[File:Pentagramma-mirificum.gif|frame|right|Sample configurations of ''pentagramma mirificum'']]
[[File:Pentagramma-mirificum.png|thumb|right|348px|Relations between angles and sides of ''pentagramma mirificum'']]
== Geometric properties ==
On a sphere, both the angles and the sides of a triangle (arcs of great circles) are measured as angles. Angles <math>A</math>, <math>B</math>, <math>C</math>, <math>D</math>, and <math>E</math> are right angles. Arcs <math>PC</math>, <math>PE</math>, <math>QD</math>, <math>QA</math>, <math>RE</math>, <math>RB</math>, <math>SA</math>, <math>SC</math>, <math>TB</math>, and <math>TD</math> are equal to <math>\pi/2</math>. In spherical pentagon <math>PQRST</math>, every vertex is the pole of the opposite side. For instance, point <math>P</math> is the pole of equator <math>RS</math>, point <math>Q</math> — the pole of equator <math>ST</math>, etc.<ref>Liquid error: wrong number of arguments (1 for 2)</ref>
== Gauss’s formulas ==
Gauss introduced the notation
<math display="block">(\alpha, \beta, \gamma, \delta, \epsilon) = (\tan^2 TP, \tan^2 PQ, \tan^2 QR, \tan^2 RS, \tan^2 ST).</math>
The following identities hold, allowing us to determine any three of the above quantities from the two remaining ones:<ref name="Coxeter"></ref>
<math display="block">\begin{alignat}{2}
1 &+ \alpha &=& \gamma\delta \\
1 &+ \beta &=& \delta\epsilon \\
1 &+ \gamma &=& \epsilon\alpha \\
1 &+ \delta &=& \alpha\beta \\
1 &+ \epsilon &=& \beta\gamma.
\end{alignat}</math>
Gauss proved the following “beautiful equality” (''schöne Gleichung''):<ref name="Coxeter"/>
<math display="block">3 + \alpha + \beta + \gamma + \delta + \epsilon = \alpha\beta\gamma\delta\epsilon = \sqrt{(1+\alpha)(1+\beta)(1+\gamma)(1+\delta)(1+\epsilon)}.</math>
It is satisfied, for instance, by numbers <math>(\alpha, \beta, \gamma, \delta, \epsilon) = (9, 2/3, 2, 5, 1/3)</math>, whose product <math>\alpha\beta\gamma\delta\epsilon</math> is equal to <math>20</math>.
'''Proof''' of the first part of the equality:
<math>\alpha\beta\gamma\delta\epsilon =</math>
<math>\alpha\beta\gamma\left(\frac{1+\alpha}{\gamma}\right)\left(\frac{1+\gamma}{\alpha}\right) =</math>
<math>\beta + \alpha\beta + \beta\gamma + \alpha\beta\gamma =</math>
<math>\beta + 1 + \delta + 1 + \epsilon + \alpha(1 + \epsilon)=</math>
<math>2 + \alpha + \beta + \delta + \epsilon + 1 + \gamma=</math>
<math>3 + \alpha + \beta + \gamma + \delta + \epsilon\quad</math> Q.E.D.
'''Proof''' of the second part of the equality:
<math>\sqrt{(1+\alpha)(1+\beta)(1+\gamma)(1+\delta)(1+\epsilon)}=</math>
<math>\sqrt{\gamma\delta \cdot \delta\epsilon \cdot \epsilon\alpha \cdot \alpha\beta \cdot \beta\gamma}=</math>
<math>\sqrt{\alpha^2\beta^2\gamma^2\delta^2\epsilon^2}=</math>
<math>\alpha\beta\gamma\delta\epsilon\quad</math>Q.E.D.
From Gauss comes also the formula<ref name="Coxeter"/>
<math display="block">(1+i\sqrt{^{^{\!}}\alpha})(1+i\sqrt{\beta})(1+i\sqrt{^{^{\!}}\gamma})(1+i\sqrt{\delta})(1+i\sqrt{^{^{\!}}\epsilon}) = \alpha\beta\gamma\delta\epsilon e^{iS},</math>
where <math>S = 2\pi - (\alpha + \beta + \gamma + \delta + \epsilon)</math>.
== Gnomonic projection ==
The image of spherical pentagon <math>PQRST</math> 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 <math>P'Q'R'S'T'</math> [[five points determine a conic|unambiguously determine]] a [[conic section]]; in this case — an [[ellipse]]. Gauss showed that the altitudes of pentagram <math>P'Q'R'S'T'</math> (lines passing through vertices and perpendicular to opposite sides) cross in one point <math>O'</math>, which is the image of the point of tangency of the plane to sphere.<ref></ref>
[[Arthur Cayley]] observed that, if we set the origin of a [[Cartesian coordinate system]] in point <math>O'</math>, then the coordinates of vertices <math>P'Q'R'S'T'</math>: <math>(x_1, y_1),\ldots</math> <math>(x_5, y_5)</math> satisfy the equalities <math>x_1 x_4 + y_1 y_4 =</math> <math>x_2 x_5 + y_2 y_5 =</math> <math>x_3 x_1 + y_3 y_1 =</math> <math>x_4 x_2 + y_4 y_2 =</math> <math>x_5 x_3 + y_5 y_3 = -\rho^2</math>, where <math>\rho</math> is the length of the diameter of the sphere.<ref></ref>
== References ==
[[Category:Spherical trigonometry]]
[[File:Pentagramma-mirificum.gif|frame|right|Sample configurations of ''pentagramma mirificum'']]
[[File:Pentagramma-mirificum.png|thumb|right|348px|Relations between angles and sides of ''pentagramma mirificum'']]
== Geometric properties ==
On a sphere, both the angles and the sides of a triangle (arcs of great circles) are measured as angles. Angles <math>A</math>, <math>B</math>, <math>C</math>, <math>D</math>, and <math>E</math> are right angles. Arcs <math>PC</math>, <math>PE</math>, <math>QD</math>, <math>QA</math>, <math>RE</math>, <math>RB</math>, <math>SA</math>, <math>SC</math>, <math>TB</math>, and <math>TD</math> are equal to <math>\pi/2</math>. In spherical pentagon <math>PQRST</math>, every vertex is the pole of the opposite side. For instance, point <math>P</math> is the pole of equator <math>RS</math>, point <math>Q</math> — the pole of equator <math>ST</math>, etc.<ref>Liquid error: wrong number of arguments (1 for 2)</ref>
== Gauss’s formulas ==
Gauss introduced the notation
<math display="block">(\alpha, \beta, \gamma, \delta, \epsilon) = (\tan^2 TP, \tan^2 PQ, \tan^2 QR, \tan^2 RS, \tan^2 ST).</math>
The following identities hold, allowing us to determine any three of the above quantities from the two remaining ones:<ref name="Coxeter"></ref>
<math display="block">\begin{alignat}{2}
1 &+ \alpha &=& \gamma\delta \\
1 &+ \beta &=& \delta\epsilon \\
1 &+ \gamma &=& \epsilon\alpha \\
1 &+ \delta &=& \alpha\beta \\
1 &+ \epsilon &=& \beta\gamma.
\end{alignat}</math>
Gauss proved the following “beautiful equality” (''schöne Gleichung''):<ref name="Coxeter"/>
<math display="block">3 + \alpha + \beta + \gamma + \delta + \epsilon = \alpha\beta\gamma\delta\epsilon = \sqrt{(1+\alpha)(1+\beta)(1+\gamma)(1+\delta)(1+\epsilon)}.</math>
It is satisfied, for instance, by numbers <math>(\alpha, \beta, \gamma, \delta, \epsilon) = (9, 2/3, 2, 5, 1/3)</math>, whose product <math>\alpha\beta\gamma\delta\epsilon</math> is equal to <math>20</math>.
'''Proof''' of the first part of the equality:
<math>\alpha\beta\gamma\delta\epsilon =</math>
<math>\alpha\beta\gamma\left(\frac{1+\alpha}{\gamma}\right)\left(\frac{1+\gamma}{\alpha}\right) =</math>
<math>\beta + \alpha\beta + \beta\gamma + \alpha\beta\gamma =</math>
<math>\beta + 1 + \delta + 1 + \epsilon + \alpha(1 + \epsilon)=</math>
<math>2 + \alpha + \beta + \delta + \epsilon + 1 + \gamma=</math>
<math>3 + \alpha + \beta + \gamma + \delta + \epsilon\quad</math> Q.E.D.
'''Proof''' of the second part of the equality:
<math>\sqrt{(1+\alpha)(1+\beta)(1+\gamma)(1+\delta)(1+\epsilon)}=</math>
<math>\sqrt{\gamma\delta \cdot \delta\epsilon \cdot \epsilon\alpha \cdot \alpha\beta \cdot \beta\gamma}=</math>
<math>\sqrt{\alpha^2\beta^2\gamma^2\delta^2\epsilon^2}=</math>
<math>\alpha\beta\gamma\delta\epsilon\quad</math>Q.E.D.
From Gauss comes also the formula<ref name="Coxeter"/>
<math display="block">(1+i\sqrt{^{^{\!}}\alpha})(1+i\sqrt{\beta})(1+i\sqrt{^{^{\!}}\gamma})(1+i\sqrt{\delta})(1+i\sqrt{^{^{\!}}\epsilon}) = \alpha\beta\gamma\delta\epsilon e^{iS},</math>
where <math>S = 2\pi - (\alpha + \beta + \gamma + \delta + \epsilon)</math>.
== Gnomonic projection ==
The image of spherical pentagon <math>PQRST</math> 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 <math>P'Q'R'S'T'</math> [[five points determine a conic|unambiguously determine]] a [[conic section]]; in this case — an [[ellipse]]. Gauss showed that the altitudes of pentagram <math>P'Q'R'S'T'</math> (lines passing through vertices and perpendicular to opposite sides) cross in one point <math>O'</math>, which is the image of the point of tangency of the plane to sphere.<ref></ref>
[[Arthur Cayley]] observed that, if we set the origin of a [[Cartesian coordinate system]] in point <math>O'</math>, then the coordinates of vertices <math>P'Q'R'S'T'</math>: <math>(x_1, y_1),\ldots</math> <math>(x_5, y_5)</math> satisfy the equalities <math>x_1 x_4 + y_1 y_4 =</math> <math>x_2 x_5 + y_2 y_5 =</math> <math>x_3 x_1 + y_3 y_1 =</math> <math>x_4 x_2 + y_4 y_2 =</math> <math>x_5 x_3 + y_5 y_3 = -\rho^2</math>, where <math>\rho</math> is the length of the diameter of the sphere.<ref></ref>
== References ==
[[Category:Spherical trigonometry]]
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