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# Pentagramma mirificum

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Pentagramma mirificum (Latin for miraculous pentagram) is a star polygon on a sphere, composed of five great circle arcs, whose all internal angles are right angles. This shape was described by John Napier in his 1614 book Mirifici logarithmorum canonis descriptio (Description of the wonderful rule of logarithms) along with rules that link the values of trigonometric functions of five parts of a right spherical triangle (two angles and three sides). 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 (arcs of great circles) are measured as angles. Angles $A$ , $B$ , $C$ , $D$ , and $E$ are right angles. Arcs $PC$ , $PE$ , $QD$ , $QA$ , $RE$ , $RB$ , $SA$ , $SC$ , $TB$ , and $TD$ are equal to $\pi /2$ . In spherical pentagon $PQRST$ , every vertex is the pole of the opposite side. For instance, point $P$ is the pole of equator $RS$ , point $Q$ — the pole of equator $ST$ , etc.

## Gauss’s formulas

Gauss introduced the notation

$\left(\alpha ,\beta ,\gamma ,\delta ,\epsilon \right)=\left({\mathrm{tan}}^{2}TP,{\mathrm{tan}}^{2}PQ,{\mathrm{tan}}^{2}QR,{\mathrm{tan}}^{2}RS,{\mathrm{tan}}^{2}ST\right).$ The following identities hold, allowing to determine any three of the above quantities from the two remaining ones:

$\begin{array}{cccccc}1+\alpha & =\gamma \delta & 1+\beta & =\delta \epsilon & 1+\gamma & =\alpha \epsilon \\ 1+\delta & =\alpha \beta & 1+\epsilon & =\beta \gamma .\end{array}$ Gauss proved the following “beautiful equality” (schöne Gleichung):

$\begin{array}{cc}\alpha \beta \gamma \delta \epsilon & =\phantom{\rule{thickmathspace}{0ex}}\\ =\phantom{\rule{thickmathspace}{0ex}}\end{array}$ It is satisfied, for instance, by numbers $\left(\alpha ,\beta ,\gamma ,\delta ,\epsilon \right)=\left(9,2/3,2,5,1/3\right)$ , whose product $\alpha \beta \gamma \delta \epsilon$ is equal to $20$ .

Proof of the first part of the equality:

$\begin{array}{cc}\alpha \beta \gamma \delta \epsilon & =\alpha \beta \gamma \left(\frac{1+\alpha }{\gamma }\right)\left(\frac{1+\gamma }{\alpha }\right)=\beta \left(1+\alpha \right)\left(1+\gamma \right)\\ =\beta +\alpha \beta +\beta \gamma +\alpha \beta \gamma =\beta +\left(1+\delta \right)+\left(1+\epsilon \right)+\alpha \left(1+\epsilon \right)\\ =2+\alpha +\beta +\delta +\epsilon +1+\gamma \\ =3+\alpha +\beta +\gamma +\delta +\epsilon \end{array}$ Proof of the second part of the equality:

$\begin{array}{cc}\alpha \beta \gamma \delta \epsilon & =\sqrt{{\alpha }^{2}{\beta }^{2}{\gamma }^{2}{\delta }^{2}{\epsilon }^{2}}\\ =\sqrt{\gamma \delta \cdot \delta \epsilon \cdot \epsilon \alpha \cdot \alpha \beta \cdot \beta \gamma }\\ =\sqrt{\left(1+\alpha \right)\left(1+\beta \right)\left(1+\gamma \right)\left(1+\delta \right)\left(1+\epsilon \right)}\end{array}$ From Gauss comes also the formula

$\left(1+i\sqrt{{}^{{}^{\phantom{\rule{negativethinmathspace}{0ex}}}}\alpha }\right)\left(1+i\sqrt{\beta }\right)\left(1+i\sqrt{{}^{{}^{\phantom{\rule{negativethinmathspace}{0ex}}}}\gamma }\right)\left(1+i\sqrt{\delta }\right)\left(1+i\sqrt{{}^{{}^{\phantom{\rule{negativethinmathspace}{0ex}}}}\epsilon }\right)=\alpha \beta \gamma \delta \epsilon {e}^{iS},$ where $S=2\pi -\left(|\stackrel{⌢}{PQ}|+|\stackrel{⌢}{QR}|+|\stackrel{⌢}{RS}|+|\stackrel{⌢}{ST}|+|\stackrel{⌢}{TP}|\right)$ is the area of pentagon $PQRST$ .

## Gnomonic projection

The image of spherical pentagon $PQRST$ 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 ${P}^{\prime }{Q}^{\prime }{R}^{\prime }{S}^{\prime }{T}^{\prime }$ unambiguously determine a conic section; in this case — an ellipse. Gauss showed that the altitudes of pentagram ${P}^{\prime }{Q}^{\prime }{R}^{\prime }{S}^{\prime }{T}^{\prime }$ (lines passing through vertices and perpendicular to opposite sides) cross in one point ${O}^{\prime }$ , 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 ${O}^{\prime }$ , then the coordinates of vertices ${P}^{\prime }{Q}^{\prime }{R}^{\prime }{S}^{\prime }{T}^{\prime }$ : $\left({x}_{1},{y}_{1}\right),\dots$ $\left({x}_{5},{y}_{5}\right)$ satisfy the equalities ${x}_{1}{x}_{4}+{y}_{1}{y}_{4}=$ ${x}_{2}{x}_{5}+{y}_{2}{y}_{5}=$ ${x}_{3}{x}_{1}+{y}_{3}{y}_{1}=$ ${x}_{4}{x}_{2}+{y}_{4}{y}_{2}=$ ${x}_{5}{x}_{3}+{y}_{5}{y}_{3}=-{\rho }^{2}$ , where $\rho$ is the length of the radius of the sphere.