Pentagramma mirificum (Latin for "miraculous pentagram") 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 (Description of the Admirable Table 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.[1]
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
\pi/2,
A
B
C
D
E.
There are ten arcs, each measuring
\pi/2:
PC
PE
QD
QA
RE
RB
SA
SC
TB
TD.
In the spherical pentagon
PQRST
P
RS
Q
ST
At each vertex of pentagon
PQRST
\angleAPT=\angleBPQ=RS, \angleBQP=\angleCQR=ST,
Napier's circles of spherical triangles
APT
BQP
CRQ
DSR
ETS
Gauss introduced the notation
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
(\alpha,\beta,\gamma,\delta,\varepsilon)=(9,2/3,2,5,1/3)
\alpha\beta\gamma\delta\varepsilon
20
Proof of the first part of the equality:
\begin{align} \alpha\beta\gamma\delta\varepsilon&=\alpha\beta\gamma\left(
1+\alpha | \right)\left( | |
\gamma |
1+\gamma | |
\alpha |
\right)=\beta(1+\alpha)(1+\gamma)\\ &=\beta+\alpha\beta+\beta\gamma+\alpha\beta\gamma=\beta+(1+\delta)+(1+\varepsilon)+\alpha(1+\varepsilon)\\ &=2+\alpha+\beta+\delta+\varepsilon+1+\gamma\\ &=3+\alpha+\beta+\gamma+\delta+\varepsilon \end{align}
Proof of the second part of the equality:
\begin{align} \alpha\beta\gamma\delta\varepsilon&=\sqrt{\alpha2\beta2\gamma2\delta2\varepsilon2}\\ &=\sqrt{\gamma\delta ⋅ \delta\varepsilon ⋅ \varepsilon\alpha ⋅ \alpha\beta ⋅ \beta\gamma}\\ &=\sqrt{(1+\alpha)(1+\beta)(1+\gamma)(1+\delta)(1+\varepsilon)} \end{align}
From Gauss comes also the formula[2]
where
APQRST=2\pi-(|\overset{\frown}{PQ}|+|\overset{\frown}{QR}|+|\overset{\frown}{RS}|+|\overset{\frown}{ST}|+|\overset{\frown}{TP}|)
PQRST
The image of spherical pentagon
PQRST
P'Q'R'S'T'
P'Q'R'S'T'
O'
Arthur Cayley observed that, if we set the origin of a Cartesian coordinate system in point
O'
P'Q'R'S'T'
(x1,y1),\ldots,
(x5,y5)
x1x4+y1y4=
x2x5+y2y5=
x3x1+y3y1=
x4x2+y4y2=
x5x3+y5y3=-\rho2
\rho