Pentagramma mirificum (Latin for ''miraculous pentagram'') is a
star polygon
In geometry, a star polygon is a type of non- convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, certain notable ones can arise through truncation operatio ...
on a
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
, composed of five
great circle
In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point.
Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geome ...
arcs, all of whose
internal angles are
right angle
In geometry and trigonometry, a right angle is an angle of exactly 90 degrees or radians corresponding to a quarter turn. If a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles. Th ...
s. This shape was described by
John Napier
John Napier of Merchiston (; 1 February 1550 – 4 April 1617), nicknamed Marvellous Merchiston, was a Scottish landowner known as a mathematician, physicist, and astronomer. He was the 8th Laird of Merchiston. His Latinized name was Ioan ...
in his 1614 book ''
Mirifici Logarithmorum Canonis Descriptio'' (''Description of the Admirable Table of Logarithms'') along with
rules
Rule or ruling may refer to:
Education
* Royal University of Law and Economics (RULE), a university in Cambodia
Human activity
* The exercise of political or personal control by someone with authority or power
* Business rule, a rule pert ...
that link the values of
trigonometric functions
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
of five parts of a
right
Rights are legal, social, or ethical principles of freedom or entitlement; that is, rights are the fundamental normative rules about what is allowed of people or owed to people according to some legal system, social convention, or ethical ...
spherical triangle
Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are gre ...
(two angles and three sides). The properties of ''pentagramma mirificum'' were studied, among others, by
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
.
Geometric properties
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
In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an interior angle (or ) if ...
is equal in measure to the opposite side. For instance,
etc.
Napier's circles of spherical triangles
,
,
,
, and
are
rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
s 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" (''schöne Gleichung''):
:
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
A gnomonic map projection is a map projection which displays all great circles as straight lines, resulting in any straight line segment on a gnomonic map showing a geodesic, the shortest route between the segment's two endpoints. This is achi ...
(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 mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a ...
; in this case — an
ellipse
In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...
. 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
Arthur Cayley (; 16 August 1821 – 26 January 1895) was a prolific British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics.
As a child, Cayley enjoyed solving complex maths problem ...
observed that, if we set the origin of a
Cartesian coordinate system
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
in point
, then the coordinates of vertices
:
satisfy the equalities
, where
is the length of the radius of the sphere.
References
External links
*
* {{YouTube, id=wQKMAx6jxGw, title=Pentagramma mirificum
Spherical trigonometry
Types of polygons