About the Geometry of a Five-Point Star | Sciencing
On this page we will derive some relations between a fundamental number and From the relation between the sides and the diagonals of a regular pentagon. In the above figure, let the point-to-point edge length be unity, the point edges be a, the edge lengths of the inner pentagon be b, the inradius of the inner. find the area of a 5 pointed star in the circle with a radius of the star regular, we DEFINE its sides to be extensions of the sides of the starting pentagon.
A 5 pointed star in a circle - Math Central
A special isosceles triangle or "golden triangle" We consider an isosceles triangle with a top angle measuring 36 degrees. Both base angles then measure 72 degrees. BD is the bisector of the angle in B. If we consider the circle with centre A and radius AB, then BC is a side of the regular decagon that is inscribed in the given circle.
A property of the regular pentagon Considering a regular pentagon with two non-intersecting diagonals and taking into account what we already derived above, it follows that the length of the side equals the greatest part of the diagonal divided by the golden section. The pentagram The five diagonals of a regular pentagon define a star-shaped figure that is called a pentagram. It has five sides and five vertices. The interior intersections of the sides aren't vertices of the pentagram.
A pentagram is an example of a non-convex polygon. A polygon is convex if and only if any line containing a side of the polygon doesn't contain a point in the interior of the polygon. As a triangle always is convex and there don't exist non-convex regular polygons with four sides, the pentagram, in this sense is the simplest of the non-convex regular polygons.
From the relation between the sides and the diagonals of a regular pentagon and considering the the second figure above, one easily states that two non-consecutive sides of a pentagram divide each other in mean and extreme ratio! Construction of a regular dodecahedron Starting with a cube we can construct a regular dodecahedron by putting appropriate "roofs" on each face of the cube.
The bases of the roofs are the faces of the cube and all edges of the roofs that aren't edges of the cube must have the same length. Moreover the adjacent faces of two meeting roofs must form a regular planar pentagon.
The length of the diagonals of the pentagons equals the length of the sides of the cube.
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The pentagram plays an important symbolic role in the 14th-century English poem Sir Gawain and the Green Knightin which the symbol decorates the shield of the hero, Gawain. The unnamed poet credits the symbol's origin to King Solomonand explains that each of the five interconnected points represents a virtue tied to a group of five: Gawain is perfect in his five senses and five fingers, faithful to the Five Wounds of Christ, takes courage from the five joys that Mary had of Jesus, and exemplifies the five virtues of knighthood.
Symbolic meaning of the pentagram
By the midth century a further distinction had developed amongst occultists regarding the pentagram's orientation. With a single point upwards it depicted spirit presiding over the four elements of matter, and was essentially "good". However, the influential writer Eliphas Levi called it evil whenever the symbol appeared the other way up.
It is the goat of lust attacking the heavens with its horns, a sign execrated by initiates. It is the sign of antagonism and fatality.
It is the goat of lust attacking the heavens with its horns. The five signs at the pentagram's vertices are astrological.
Another pentagram from Agrippa's book. This one has the Pythagorean letters inscribed around the circle. The occultist and magician Eliphas Levi 's pentagram, which he considered to be a symbol of the microcosmor human.
The apotropaic use of the pentagram symbol in German folklore called Drudenfuss in German is referred to by Goethe in Faustwhere a pentagram prevents Mephistopheles from leaving a room but did not prevent him from entering by the same way, as the outward pointing corner of the diagram happened to be imperfectly drawn: I must confess, my stepping o'er Thy threshold a slight hindrance doth impede; The wizard-foot [Drudenfuss] doth me retain.
The pentagram thy peace doth mar?
To me, thou son of hell, explain, How camest thou in, if this thine exit bar? Could such a spirit aught ensnare?What's With the Pentagrams at Trump's Executive Order Signing?
Observe it well, it is not drawn with care, One of the angles, that which points without, Is, as thou seest, not quite closed. East Asian symbolism[ edit ] Wu Xing Chinese: The five phases are: