Calculations at a unicursal hexagram. This is made of two long and four short diagonals of a hexagon. Other than a regular hexagram, this can be drawn in one go, without lifting the pen. The short diagonal b is split into three pieces of different length, the long diagonal c is split into four pieces of equal length. Both large spikes have an angle of 60°, the four small spikes are rectangular to the large ones. The area is calculated as that of a rhombus with the side length b1+b2 plus the equal right triangles of the four small spikes. Enter one value and choose the number of decimal places. Then click Calculate.
Formulas:
b = √3 * a
b1 = b/2 = √3 / 2 * a
b2 = b/6 = √3 / 6 * a
b3 = b/3 = √3 / 3 * a
c = 2 * a
c' = c/4 = a/2
p = 4b1 + 4b3 + 4c' = ( 2 + 10/3 * √3 ) * a
A = ( b1 + b2 )² * sin(60°) + 2b2c' = 5/6 * √3 * a²
Edge length, diagonals, sections and perimeter have the same unit (e.g. meter), the area has this unit squared (e.g. square meter).
The unicursal hexagram is point-symmetrical to its center and axially symmetrical to the line through the two large points and to the perpendicular line through the small concave corners. It is rotationally symmetrical at an angle of 180 degrees or multiples thereof.
The ability to draw a hexagon without lifting the pen is a modern discovery. The first known appearance of this form in a more general version is in the Figura Amoris by Giordano Bruno as part of an illustration of a text in 1588. Also the more general form was described mathematically by Blaise Pascal in the work Hexagrammum Mysticum, published in 1639. As a symbol, it was mainly used in the esoteric field; for example by the magical secret society Hermetic Order of the Golden Dawn and from there adopted by Aleister Crowley for his Thelema movement. Occasional use in pop culture probably alludes to this occult and seemingly meaningful connection.