Everything that we see in the world is a 3-dimensional form that can be described using geometric shapes. Simply put, geometry is the study of spatial patterns or structures. Artists also investigate different patterns and shapes that they see around them. Symmetry and proportion are studied and explored by both mathematicians and artists. These patterns occur in nature both in the micro and macro world – the atomic and nucleus structures, crystals as well as the galaxies, mountains, rivers, trees and herbs, animals, birds and insects. The same can be seen in man-made structures – architecture, bridges, domes, huts, textiles as well as in music, dance, paintings and sculpture.
So we thought this festive season, let’s make something that combines aesthetics with mathematical pattern. This resulted in a model of Kepler’s Stella Octangula, an 8-pointed star, and the 5-pointed star based on the Pentagram as decorative art pieces to spread hope, happiness and cheer for Christmas and the coming new year!
In 1619, Kepler defined stellation for polygons and polyhedra, as the process of extending edges or faces until they meet to form a new polygon or polyhedron. He stellated the regular octahedron to obtain the stella octangula, a regular compound of two tetrahedra.
The word stellation comes from the Latin stellātus, ‘starred’, which in turn comes from Latin stella, ‘star’.
The stella octangula is a polyhedron compound composed of a tetrahedron and its dual (a second tetrahedron rotated with respect to the first). The solid common to both tetrahedra is an octahedron (fig 1), which is another way of saying that the stella octangula is a stellation of the octahedron (in fact, the only stellation).
The edges of the two tetrahedra in the stella octangula form the 12 polyhedron diagonals of a cube (as in fig 2).
The Pentagram Star
Pentagram is a five-pointed star, with all lines the same length and all angles the same. The pentagram is the stellation of a pentagon.Pentagram in literature:
Dan Brown’s Da Vinci Code discussed the meaning of Pentagram. According to the book, the pentagram actually belonged to the ancient religion of Pagan Sun Worship, and was tied to the planet Venus. It is claimed every 8 years Venus traces a perfect pentagram across the sky. Fancy wearing a Stella Octangula around your neck?
A wireframe version of the stella octangula is sometimes known as the merkaba and is said to be imbued with mystic properties. The merkaba is also used in the form of jewellery and as a tool for healing and self balance.
3. http://mathworld.wolfram.com/StellaOctangula.html Our model of 5-pointed star based on the Pentagram Our model of 8-pointed Stella Octangula