Star tetrahedron

Star tetrahedron

There are two tetrahedra in the cube that merge to form a star tetrahedron.

The star tetrahedron , also known as the star body to the octahedron (even if it is not a star body , since the same number of surfaces do not meet in all corners) and as the Kepler star, is an eight-pointed star and belongs to the non- convex deltahedra . It is a multi-faceted body that is created by merging two point-symmetrical tetrahedra . Named by Johannes Kepler (hence the name "Keplerstern") in 1609, this is both the simplest regular composite polyhedron and the simplest non- convex uniform polyhedron. He was first depicted by Leonardo da Vinci in Luca Pacioli's De Divina Proportione 1509.

The outer corner points of the body describe a cube , while the intersection of the two tetrahedra represents an octahedron , the edges of which in turn represent the inner edges of the star tetrahedron.

The graphic artist MC Escher used the star tetrahedron as a motif for the picture Doppelplanetoid : One tetrahedron has the shape of a castle inhabited by humans, while the other represents a world penetrated by the first and inhabited by dinosaurs.

The star tetrahedron is the first stage of the convex shape of the Sierpinski octahedron . The eight small tetrahedra can be made into star tetrahedra again, and this process can be repeated, so that finally a fractal is created, which approximates the shape of a hexahedron .

Formulas

Sizes of a star tetrahedron with edge length a or b = a / 2
Volume  ≈ 0.18 a 3	${\displaystyle V={\frac {a^{3}}{8}}{\sqrt {2}}=b^{3}{\sqrt {2}}}$
Surface area  ≈ 2.6 a 2	${\displaystyle O={\frac {3}{2}}a^{2}{\sqrt {3}}=6\,b^{2}{\sqrt {3}}}$
Umkugelradius  ≈ 0.61 a	${\displaystyle R={\frac {a}{4}}{\sqrt {6}}={\frac {b}{2}}{\sqrt {6}}}$

• The volume of the star tetrahedron is equal to the sum of the volumes of one octahedron and eight attached tetrahedra, each with half the edge length . It fills half of the bounding cube ( ). ${\displaystyle a/2}$ ${\displaystyle V={\frac {a^{3}}{4}}{\sqrt {2}}=2b^{3}{\sqrt {2}}}$
• The spherical radius of the star tetrahedron corresponds to that of a single tetrahedron.

Application in art

• 

  Wall painting of a star tetrahedron in the Boris and Gleb Church in Kidekscha

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  Another wall painting of a star tetrahedron in the Boris and Gleb Church

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  Jovilliers Abbey ( Stainville , Lorraine)

Web links

Commons : Star Tetrahedron  - collection of images, videos and audio files

• Eric W. Weisstein : Star Tetrahedron . In: MathWorld (English).

Individual evidence

1. ↑ MC Escher: Double Planetoid .
2. ↑ Keplerian Fractals ( Memento of the original from March 16, 2015 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. 
3. ↑ Approaching a Fractal Cube by a series of non-convex polyhedra
4. ↑ Keplerian fractals