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Stellated octahedron

Stellated octahedron 350x350

The presented image of this polyhedron illustrates exactly its second name - the compound of two tetrahedra. So you can imagine a red tetrahedron pointing up through which a beige tetrahedron goes down.

However, mathematicians prefer to call the polyhedron as stellated octahedron.

The stellated octahedron could be recognized as a regular polyhedron, since all its faces are regular triangles of the same size and all angles between them are equal. But in fact, this geometric solid is not the sixth regular polyhedron on par with the five known Platonic solids. The reason is that the word convex is present in the definition of a regular polyhedron, that is, all faces must lie on one side of the plane passing through any of them.

Discovery history

stella octangula
The star octahedron was first depicted in 1509 in a book De divina proportione.The author of which was the mathematician Luca Pacioli (1445-1514). And the illustration for the book belongs to the hand of Leonardo da Vinci.
The stellated octahedron was made in the form of eight frame tetrahedra interconnected.
 
Then, almost 100 years later, the polyhedron was rediscovered by Johann Kepler, and named after him Stella octangula, the octagonal star.
 
It is this illustration of a stellate octahedron that is controversial about how this polyhedron was discovered.
 
zv okt 1

The first possible variant is to attach 8 tetrahedra to the sides of the octahedron
 
Or continue the planes parallel to the sides of the octahedron, getting the same 8 tetrahedra at the intersection of the planes. In other words, create a star-shaped octahedron.

zv okt 2

Second way: Combine two tetrahedrons.
What clearly illustrates the color of the faces in two colors - red and beige.

 

объединение двух тетраэдров
 

Video. Rotation of a polyhedron

Video. Polygon transform

Video from our partners - the ART KOSEKOMA team, clearly demonstrates how you can transform an octahedron into a star form and get two tetrahedrons.

First, the faces of the octahedron, drawing out, make a star form. Then the video clip shows how a stellated octahedron can be decomposed into two tetrahedrons.

Polyhedron properties

звездчатый октаэдр вписан в куб

2. An interesting feature of the polyhedron. If all the pointed tops are interconnected, the intersection lines correspond exactly to the edges of the cube.
Thus, the stellated octahedron can be inscribed in a cube.

 

тень звездчатого октаэдра
звездчатый октаэдр и звезда давида

3. If you look at the polyhedron from above, or at the shadow cast, the outlines of the pattern will create a regular hexagonal star.
The hexagonal star in the form of two intersecting triangles is the oldest symbol, which is referred to as the Star of David (another name is the Seal of King Solomon).

Net 1

net Stellated octahedron
The first of the proposed assembly options is to glue a stellated octahedron from a single net. This option is most often found in educational literature. However, the limitation is the size of the sheet. In the event that this is an A4 sheet, then the dimensions of the finished polyhedron are very modest.
 

Net 2

The second option, when separately collected octahedron and 8 blanks in the form of pyramids. Then, to the sides of the octahedron, all blanks are alternately glued. This option is preferable for beginner modellers.

Net 3

net Stellated octahedron4
On the pages of the Internet, we met a very unusual version of the assembly, also in the form of a single net.
 

Stellated octahedron from the set "Magic Edges"

For the assembly of a polyhedron, we can offer you ready-made nets - cut and folded.
To do this, you need to use the details of the set Magic Edges 15.
In addition, in the release itself you will find information about the structure of the polyhedron.

 
stellated octahedron 15 400 2
 
The rotation of the finished polyhedron assembled from these parts:
 

 

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