The stellated octahedron is the first iteration of the 3D analogue of a Koch snowflake.Ī compound of two spherical tetrahedra can be constructed, as illustrated.

It can be seen as a net of a four-dimensional octahedral pyramid, consisting of a central octahedron surrounded by eight tetrahedra.Ī single diagonal triangle facetting in red.

It can be seen as a being a tetragram, a compound of two dual digons, and the tetrahedron seen as a digonal antiprism, this can be seen as a compound of two digonal antiprisms.

It is a facetting of the cube, sharing the vertex arrangement.

In this construction it has the same topology as the convex Catalan solid, the triakis octahedron, which has much shorter pyramids.

It can be obtained as an augmentation of the regular octahedron, by adding tetrahedral pyramids on each face.It is also a regular polyhedron compound, when constructed as the union of two regular tetrahedra (a regular tetrahedron and its dual tetrahedron).The only stellation of a regular octahedron, with one stellation plane in yellow. It is a stellation of the regular octahedron, sharing the same face planes.The stellated octahedron can be constructed in several ways: The first stage of the construction of the Koch Snowflake is a single central tetrahedron, and the second stage, formed by adding four smaller tetrahedra to the faces of the central tetrahedron, is the stellated octahedron. It can also be seen as one of the stages in the construction of a 3D Koch snowflake, a fractal shape formed by repeated attachment of smaller tetrahedra to each triangular face of a larger figure. This can be generalized to any desired amount of higher dimensions the four-dimensional equivalent construction is the compound of two 5-cells. It can be seen as a 3D extension of the hexagram: the hexagram is a two-dimensional shape formed from two overlapping equilateral triangles, centrally symmetric to each other, and in the same way the stellated octahedron can be formed from two centrally symmetric overlapping tetrahedra. It is also the least dense of the regular polyhedral compounds, having a density of 2.

It is the simplest of five regular polyhedral compounds, and the only regular compound of two tetrahedra. It was depicted in Pacioli's De Divina Proportione, 1509. It is also called the stella octangula (Latin for 'eight-pointed star'), a name given to it by Johannes Kepler in 1609, though it was known to earlier geometers. The stellated octahedron is the only stellation of the octahedron. Seen as a compound of two regular tetrahedra (red and yellow) Therefore, there are five ways to arrange regular polygons around a vertex to form a net, which can be folded to form a concave three-dimensional figure.(Redirected from Star Tetrahedron) Stellated octahedron There are three possible ways we can form a three-dimensional vertex, with equilateral triangles, squares, and pentagons. The only figures that can form the Platonic solids are triangles, squares, and pentagons. The reason for the second condition is that if the angles formed at a vertex are equal to 360°, the figures would be flat.Ĭonsidering this, it turns out that only the 5 figures known as Platonic solids meet these conditions, as we can see in the following table: Platonic solid The reason for the first condition is that if only two faces meet at each vertex, it is not possible to form a closed three-dimensional figure. Interior angles that meet at a vertex must be less than 360°.At least 3 faces must meet at each vertex of the Platonic solid.In turn, for this to be possible, the figure must meet the following conditions: For a three-dimensional figure to be a Platonic solid, it must be composed of congruent regular polygons.

Tall octahedron template