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# Cuboctahedron

This polyhedron is obtained by successively cutting off each of the vertices of the octahedron or cube.

A cuboctahedron is a semi-regular convex polyhedron with two properties:
1. All faces are regular polygons of two types, a triangle and a square;
2. For any pair of vertices, there is a symmetry of the polyhedron (that is, a motion that translates the polyhedron into itself) that transforms one vertex to another.

# The cuboctahedron is one of the 13 solids of Archimedes.

Archimedean solids are semiregular polyhedra in the sense that their faces are regular non-similar polygons while the condition of one of the types of spatial symmetry (tetrahedral, octahedral or icosahedral) is preserved.

## Mathematical characteristics of the cuboctahedron A cuboctahedron can be placed in a sphere (inscribed), so that each of its vertices will touch the inner wall of the sphere.

The radius of the described sphere of the cuboctahedron

R=a

where a is the side length.

It is not possible to place the sphere inside the cuboctahedron so that it touches all sides simultaneously.
To fulfill this condition, the sphere must touch all squares and triangles.
The distances from the center of the sphere to the middle of the triangles and to the middle of the squares are different. The surface area of the cuboctahedron.

The surface area can be defined by the following formula:  The volume of the cuboctahedron is determined by the following formula:  A cubooctahedron can be divided by a plane into two equal parts.

In this case, the cross section will have the shape of a regular hexagon.
The following four cases of section the cubooctahedron are possible:     The famous mathematical problem "On the Seven Königsberg Bridges" or the Euler Path can be reformulated and considered on the example of the edges of a cubooctahedron.

How to consistently bypass all the edges of a cubooctahedron? One edge cannot be touched twice.

The figure shows an animated example of solving this problem.

One of the names of the task is the Euler Path. In honor of the famous mathematician Leonard Euler.

## Cuboctahedron nets

The figures shows the shape nets of a cuboctahedron:

To build a model, you can download the pattern in pdf format and print it on an A4 sheet.

We suggest making a cubo-octahedron, where the faces formed by the cube are brown and the faces formed by the octahedron are gray.
If you intend to print it on a color printer, download the color net.

If you intend to use colored paper, use this net.

In addition, there are two versions of color for the polyhedron, when each of the adjacent faces is painted in its own color. The same colors do not border with each other.

We draw your attention to two options for painting the dodecahedron using five colors.

The first option is the faces formed by a cube of the same color, the faces formed by the octahedron are colored in different colors. Moreover, the opposite faces of the octahedron have the same colors: download net The second option is the faces formed by the octahedron of the same color, the faces formed by the cube are colored in different colors. Moreover, the opposite sides of the cube have the same colors: download net ## The video "Build a polyhedron from a single net"

This video from our partners, the ART KOSEKOMA team, clearly demonstrates how the pattern is converted into a geometric shape:

## Cuboctahedron from the "Magic Edges" set

To assemble polyhedra, we can offer you ready-made patterns that are cut and folded.
To do this, you need to use the details of the set Magic Edges 18.
In addition, in the release itself you will find information about the structure of the polyhedron.

Polyhedra, from the set Magic Edges 18:

Assembling a cuboctahedron from the set "Magic Edges"

The rotation of the finished polyhedron assembled from these parts:

Detailed assembly from Alexei Zhigulev (youtube channel - Origami)

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