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The truncated tetrahedron is an Archimedean polyhedron with four equilateral triangles and four regular hexagons. It was drawn by Leonardo for Luca Pacioli's book 'De Divina Proportione' and studied by Kepler in 'Harmonices mundi'.

Leonardo da Vinci: Drawing of a truncated tetrahedron made to Luca Pacioli's De divina proportione.
Leonardo da Vinci made several drawings of polyhedra for Luca Pacioli's book 'De divina proportione'. Here we can see an adaptation of the truncated tetrahedron.
Truncated tetrahedron: Kepler | matematicasVisuales

You can read Kepler's book 'Harmonices Mundi' in Posner Library Collection.

Truncated tetrahedron: | matematicasVisuales

We know how to calculate the volume of a tetrahedron. To calculate the volume of a truncated tetrahedron the main idea is:

Truncated tetrahedron: | matematicasVisuales

We want to calculate the volume of a truncated tetrahedron of side length 1. We can consider a tetrahedron of side length 3.

Truncated tetrahedron: | matematicasVisuales

We need to remember that the volume of a tetrahedron of side length a is:

Then we start with a tetrahedron of side length 3. Its volume is:

Truncated tetrahedron: | matematicasVisuales

Now we need the volume of a tetrahedron of side length 1:

Truncated tetrahedron: | matematicasVisuales

The volume of a truncated tetrahedron of side lenght 1 is:

You can control how deep is the truncation:

Truncated tetrahedron: | matematicasVisuales

You get the octahedron as an extreme case:

Truncated tetrahedron: | matematicasVisuales

You can play with transparency to get beautiful projections:

Truncated tetrahedron: | matematicasVisuales

Truncated tetrahedron: | matematicasVisuales

Resources: How to build polyhedra using paper and rubber bands
A very simple technique to build complex and colorful polyhedra.
Resources, How to build polyhedra with paper and rubber bands: Truncated tetrahedron | matematicasVisuales
Resources, How to build polyhedra with paper and rubber bands: Truncated tetrahedron| matematicasVisuales
Resources, How to build polyhedra with paper and rubber bands: Truncated tetrahedron | matematicasVisuales

Durer was the first to publish a plane net of a truncated tetrahedron:

Truncated tetrahedron: Durer was the first to publish a plane net of a truncated tetrahedron | matematicasVisuales
Truncated tetrahedron: | matematicasVisuales
Truncated tetrahedron: | matematicasVisuales
Truncated tetrahedron: | matematicasVisuales
Truncated tetrahedron: | matematicasVisuales

REFERENCES

W.W. Rouse Ball and H.S.M. Coxeter - 'Matematical Recreations & Essays', The MacMillan Company, 1947.
Magnus Wenninger - 'Polyhedron Models', Cambridge University Press.
Peter R. Cromwell - 'Polyhedra', Cambridge University Press, 1999.

MORE LINKS

Resources: How to build polyhedra using paper and rubber bands
A very simple technique to build complex and colorful polyhedra.
The volume of the tetrahedron
The volume of a tetrahedron is one third of the prism that contains it.
Plane developments of geometric bodies: Tetrahedron
The first drawing of a plane net of a regular tetrahedron was published by Dürer in his book 'Underweysung der Messung' ('Four Books of Measurement'), published in 1525 .
Volume of an octahedron
The volume of an octahedron is four times the volume of a tetrahedron. It is easy to calculate and then we can get the volume of a tetrahedron.
Regular dodecahedron
Some properties of this platonic solid and how it is related to the golden ratio. Constructing dodecahedra using different techniques.
The volume of a truncated octahedron
The truncated octahedron is an Archimedean solid. It has 8 regular hexagonal faces and 6 square faces. Its volume can be calculated knowing the volume of an octahedron.
The volume of an stellated octahedron (stella octangula)
The stellated octahedron was drawn by Leonardo for Luca Pacioli's book 'De Divina Proportione'. A hundred years later, Kepler named it stella octangula.
The volume of a cuboctahedron
A cuboctahedron is an Archimedean solid. It can be seen as made by cutting off the corners of a cube.
The volume of a cuboctahedron (II)
A cuboctahedron is an Archimedean solid. It can be seen as made by cutting off the corners of an octahedron.
The icosahedron and its volume
The twelve vertices of an icosahedron lie in three golden rectangles. Then we can calculate the volume of an icosahedron
Plane developments of geometric bodies: Dodecahedron
The first drawing of a plane net of a regular dodecahedron was published by Dürer in his book 'Underweysung der Messung' ('Four Books of Measurement'), published in 1525 .
Chamfered Cube
You can chamfer a cube and then you get a polyhedron similar (but not equal) to a truncated octahedron. You can get also a rhombic dodecahedron.
Leonardo da Vinci: Drawing of a truncated octahedron made to Luca Pacioli's De divina proportione.
Leonardo da Vinci made several drawings of polyhedra for Luca Pacioli's book 'De divina proportione'. Here we can see an adaptation of the truncated octahedron.
Leonardo da Vinci: Drawing of a cuboctahedron made to Luca Pacioli's De divina proportione.
Leonardo da Vinci made several drawings of polyhedra for Luca Pacioli's book 'De divina proportione'. Here we can see an adaptation of the cuboctahedron.
Leonardo da Vinci: Drawing of an stellated octahedron (stella octangula) made to Luca Pacioli's De divina proportione.
Leonardo da Vinci made several drawings of polyhedra for Luca Pacioli's book 'De divina proportione'. Here we can see an adaptation of the stellated octahedron (stella octangula).
Leonardo da Vinci: Drawing of a dodecahedron made to Luca Pacioli's De divina proportione.
Leonardo da Vinci made several drawings of polyhedra for Luca Pacioli's book 'De divina proportione'. Here we can see an adaptation of the dodecahedron.
The truncated octahedron is a space-filling polyhedron
These polyhedra pack together to fill space, forming a 3 dimensional space tessellation or tilling.
Chamfered Cube
You can chamfer a cube and then you get a polyhedron similar (but not equal) to a truncated octahedron. You can get also a rhombic dodecahedron.
Resources: Modular Origami
Modular Origami is a nice technique to build polyhedra.
Resources: Tensegrity
Examples of polyhedra built using tensegrity.
Resources: Building polyhedra using tubes
Examples of polyhedra built using tubes.
Resources: Building polyhedra using Zome
Examples of polyhedra built using Zome.