matematicas visuales visual math

Truncating a polyhedron means that the corners (or the edges) are cut off.

We have already study several cases of truncating the vertices of regular polyhedra.

Truncated tetrahedron
The truncated tetrahedron is an Archimedean solid made by 4 triangles and 4 hexagons.
Truncations of the cube and octahedron
When you truncate a cube you get a truncated cube and a cuboctahedron. If you truncate an octahedron you get a truncated octahedron and a cuboctahedron.

In this page we are going to see the truncation of the edges of a cube and we are going to get a new polyhedron called the chamfered cube.

Chamfered cube: wood | matematicasVisuales
Chamfered cube: chamfering only a little | matematicasVisuales
Chamfered cube: equilateral | matematicasVisuales

You can get a polyhedron with all the edges of equal length. Then the twelve hexagons are equilateral but no equiangular.

The chamfered cube is a polyhedron that is similar to the truncated octahedron. But the truncated octahedron have twelve hexagons that are regular.

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.
Chamfered cube: it is similar to the truncated octahedron. But the truncated octahedron have twelve hexagons that are regular| matematicasVisuales

Hexagons in a truncated octahedron are equilateral and equiangular.

To see that the hexagons in a chamfered cube are not equiangular we are going to calculate these angles.

Chamfered cube: hexagons in a chamfered cube are not equiangular | matematicasVisuales

Angles of this polyhedron are related with angles we found studying the DinA4 rectangle.

Chamfered cube: DinA rectangle | matematicasVisuales

You can calculate angle A:

Chamfered cube: angles | matematicasVisuales
Chamfered cube: angles | matematicasVisuales

Then angle B:

Chamfered cube: angles | matematicasVisuales

In fact

Chamfered cube: angles | matematicasVisuales

As a limiting case you get a polyhedron with very interesting properties: the rhombic dodecahedron.

Chamfered cube: rhombic dodecahedron | matematicasVisuales

The rhombic dodecahedron and the cuboctahedron are dual polyhedra.

The chamfered cube with all edges of equal length is also called truncated rhombic dodecahedron because you can get this polyhedron by truncating the six vertices of orden 4 of a rhombic dodecahedron is such a way that the truncated vertices become squares and the rhombic faces become equilateral but not regular hexagons.

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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.
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.
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 .
Regular dodecahedron
Some properties of this platonic solid and how it is related to the golden ratio. Constructing dodecahedra using different techniques.
Plane developments of geometric bodies (1): Nets of prisms
We study different prisms and we can see how they develop into a plane net. Then we explain how to calculate the lateral surface area.
Plane developments of geometric bodies (3): Cylinders
We study different cylinders and we can see how they develop into a plane. Then we explain how to calculate the lateral surface area.
Plane developments of geometric bodies (5): Pyramid and pyramidal frustrum
Plane net of pyramids and pyramidal frustrum. How to calculate the lateral surface area.
Plane developments of geometric bodies (7): Cone and conical frustrum
Plane developments of cones and conical frustum. How to calculate the lateral surface area.
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 .
Hexagonal section of a cube
We can cut in half a cube by a plane and get a section that is a regular hexagon. Using eight of this pieces we can made a truncated 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
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 truncated octahedron is a space-filling polyhedron
These polyhedra pack together to fill space, forming a 3 dimensional space tessellation or tilling.
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.
Stellated cuboctahedron
The compound polyhedron of a cube and an octahedron is an stellated cuboctahedron.It is the same to say that the cuboctahedron is the solid common to the cube and the octahedron in this polyhedron.
Truncated tetrahedron
The truncated tetrahedron is an Archimedean solid made by 4 triangles and 4 hexagons.
Resources: How to build polyhedra using paper and rubber bands
A very simple technique to build complex and colorful polyhedra.
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.
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 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.
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).