matematicas visuales visual math
3d Printing: Cube and Octahedron
The Cube and the Octahedron are dual polyhedra

In this page we are going to build the cube and the octahedron using a 3d printer. Then we are going to see that these two polyhedra are dual polyhedra.

The Cube

The cube is a very well known polyhedron.

Building polyhedra 3d printing: The cube and the octahedron are dual polyhedra | matematicasVisuales

To model these vertices I used OpenSCAD, a free wonderful program.

Building polyhedra 3d printing: The cube and the octahedron are dual polyhedra | matematicasVisuales
Building polyhedra 3d printing: The cube and the octahedron are dual polyhedra | matematicasVisuales

We can count its faces (C), edges (A) and vertices (V):

To calculate the circumradius (the radius of a circumsphere touching each of the cube's vertices) we can start calculating the diagonal d of one face:

Building polyhedra 3d printing: The cube and the octahedron are dual polyhedra | matematicasVisuales
Building polyhedra 3d printing: The cube and the octahedron are dual polyhedra | matematicasVisuales
DIN A ratio: trigonometry, angles | matematicasvisuales

Then the diagonal of the cube (the space diagonal) D is:

Building polyhedra 3d printing: The cube and the octahedron are dual polyhedra | matematicasVisuales
DIN A ratio: trigonometry, angles | matematicasvisuales

This section of the cube is related with the DinA size of paper:

Standar Paper Size DIN A
There is a standarization of the size of the paper that is called DIN A. Successive paper sizes in the series A1, A2, A3, A4, and so forth, are defined by halving the preceding paper size along the larger dimension.

One diagonal of this rectangle:

DIN A ratio: trigonometry, angles | matematicasvisuales
DIN A ratio: trigonometry, angles | matematicasvisuales

You can calculate D as a basic application of the Pythagorean Theorem:

DIN A ratio: trigonometry, angles | matematicasvisuales

Then, the circumradius is:

The inradius of a cube is (the radius of the insphere or inscribed sphere that is tangent to each of the cube's faces):

The Octahedron

Building polyhedra 3d printing: The cube and the octahedron are dual polyhedra | matematicasVisuales

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.
Plane developments of geometric bodies: Octahedron
The first drawing of a plane net of a regular octahedron was published by Dürer in his book 'Underweysung der Messung' ('Four Books of Measurement'), published in 1525 .

As always, I used OpenSCAD, a free wonderful program, to model these vertices:

Building polyhedra 3d printing: The cube and the octahedron are dual polyhedra | matematicasVisuales
Building polyhedra 3d printing: The cube and the octahedron are dual polyhedra | matematicasVisuales
Building polyhedra 3d printing: The cube and the octahedron are dual polyhedra | matematicasVisuales

We can count its faces (C), edges (A) and vertices (V):

Now we are going to calculate le circumradius of an octahedron.

Building polyhedra 3d printing: The cube and the octahedron are dual polyhedra. Circumsphere of an octahedron | matematicasVisuales

We can see the height of these two pyramides as the diagonal of a square.

Octahedron: an octahedron diagonal | matematicasvisuales
Octahedron: built plastic tubes | matematicasvisuales
The height of an octahedron is the diagonal of a square | matematicasvisuales

The diagonal of a square of edge length 1 is:

The circumradius of an octahedron of edge length a is:

The inradius of an octahedron is:

Building polyhedra 3d printing: The cube and the octahedron are dual polyhedra | matematicasVisuales
The Cube and the Octahedron are dual polyhedra

The vertices of the cube correspond with the faces of the octahedron and vice versa. And both have the same number of edges.

Then we say that the cube and the octahedron are dual polyhedra.

One way to contruct a dual polyhedron of a regular polyhedron is to choose the center of the faces and connect each point with the points of its neighboring faces.

We are going to do this with the cube and the octahedron.

An Octahedron inside a Cube

Building polyhedra 3d printing: The cube and the octahedron are dual polyhedra | matematicasVisuales

Building polyhedra 3d printing: The cube and the octahedron are dual polyhedra | matematicasVisuales

The octahedron has edge length l and the edge length of the cube is L. Then the circumradius of the octahedron must be equal to the inradius of the cube:

Building polyhedra 3d printing: The cube and the octahedron are dual polyhedra | matematicasVisuales
Building polyhedra 3d printing: The cube and the octahedron are dual polyhedra | matematicasVisuales
A Cube inside an Octahedron

Building polyhedra 3d printing: The cube and the octahedron are dual polyhedra | matematicasVisuales

The cube has edge length l and the edge length of the octahedron is L. Then the circumradius of the cube must be equal to the inradius of the octahedron:

Building polyhedra 3d printing: The cube and the octahedron are dual polyhedra | matematicasVisuales

Building polyhedra 3d printing: The cube and the octahedron are dual polyhedra | matematicasVisuales

There is an easy way to see the relation between the edge lengths of the inscribed cube l and the octahedron L. See this picture:

Building polyhedra 3d printing: The cube and the octahedron are dual polyhedra | matematicasVisuales
Building polyhedra 3d printing: The cube and the octahedron are dual polyhedra | matematicasVisuales

The four edges of the square on the middle pass through four barycenters of four faces of the big octahedron.

Building polyhedra 3d printing: The cube and the octahedron are dual polyhedra | matematicasVisuales

REFERENCES

OpenSCAD a free wonderful program to model shapes in three dimensions.
Magnus Wenninger - 'Polyhedron Models', Cambridge University Press.
Hugo Steinhaus - Mathematical Snapshots - Oxford University Press - Third Edition (p. 197)
Peter R. Cromwell - 'Polyhedra', Cambridge University Press, 1999.
H.Martin Cundy and A.P. Rollet, 'Mathematical Models', Oxford University Press, Second Edition, 1961 (p. 87).
W.W. Rouse Ball and H.S.M. Coxeter - 'Matematical Recreations & Essays', The MacMillan Company, 1947.

MORE LINKS

Standar Paper Size DIN A
There is a standarization of the size of the paper that is called DIN A. Successive paper sizes in the series A1, A2, A3, A4, and so forth, are defined by halving the preceding paper size along the larger dimension.
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.
Plane developments of geometric bodies: Octahedron
The first drawing of a plane net of a regular octahedron was published by Dürer in his book 'Underweysung der Messung' ('Four Books of Measurement'), published in 1525 .
Resources: Building polyhedra gluing discs
Simple technique to build polyhedra gluing discs made of cardboard or paper.
Resources: How to build polyhedra using paper and rubber bands
A very simple technique to build complex and colorful polyhedra.
Resources: Building polyhedra gluing faces
Using cardboard you can build beautiful polyhedra cutting polygons and glue them toghether. This is a very simple and effective technique. You can download several templates. Then print, cut and glue: very easy!
Resources: The golden rectangle and the icosahedron
With three golden rectangles you can build an icosahedron.
Resources: Building Polyhedra with cardboard (Plane Nets)
Using cardboard you can draw plane nets and build polyhedra.
Resources: Modular Origami
Modular Origami is a nice technique to build polyhedra.
Resources: Building polyhedra using tubes
Examples of polyhedra built using tubes.
Resources: Tensegrity
Examples of polyhedra built using tensegrity.
Resources: Building polyhedra using Zome
Examples of polyhedra built using Zome.
Building polyhedra. Basic techniques: Taller de Talento Matemático de Zaragoza (Spanish)
Material for a sesion about polyhedra (Zaragoza, 13th Abril 2012).
Construcción de poliedros. Cuboctaedro y dodecaedro rómbico: Taller de Talento Matemático de Zaragoza 2014 (Spanish)
Material for a sesion about polyhedra (Zaragoza, 9th May 2014). Simple techniques to build polyhedra like the tetrahedron, octahedron, the cuboctahedron and the rhombic dodecahedron. We can build a box that is a rhombic dodecahedron.
Cube, octahedron, tetrahedron and other polyhedra: Taller de Talento Matemático Zaragoza,Spain, 2014-2015 (Spanish)
Material for a sesion about polyhedra (Zaragoza, 7th November 2014). We study the octahedron and the tetrahedron and their volumes. The truncated octahedron helps us to this task. We build a cubic box with cardboard and an origami tetrahedron.
Duality: cube and octahedron. Taller de Talento Matemático de Zaragoza, Spain. 2015-2016 XII edition (Spanish)
Material for a sesion about polyhedra (Zaragoza, 23rd Octuber 2015) . Building a cube with cardboard and an origami octahedron.
The Cuboctahedron and the truncated octahedron. Taller de Talento Matemático de Zaragoza, Spain. 2016-2017 XIII edition (Spanish)
Material for a sesion about polyhedra (Zaragoza, 21st October 2016). Instructions to build several geometric bodies.
Resources: Acona Biconbi, designed by Bruno Munari
Italian designer Bruno Munari conceived 'Acona Biconbi' as a work of sculpture. It is also a beautiful game to play with colors and shapes.
Sections on a tetrahedron
Special sections of a tetrahedron are rectangles (and even squares). We can calculate the area of these cross-sections.
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
Regular dodecahedron
Some properties of this platonic solid and how it is related to the golden ratio. Constructing dodecahedra using different techniques.