3d Printing: Icosahedron and Dodecahedron
Icosahedron and Dodecahedron are dual polyhedra
In this page we are going to build the icosahedron and the dodecahedron using a 3d printer. Then we are going to see that these two polyhedra are dual polyhedra.
The Icosahedron
The icosahedron is a beautiful polyhedron.
With three golden rectangles you can build an icosahedron.
To model these vertices I used OpenSCAD, a free wonderful program.
We can count its faces (C), edges (A) and vertices (V):
The twelve vertices of an icosahedron lie in three golden rectangles. Then we can calculate the volume of an icosahedron
To calculate the circumradius (the radius of a circumsphere touching each of the icosahedron's vertices):
Then, the circumradius is:
From Euclid's definition of the division of a segment into its extreme and mean ratio we introduce a property of golden rectangles and we deduce the equation and the value of the golden ratio.
The inradius (the radius of the insphere or inscribed sphere that is tangent to each of the faces) of a icosahedron is :
The Dodecahedron
Some properties of this platonic solid and how it is related to the golden ratio. Constructing dodecahedra using different techniques.
One eighth of a regular dodecahedon of edge 2 has the same volume as a dodecahedron of edge 1.
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 .
As always, I used OpenSCAD, a free wonderful program, to model these vertices:
We can count its faces (C), edges (A) and vertices (V):
Now we are going to calculate le circumradius of a dodecahedron.
The circumradius of an dodecahedron of edge length a is:
The inradius of a dodecahedron is:
The Icosahedron and the Dodecahedron are dual polyhedra
The vertices of the icosahedron correspond with the faces of the dodecahedron and vice versa.
And both have the same number of edges.
Then we say that the icosahedron and the dodecahedron 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 icosahedron and the dodecahedron.
An icosahedron inside a dodecahedron
Kepler's Harmonices Mundi
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:
A dodecahedron inside an icosahedron
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:
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:
The four edges of the square on the middle pass through four barycenters of four faces of the big octahedron.
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.
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Building cubes and octahedra using 3d printing. Cube and Octahedron are dual polyhedra.
MORE LINKS
Building tetraedra using 3d printing. The tetrahedron is a self-dual polyhedron. The center of a tetrahedron.
The twelve vertices of an icosahedron lie in three golden rectangles. Then we can calculate the volume of an icosahedron
Some properties of this platonic solid and how it is related to the golden ratio. Constructing dodecahedra using different techniques.
One eighth of a regular dodecahedon of edge 2 has the same volume as a dodecahedron of edge 1.
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 .
A Cube can be inscribed in a Dodecahedron. A Dodecahedron can be seen as a cube with six 'roofs'. You can fold a dodecahedron into a cube.
Simple technique to build polyhedra gluing discs made of cardboard or paper.
A very simple technique to build complex and colorful polyhedra.
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!
With three golden rectangles you can build an icosahedron.
Using cardboard you can draw plane nets and build polyhedra.
Modular Origami is a nice technique to build polyhedra.
Examples of polyhedra built using tubes.
Examples of polyhedra built using tensegrity.
Examples of polyhedra built using Zome.
Material for a session about polyhedra (Zaragoza, 13th Abril 2012).
Material for a session 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.
Material for a session 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.
Material for a session about polyhedra (Zaragoza, 23rd Octuber 2015) . Building a cube with cardboard and an origami octahedron.
Material for a session about polyhedra (Zaragoza, 21st October 2016). Instructions to build several geometric bodies.
Italian designer Bruno Munari conceived 'Acona Biconbi' as a work of sculpture. It is also a beautiful game to play with colors and shapes.
Microarquitectura is a construction game developed by Sara San Gregorio. You can play and build a lot of structures modelled on polyhedra.
