]> Constructing an Icosahedron
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Created 2002-01-01 Modified 2007-01-01
Chelton Evans

Constructing an Icosahedron home

Definitions
Planar Diagrams of the Icosahedron
Construction
Duality

Definitions

polyhedron
surface composed of plane polygon faces
regular polyhedron
polygons are same and regular
plotonic solids
tetrahedron(3*4), cube(4*6), octahedron(3*8), dodecahedron(5*12), icosahedron(3*20)

Planar Diagrams of the Icosahedron

Consider a icosahedron. A pattern of 5 triangles meeting at a point emerges. Lets call this a face pattern. Projecting the face pattern to a plane yields 6 points suchthat a pentagon surrounding a point at the origin.

The icosahedron can be represented by a planar diagram that separates the surface in two. The verticies of two opposite face patterns cover the verticies of the icosahedron exactly, so the icosahedron can be constructed from the verticies of two opposite face patterns.

The icosahedron can be described by two sets of verticies, 6 in each set. The dashed lines indicate that the points coinside.

Construction

The problem of constructing the icosahedron is now reduced to finding one set of vertices of a face pattern because the other face pattern can be obtained by mirroring and rotated the other one. Since the base of the face is a pentagon, complex numbers were used to generate the pentagon's points. See complex numbers for geometric constructions . So all that remains is find the height of the central point in the face pattern to determine completely the geometry of the face pattern.

Orient the face pattern, let the middle point of the face pattern lie on the origin and be the highest point. Let the other 5 points lie in a plane perpendicular to the middle point. Consider a facet with edge length a and radius r. Since a and r are known we can express h as a function of a and r.

h = ( a 2 - r 2 ) 1 2

Now the face pattern needs to be translated up as the origin of the icosahedron will be placed at the center of the object. Looking at an existing icosahedron, two opposite faces are parallel. Further there exists a face perpendicular to the plane of 5 points. This face is in the middle band of triangles in the second planar diagram. Since two of its points are on the plane, the height of the translation - labeled x can be calculated.

x = 3 1 2 a 2

To construct the first set of verticies, construct 5 equally spaced points about the origin, at a distance r from O. Their distance from non-origin neighbours is a. Place a point above the origin at h. Translate the 6 points by x/2 above O. Mirror in the horizontal plane for the other set of points. Join neighbouring vertices by edges.

For the 3 dimensional representation in VRML format, see icosahedron.wrl.

Duality

The icosahedron has a dual geometric object, which naturally enough is also a platonic solid - the dodecahedron. To get one object form the other, take the midpoints of a face and construct a vertex for the new geometric object. Connect the verticies to their nearest neighbours with an edge and vola the new geometric object is constructed. (Note: taking the midpoints of a face is the same as finding the faces center of gravity or the intersection of lines perpendicular to the edges at the point of bisection )

plotonic solid dual faces verticies
icosahedron dodecahedron 20 12
dodecahedron icosahedron 12 20
cube octahedron 8 6
octahedron cube 6 8
tetrahedron tetrahedron 4 4

<TODO>
Find an algebraic 3D representation for the dodecahedron or icosahedron. From this construct the dual. Scaling should be easily considered (rescale).
Write a program that given a geometric object constructs the dual.