Project 12: Polyhedra of Point Groups

Steffen Weber, May 1998 

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Creating a Polyhedron
Click/Drag any point  in the Wulff net on the left. The dots you see moving around are generated by the selected point group (a set of rotational  symmetry operations). Each dot is the projection of a plane normal (that intersects the stereographic hemisphere) onto the equator plane (represented by the Wulff net) A solid dot represents a normal pointing out of the screen, and an empty  circle represents one pointing away from you. When you release the mouse the corresponding polyhedron is calculated and displayed in the right window.
When you drag the dots in the Wulff net you can see that the number of poles (planes) changes as you move to special symmetry centers. Different numbers of planes means different types of polyhedra. Therefore you can  create several types of  polyhedra in any of the point groups with higher symmetry (eg: cubic, icosahedral).

In order to rotate the right figure around its x-axis use the right mouse key and the left mouse key to rotate around the y-and z-axis.

right mouse button: canvas color
left mouse button: polyhedra color

Please note that for the pyramidal forms I added the basal plane (00-1), in order to obtain a closed form. This basal plane is NOT a result of the chosen point group.

Further reading
tutorial on the stereographic projection (explains also the meaning of the Wulff net)

Be aware that the calculation of general icosahedral polyhedra with 60 or 120 planes in the  point groups 235 & m-3-5 may take quite a while even on fast computers. (but it works!)

Link:Solid Geometry a computer program ( Hope Paul Productions) for generating and printing shapes that can be cut and glued to make 3D bodies.