Stereographic Projection of Some Double Groups

group

dihedral-3

octahedral

icosahedral

figure

2D

3D

μ :

r

1

0.618034

θ

1

0.25

μ :

r

2

1

θ

2

0.25

sphere zoom

3

in 2D, show vectors?

yes

no

Stereographic projection provides geometric insight into the double cover

SU(2)~SO(3)

. Each rotation of the sphere corresponds to exactly two linear transformations of homogeneous coordinates

μ

1

,

μ

2

. The projection remains a bijection because the Möbius transformation of a complex plane coordinate

z=

μ

1

/

μ

2

retains only the relative sign of

μ

1

,

μ

2

[1]. Taking a linear perspective, you can view points in the plane as the cosets of inversion by a

2π

rotation. Plotting complex vectors

μ

1

,

μ

2

off each plane coordinate

z

reveals the hidden coset structure, which sometimes gets overlooked. The double groups, also called binary groups, contribute a foundational element in the analysis of quintic equations [2] and for quantum mechanics [3]. Furthermore, there is an aesthetic value in these dynamic images, in which variation of parameters appears to create a whirling dance of the points across the plane.