Stereographic Projection of a 3-Sphere

A four-dimensional hypersphere of radius
r
and center
(a,b,c,d)
has equation
2
(x-a)
+
2
(y-b)
+
2
(z-c)
+
2
(w-d)
=
2
r
. The unit 3-sphere is the four-dimensional hypersphere of radius
1
and center
(0,0,0,0)
. This Demonstration lets you rotate the unit 3-sphere around its six directional planes (
x
-
y
,
x
-
z
,
y
-
z
,
x
-
w
,
y
-
w
,
z
-
w
), showing the stereographic projection of its parallels, meridians, and hypermeridians in three-dimensional space. Although the circles seen in 3D seem to intersect themselves when the hypersphere is rotated, in 4D, the 3-sphere does not self-intersect.

Permanent Citation

John Na
​
​"Stereographic Projection of a 3-Sphere"​
​http://demonstrations.wolfram.com/StereographicProjectionOfA3Sphere/​
​Wolfram Demonstrations Project​
​Published: August 6, 2014