The Hurwitz Prime sphere

The Lipschitz primes (integer coordinates) are yellow, the Hurwitz primes (half integer coordinates) are red. These primes together form the quaternion primes. They are located in the four dimensional Quaternion space, projected into the three dimensional space by looking at the spacial last components (b,c,d) of the quaternion (a,b,c,d) = a + i b + j c + k d, then projecting onto the unit sphere. On the coordinate planes, where one of the coordinates is zero, there are by definition only Lipschitz primes.
Hurwitz Prime Sphere