The Hopfbrot is a four-dimensional fractal obtained by applying a spherical power construction to
a parameterization of the 3-sphere
adapted to the Hopf map (Knill 2023). In the corresponding
Hopf coordinates, a point of the 3-sphere is written as
the pair of complex numbers 
. Adding the
same angle to 
and 
traces one circular fiber
while leaving its image under the Hopf map unchanged.
The power construction below multiplies these Hopf-coordinate
angles. Write
 |
(1)
|
where 
and ![phi in [0,pi/2]](/images/equations/Hopfbrot/Inline5.svg)
. For a nonzero vector 
, where 
, and an integer 
, define
 |
(2)
|
and set 
. The degree-
Hopfbrot is the set of vectors 
for which the orbit
of 
under
 |
(3)
|
is bounded. As with the Mandelbulb, this power depends on the chosen coordinate parameterization rather than on quaternion multiplication.
The figure above shows the 
three-dimensional slice of a degree-2 Hopfbrot approximation
obtained after five iterations with escape radius
2. In the figure, the coloring runs from blue through cyan and green to yellow as
the fourth coordinate 
increases. It does not encode escape time.
Setting 
or 
gives two traces of codimension
one that are Mandelbulbs. Setting 
gives a trace of codimension
two that is the Mandelbrot set (Knill 2023). The
name joins "Hopf," referring to Heinz Hopf and the Hopf
map, with the ending of "Mandelbrot." Knill (2023) reports that Paul
Nylander coined "Hopfbrot."
