The Mandelbox is the escape-time fractal consisting of the parameters for which an iteration based on box and ball folds has a bounded map orbit. It was introduced by Tom Lowe in 2010 (Lowe 2021). In the figure above, the left panel shows a ray-marched perspective projection of the three-dimensional scale-2 Mandelbox, while the right panel shows the corresponding three-dimensional surface.
For a vector 
, define the box fold 
componentwise by
 |
(1)
|
Here, 
denotes the 
th coordinate of the vector 
. Thus the box fold reflects
each coordinate lying outside the hypercube ![[-1,1]^d](/images/equations/Mandelbox/Inline6.svg)
across the nearest face
of the hypercube. For 
, define the ball fold 
by
 |
(2)
|
where 
is the Euclidean
norm. Given parameters 
and 
, start with 
and apply the iteration
![v_(k+1)=sS_r[fB(v_k)]+c.](/images/equations/Mandelbox/NumberedEquation3.svg) |
(3)
|
The Mandelbox is the set of parameters 
for which the resulting map
orbit is bounded. The standard Mandelbox uses 
, 
, and 
. The construction works in any finite dimension,
although the three-dimensional Mandelbox is most commonly rendered (Helt 2018, Lowe
2021).
