The graph we are examining is the convergence graph of the iterative procedure Z -> (e^(i pi/2)) Z^2 + (e^(-i pi/2)) (-0.745,0.113,0.01,0.01), where Z is a complex quaternion four-vector of the form (a, b i, c j, d k).
The three-dimensional graph of the Quaternion Julia set is represented as a pixel volume spanned by (1, i, j, 0). The function is sampled over a finite grid in (r, i, j). Here is a 3-D rendering of the object:
In FractalMUD, we view the Julia set as a two-dimensional cross-section. We navigate FractalMUD using finite two-dimensional windows in our Web browsers. The system uses HTML as a program data stack to store the current state.
The "Zoom" button allows us to sample the graph at three times the detail at the next level. The buttons marked "YZ", "ZX" and "XY" allow us to turn our view 90 degrees to our current view. The "+X", "-X", "+Y", "-Y", "+Z", "-Z" buttons allow us to traverse the graph in the current viewing plane.
By turning our view 90 degrees, traversing left-right or up-down and turning again, we may explore regions displaced in 3-space from our original "X-Y" "flatland".
Using these controls, we can navigate the 3-dimensional graph at arbitrary detail with some idea of relative positions in 3-space. The network of "chambers" we view is generated on-the-fly. When you come to a chamber which has not been viewed before, the server program generates the image for the view you will see.
We rely on the finite bandwidth of the network to limit CPU load. The MUD Keeper daemon keeps an eye on disk capacity, ages and removes images which have not been visited recently.
The Oracle is an extension on Mathart Automatic Poetry. Warning: the oracle is uncensored.
The Wisdom of the last ten Pilgrims is stored in each chamber.
See Stewart Dickson's 3-D Fractal sculpture for direct 3-D sculpture of Quaternion Julia sets. We can do the Fractal Zoom in three physical dimensions.