The Basic Plot
- Solve corners
- Solve centers
- Solve edges
To begin with, find one set of 4 corners where all 4 colours are different. This set of 4 corners is like your solved white centers on the 4x4x4 cube. It doesn't matter which 4 colours, either.
Next, opposite those 4 corners, place the other 4 colours. Do this just as for solving the yellow centers on a 4x4x4 cube. This involves simple moves.
Once those are done, hold the puzzle with those two sets of corners on the left and right. Work your way around the other sets of corners, placing new corners with EPS and similar moves.
It's all very simple, and this video will show how.
If you find yourself needing to swap the last two corners, simply carry out a corner piece series turning the outer layer for the upper face turns, but the inner layers for the left and right face turns. This will cycle 3 corner pieces and leave everything else intact.
Step 2: Solve Centers
The centers are the easiest and shortest stage of the solve. Simply carry out corner piece series using all outer layers.
Help?! I Need To Swap Two Centers!
If you encounter the situation of needing to swap two centers, this means you need to turn an outer face one turn. Then, re-solve the corner pieces. Finally, re-solve the centers, which will now be in a 3-cycle, rather than a swap. This video will make clear all of the above, including a center swap.
Step 3: Solve Edges
The last stage of the solve is to place the edges. We'll use a pure edge 3-cycle which is an adaptation of the corner piece series.
This video will show exactly how to solve the edges.
And finally, a video showing the endgame.
And that's it. Your 4x4x4 Octahedron is now solved. I trust this site has been helpful. If you have any questions or want some clarifications, please use the comments to do so.