My technical wording is bad. But, let me put the picture as clear as possible.
My idea is that if we can make magentic flux or field lines in a desired shape, we can use those field lines to align magnetic particles to create a shape. Lets say a cube. The field lines are in the shape of a cube and in such an environment, if we release iron granular particles, they might get arranged in 3D. If this is possible, we can make a new way of 3D printing.
Let me know if this is possible.
There are a few fundamental constraints for magnetic field lines. First, they have to close because there are no magnetic monopoles. Second, magnetic field lines, as electric field lines, can't cross. Apart from that you can use the Biot-Savart law to calculate the magnetic field created by an electric current flowing through an arbitrarily shaped wire. In fact this approach is used to calculate for example the complex magnetic fields in a stellarator to achieve plasma confinement. This sounds like a simple task because the physics of Biot-Savart's law is so old, but it is not at all simple. Basically one is interested in solving an inverse problem where you know what magnetic field you need to achieve something like plasma confinement or what you have in mind and then you ask how do I have to design the flow of the current which creates the required field. The inverse problem is not easy, but it can be done. You could certainly learn a lot from the nuclear fusion community interested in the confinement of hot plasma. They indeed use Biot-Savart. See for example Garabedian P R, PNAS 2008;105:13716-13719 .
Easy answer is 'yes'. Take a long solenoid with small diameter and deform it as you wish. The magnetic field directed along its original symmetry axis will follow the shape changes. This, however, has almost nothing to do with 3D printing. First, because the desired shape of field lines will be valid within the solenoid only, not outside it. Second, the magnetic field doesn't produce any force, if it is uniform. Magnetic bodies will only reorient themselves in such a field (rotate), not relocate. For relocation you need the magnetic field gradient. This is how magnetic traps are working (see, for example, 'magnetic trap' in Wikipedia). In addition: while it is relatively easy to compute what will happen with paramagnetic or diamagnetic particles, then the same task for ferromagnetic bodies is much more complicated (unless they are small enough to be in single-domain state, and this only happens within nanometer range, say below 30 nm). Anyway, ferromagnetic particles making a cube will not last long, they will strongly repel each other.
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