Would an octahedral magnet work as a magnetic bottle? First, let me see if I understand all this:
Charged particles are curved by magnetic fields, so if the magnetic field is strong enough, the particles "follow" the magnetic field lines, in N or S direction, by spiraling around them:
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If the magnetic field lines get closer together, it can create a magnetic mirror, which affects the rotation of the particles such that they turn back and go the opposite way:
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If you combine 2 magnetic mirrors, you get a magnetic bottle, which can confine charged particles by both spiraling and reflection off the mirrors:
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So my question is whether it's possible to combine multiple magnetic bottles/mirrors into other, more "spherical" configurations, and still have it confine particles in the center, such as an octahedral magnet (brown arrows are current, magnets show field direction):
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or a cuboctahedral magnet, or other shapes that can be painted with 2 colors, so the field direction alternates from one face to the next (to avoid trying to create a magnetic monopole field, which obviously wouldn't work).
 A: I have been writing, researching and explaining the Polywell for five years. Here is my blog: http://thepolywellblog.blogspot.com/. I do not know everything. Endolith and I have been communicating for a couple of months about his design. I think he wanted an analysis of the rings, rather than just a yes or no. Communication can be tricky – so I am going to stick to what I found in the WB6 design. This machine evolved from experiments – so I trust that it works.
If you look at WB6 you can see several principals. These are likely goals we want to aim for any ring design in order to get electron containment. Here are some:


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*Uniform strength in the magnetic field.

*Uniform symmetry in the magnetic field.

*Smooth rings, allowing for plasma recirculation and reduction of arching.

*Electrically insulated or “magnetically transparent” materials elsewhere.

*Reducing plasma counter streaming.


Uniform field strength & symmetry
The rings are a symmetric structure. It has a center, 8 corners, 6 sides and 8 joints. We know in the center there is no magnetic field. This is the null point. Electrons or ions move straight through this region. We know this from theory, papers and simulations (http://www.youtube.com/watch?v=ao0Erhsnor4). This straight motion hurts electron containment – overtime, it will scatter the electrons.
If you want to find the magnetic field you apply the Biot-Savart law at each point (joint, corner, axis). I developed simple equations for the field at each point.  These were integrate into MATLAB and EXCEL code, as well as compared against publications.  I applied these equations to the WB6 design. I then moved the rings outward. I recalculated the field as the rings moves. I plotted the results and they were surprising. WB6 was designed so the field at the axis and corner were close, if not the same. I would bet that ideally, we would want identical field strength, if it were physically possible.
Bottom line: if your an electron in the center and you move any direction outward, you should encounter a field uniform in strength and symmetric. It analogous to an above ground pool - the walls must be uniform in height, otherwise water leaks out. The design must strive for uniform magnetic strength and direction, otherwise more electrons will leak out.
Smooth Rings
Rider, Nevins, Dolan and Bussard all talk about recirculation.  It was recirculation which prompted Endoliths design.  Recirculation means encouraging material to move back and forth, or in and out of the ring structure, without hitting something.  That is one reason why I am in favor of an open design with no sharp edges or corners.  This might sound simple, but both Convergent Scientifics Inc. patent and the Iranian 2011 publication ignores this rule.
We have seen free recirculation of plasma as a key fusion principal since John Lawson wrote his seminal work in the 1950's.  This is one of the reasons the Tokamak machine uses it.  Lawson wrote a key energy balance equation for fusion power: 
Power = (Fusion - radiation loss - conduction loss) * Machine efficency
Conduction and recirculation are opposites.  Conduction loss is when ions or electrons touch a surface and are lost.  They take their energy with them away from the machine.  We need to minimize conduction loss.  This means rings which are smooth.  It means fields which are curved into circles and do not run into surfaces.  Curved fields are not perfect, plasma drifts outward from the center tracks to the edges over time.  Tokamaks suffer from this "drift velocity" as well.  But its where we need to go with ring design.  The rings must also be smooth to avoid arching.  If their is a bump, charge can buildup here.  This can lead to sparks between surfaces. 
Electrically insulated or “magnetically transparent” materials
In practice, the rings must be held with struts, connectors and legs.  These parts hurt the ideal design, but cannot be avoided.  Here, we must choose materials which avoid hurting operation.  "Magnetically transparent" materials are ones which have a magnetic permeability of ~1.  This means magnetically, they act like vacuum.  A good example is Teflon.  Teflon was used by Mark Suppes and the Sydney team for their ring design. Pure Teflon has an extremely low electrical conductivity. This will help against arching.  The material is also cheap and easily machinable.  It has some problems with out-gassing in a vacuum, there maybe lots of pockets of gas inside which need removal.  Bussard also described electrically insulated surfaces.  This was mainly to eliminate arching and conduction loss.  There was arching in WB6 tests. 
counter streaming
When charged particles move past one another in streams, it can lead to instabilities.  Marshall Rosenbluth developed most of the understanding on this topic.  He modeled (+) and (-) streams of particles passing one another, and showed that the streams devolve into a disorganized mess.  The rings try to mitigate this problem by point the field in through the axis and out everywhere else.  Fields pointing in opposite directions never run parallel to one another.  This will not stop counter streaming or plasma instabilities in general from being an issue.  I expect researchers will need to deal with this going forward.
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I see the octahedron as having two problems. One, electrons will stream past one another, in opposite directions.  That could lead to problems.  Two, the magnetic fields are not symmetric.
