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I'm probably the only person in the world that makes rehearsal cubes like this (if I'm not, that's cool)  So, thought I would share if this is of some use to others.

I use 1/4 inch Luan to create a sort of a panelized system along with a knot free piece of fir or yellow pine ripped down to 3/4" by 3/4".  Those then become the legs for a plywood top which is inserted between the Luan sides and rests on the 3/4" legs.  They are light and easy to carry.  If the cube is 18 by 18 (inches) or less, I usually just use 1/2 inch cabinet grade plywood for the top.  (BTW)

Here are some photos.  

Let me know if there is some interest and I will make a mock up and share the process.  

Best

Seeing your process with photos would definitely be fascinating-count me in if you decide to mock it up and share!

Michael: these look great - I'd love to see how. you make them. also, how much weight can they support? Can a student stand on them? 

I've used this particular method for about 20 years or so with multiple students standing, sitting or weight sharing while using the boxes for levels.  I don't think these are suitable for dance, however, as the design does not respond well to lots of lateral force.

I'll make a model or sample cube here in the next couple of weeks and share back.

For dance-specific needs, though, you're right about lateral movements being crucial. You might want to consider reinforcing the joints or exploring materials that can handle more dynamic shifts in weight.