It makes me wonder if there's a better shape than a cube for this. You'd want to be non-circular to walk up things, but you'd want a circular edge for rolling down them. I'm imagining something like two hoops forming a sphere: rotate 90 degrees to switch between roll mode and walk mode.

That’s an interesting idea. I have no good answer to your question, but something like a dodecahedron comes to mind, as a ‘trade-off’ between a cube and a sphere.

If the edges were all made as small arcs with extra actuators to rotate them either flat along the surface or outwards to form something that can roll. Might need to do the same for diagonals to get it to roll smoothly?

I like the actuated edges idea. I'm imagining it using an edge to hook the top of a step, pull itself half way up with edge actuation, and then building up momentum and slamming a brake to kick itself onto the stair.

In middle school I imagined a toy that would be a top with a similar mechanism as this cube. It could spin it's outer shell and then tilt and engage a "wheel" hoop, just a bit above the the tops tip, exchange some of of it's angular momentum for linear, right itself and build up angular momentum again. This would enable a fully enclosed, no external moving parts RC vehicle.

It seems such a thing is possible now.

Knowing and seeing how it works takes away a lot of the magic, but imagine finding a completely featureless whirring metal cube... that then jumps up onto an edge, then onto a corner, and starts spinning. And is capable of doing this on any random surface.

I bet that would puzzle even most people who consider themselves familiar with physics and when the cube suddenly jumps back from a corner onto a face, clearly showing that the jumping isn't just some hidden mechanism that extends, make some seriously consider the possibility of magic.

Amazing. Now it just needs a nice wood enclosure and matching induction charging pedestal. Kickstarter please take my money!