Understanding Geeky T-Shirt

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Some time ago, Henry Reich of MinutePhysics guest-starred on the Vsauce video "Guns in Space". At around the three-minute mark, he appeared on-screen, wearing a shirt with a bunch of symbols connected by arrows. I think I understand some of the connections:

1) The coffee mug becomes a donut because both shapes are homeomorphic.
2) The donut is next to the rectangle to illustrate the idea of a wraparound.

That's all I've got. Could someone please explain the rest?

My assumption was that the six steps show a progression of entropy.

Not likely. The pics dont seem to have anything to do with that.

Its some concept in geometry.
Dont quite grasp it, but it illustrates some progression in the number of dimensions, or in the number of sides of the figures.

The first thing is a coffee mug-which is a cylinder-which in turn is a circle moving along a straight axis.

The second thing is a doughnut-which is a torus-which in turn is a circle moving along a circular axis.

But then the doughnut becomes a flat rectangle. How that happens I dont get. Then that rectangle gets broken into two irreglular halves. And then those two halves become concentric rings imploding, and then finnally those imploding concentric rings reverse direction and EXplode.

I kinda get how each pair of side by side figures relate. But how each pair relates to the pair beneath is not obvious to me.

I think the first couple are right in the original post - the third figure illustrates a space that wraps on two axes, just as on a torus you end up back at the same spot going around the big loop, or around a "cross section".

But visually, the third figure is also a rhombus.

The split into two parts is not arbitrary, I think - splitting a rhombus in a particular way is used to form Penrose tiles. A Penrose tiling is a way to tile a space, with a fixed number of tile shapes, yet the pattern of the tiling never repeats itself (Wikipedia shows it well). Which is what the fifth figure seems to show.

Can't quite make out the last figure - it may be illustrating the kinds of rotational symettry in the Penrose tiling, but I can't see it clearly enough to be at all sure about that.