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Comment by cluckindan | Hacker News Reader

A sphere can be approximated by a polyhedron. Somewhat obviously, all such polyhedra would seem to have the Rupert property. This new Nopert seems to differ in one key detail: some of the vertices near the flat top/bottom are at a shallower angle to the vertical axis than the vertices below/above them.

Can you pass the T-shaped tetromino through itself?

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The T-shaped tetromino is not convex, so not part of the conjecture. There are many nonconvex shapes that don't have the Rupert property.

Nevertheless, the t-shaped tetronimo (assuming four glued cubes) has a shadow shaped like a bar of length two. I believe that such a shadow will pass through a bar of length three, with a tilt similar to the cube's.

If the remaining edge has exactly zero thickness , it means it doesn’t fit. I think that would be the case in that example?

It fits if it is tilted. If the 3-cube bar of the T is tilted 45° there will be a 3×√2 rectangle part of the shadow, which the 1×2 shadow fits through.

Somewhat unrelated question, a lot of the folks replying to the parent comment read to me like they're really good at visualizing things in their minds eye, when you talk like this is it because you can think about math really well? Can you visualize what you're saying? Sorry if this question doesn't make sense!

I do not have a mind's eye*, outside of the dimmest, briefest flash of people's faces if I've known them for several years. I do have a peculiarly strong sense of imaginary touch, which gets used when contemplating problems like these. Also, a significant component of my job is arranging things in 3-space. I'd be one of those people who say "I'm good at tetris" while packing a moving truck, but I am not actually good at tetris per se.

* https://en.wikipedia.org/wiki/Aphantasia

Thank you for your reply. Extremely interesting to me. My thinking and memory is pure motion picture and sound, it makes me think that I could be good at geometry as I can do spatial thinking well, however the downside of my thinking style is I've never been able to find a framework that allows me to hold fine detail symbols in space and work with them usefully, I suppose hence dyslexia + dyscalculia diagnosis. Maybe people like me who are in math use whiteboards and notes a lot or something? Maybe people like me don't go into maths so much. Imagination of touch is also very interesting, I've read about kinetic leaners before, I like to touch things when I'm learning as it helps with the recall later, but absolutely zero sense of touch is present in the recall. Sorry for the massive tangent, I'm just very curious about this stuff!

I'm a mathematician, so I can think about the maths well, and I figured out that particular maths by visualising the shapes and seeing the dimensions and the way the shadows fit together.

Cool. I appreciate your reply. If by chance you happened to do any art of any type by chance, I would be extremely interested in viewing it. :)

I don't think that's the case, how can you turn a bar of length two but still have it fit within the width of the bar of length three?

The long side is three cubes long, the short side two. You can easily move the short side through the long side's shadow if you tilt the latter so it becomes wider than a cube's side.

You are of course right, thanks.

That’s true. I digress, but you could give tetrominos convex hulls, and the result would still be somewhat Tetris-compatible.

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