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Prince Rupert's cube

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A video is worth a thousand words of theory:

https://www.youtube.com/watch?v=-2jjgHsxEu4

That was very good (thank you!), but it led me to this, which was even better:

https://www.youtube.com/watch?v=rAHcZGjKVvg

I wish more of mathematics was visual. That's the mode of thinking I employ the most, and I excel at spatial problems.

I wish higher level mathematics and physics paid more attention to the use of visual schematics and diagrams. So much can be communicated with them. Words often pale in comparison.

I would like to recommend the 3b1b's[1] YouTube channel.

His procedurally generated videos really make you think of certain math that looked unintuitive or hard to explain in a truly intuitive, visual way.

His series on linear algebra was very eye opening to me, as well as many of his videos about subjects I was initially failing to understand.

[1] https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw

Thanks so much for sharing! This looks fantastic.

drop https://m.youtube.com/watch?v=lt-zvsGvtqg

After watching the video, I feel like this follows intuitively from the fact that sqrt(2) > 1.

The biggest possible difference between Prince Rupert's cubes is 3%, which is much smaller than the difference between sqrt(2) and 1.

FWIW, `(sqrt(2)-1)^3` is `0.071067831`, which while not equal to `(1.0606601-1)` is fairly close.

Another nice puzzle:

A cube looks like a square from three orthogonal directions. A cylinder can look like a square from infinitely many directions, but they are all coplanar. Can you find a convex shape that looks like a square from more than three directions, without all of them being coplanar? In particular, can you find a convex shape that looks like a square from two distinct sets of three orthogonal directions? Can you find all such shapes?

(Rot13 as not to spoil the riddle)

Ubj nobhg n grgenurqeba? Begubtencuvp cebwrpgvba nybat gur yvar pbaarpgvat nal rqtr zvqcbvag gb gur bccbfvgr rqtr zvqcbvag tvirf n fdhner, lvryqvat 3 fhpu cebwrpgvbaf (cyhf gurve bccbfvgr pbhagrecnegf). Abg fher vs gurer ner bgure fhpu funcrf (vtabevat gevivny zbqvsvpngvbaf bs gur grgenurqeba) - qvq lbh znantr gb svther guvf cneg bhg?

A tetrahedron looks like a square from three orthogonal directions, same as a cube. It's possible to get more.

I'm thoroughly impressed by your geometric abilities! I didn't know that, and took me a while to check. Any hints on the puzzle, and as a sidequestion, what tools do you use to figure this kind of question? Just imagination, vector algebra, elementary trigonometry?

Huh? There's nothing to be impressed about. You can stick a tetrahedron inside a cube so it creates the same square shadows in all three directions: https://i.stack.imgur.com/oAUnH.gif

I know a lot of math, but for this puzzle, drawing stuff on paper is enough. Here's a hint: if you cut off one corner of the cube, all shadows are still square. How much can you cut? Can you cut some corners strategically to make at least one new square shadow while keeping all the old ones? How many square shadows can you get?

Do mathematicians consider this an open problem or does there exist a well known solution? And what about a name for this problem, does it have one already?

Nah, it's nothing as grand as that. I saw it on facebook a few months ago and solved it in about ten minutes, just wanted to share the fun.

...do you know the story of George Dantzig?

https://en.wikipedia.org/wiki/George_Dantzig#Mathematical_st...

I don't mean to say that it seems applicable here - but I also don't mean to say that it doesn't seem applicable here.

Orthographic or perspective projection?

I only know the solution for orthographic. But it seems like perspective would be too easy, just make any shape with many square sides, and put the cameras close to the sides.

Cylinders only look like squares in orthographic projection, so my guess is that that's the intention.

Am I allowed to invoke hypercubes because that would cut this knot nicely.

An octahedron?

An octahedron looks like a square from three orthogonal directions, same as a cube. It's possible to get more.

Is the question posed only in relation to three dimensional space?

Yeah, it's the kind of shape you can carve from a block of cheese in a minute.

Another interesting thing named after Prince Rupert is the Prince Rupert's drop for anyone interested: https://www.youtube.com/watch?v=k5MORochIDw

I recommend trying them with capillary tubes, you can melt them with a cigarette lighter

I knew of Prince Rupert’s drop before[1]. Reading about Prince Rupert’s cube sent me down the rabbit-hole of reading up on Prince Rupert himself.

Why do we no longer have Renaissance men/women today, contributing to the sciences, philosophy and the arts? What did we lose?

[1] https://en.m.wikipedia.org/wiki/Prince_Rupert%27s_Drop

> Why do we no longer have Renaissance men/women today, contributing to the sciences, philosophy and the arts? What did we lose?

A lot of the low-hanging fruit has already been claimed, and it's very difficult now for amateurs to make discoveries in mathematics and physics. There are people today doing amazing research, it just takes a career and a team (and in many cases expensive equipment) to do so.

If by "Renaissance men/women" you mean "well-versed/engaged in many topics", well, you just need to open your eyes. There are many folks like this now, likely millions. It was a bit more of a rarity a few hundred years ago, so it was much easier for these individuals to be documented/preserved.

The more complex (advanced) a society becomes, the more specialized its individual members have to become so the longer it takes to produce a unit for a specific function (basic education of people now takes 18 years of their life). With the exponential increase in scientific output (http://pespmc1.vub.ac.be/POS/turchFigs/IMG.FIG14.1.GIF) it becomes ever harder for an individual to keep up with multiple fields (in fact, I suspect that as soon as a field becomes _too_ big it automatically branches into new fields because of this).

It would have been possible to read every book in existence on mathematics in a lifetime in the 17th century, but not in the 21st century.

I'd wager that there weren't all that many renaissance men/women back in the day either, but some of the ones that were there are now well known.

In any case, there are plenty of people contributing in a wide array of fields who are alive today. How about Franklin Story Musgraves, who has academic degrees in six different fields, including medicine, computer programming and literature. He also went into space for NASA no less than six times, served in numerous conflicts while he was in the air force, was briefly employed as a mathematician for Kodak, practiced and taught clinical medicine and is/was a consultant for Disney. I found him with the briefest of google searches for "modern renaissance men" :)

https://en.wikipedia.org/wiki/Story_Musgrave

Interesting that the optimal solution is not "slide it along the space diagonal" but rather "slide it parallel to a face".

I might be wrong about this, but as far as I can see the larger cube slides parallel to an edge and not a face. On top of that, I think the cube passing through would still be sliding parallel to an edge, even if it were to slide along a space diagonal, so those two wouldn't be mutually exclusive. Unless I have failed to visualize it right ok my head.

Mathologer on Rupert's cube: https://www.youtube.com/watch?v=rAHcZGjKVvg

looks like a fun model to 3d-print

So, cubes have a margin of 6% play, if you need to pass any other cube through a given cube.

Wow, a downvote for paraphrasing a wikipedia article.

  Its side length is approximately 6% larger than 
  that of the unit cube through which it passes.

I fail to see any error, based on that verbatim quote.

Good job. Keep downvoting.

I reckon you could stretch this to a 30 minute whiteboard session to answer why (some) manhole covers are round.

And then fall in the Reuleaux triangle rabbit hole.

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