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