Apparently Disney Shorts are not for everybody. I live in Romania, and Disney has apparently decided we're not good/rich/important/honest enough over here to bless us with this clip -- or, in fact, any of the other Disney Shorts. It's a money-driven world, after all.
"The result of his work over 20 years is that his footprints are now marked an inch and a half deep into the wood." Agreed. Even as an art project, if you only had that to show after 20 years of work, it's still way underwhelming.
I disagree. The album was certainly a live performance recorded directly onto the medium that's being showcased. Instead, they recorded the voices using some digital equipment, post-processed it using some other digital equipment, and then copied the final result onto the medium that's being showcased, after which they supposedly destroyed the digital originals. As such, I think the original wording is appropriate: there is a single copy (of the original recording) out there.
Fair enough, I wasn't deploring the lack of, or advocating for knowledge regarding triangular numbers among the masses -- I was simply observing a (neat?) fact.
Yes, well, you're actually proving my point. If I mentioned square numbers in a conversation you wouldn't think to point out that n * n is used specifically in Einstein's famous E = mc^2 equation, or whatever other use you heard for square numbers -- they're so widely used today that it would seem silly for someone to do that.
Triangular numbers on the other hand are so seldom used that you immediately remembered exactly where you used them before. I, for one, remember them for being used to calculate the sum of all natural numbers from 1 to n; others might remember them for being the number of handshakes in a group, the number of matches played in a simplistic tournament (as above) or the number of edges in a complete graph (as you do) -- these last three are all the same thing, by the way. But the point is you wouldn't think to say anything like this regarding square numbers, specifically because of their ubiquity.
This is one of the precious few classes of problems that would've been more easily solved by ancient knowledgeable men than their modern counterparts: we rarely hear mention of triangular numbers, whereas the ancients held them at equal rank with the square numbers we're so comfortable with today. If I asked you what's the side of a square which holds 100 unit squares you'd offer the answer instantly; the problem above, however, seems a lot more counter-intuitive.
This is not as original as you might think -- it's probably inspired by the Invis Mx, a commercial product that uses the same principle to assemble screws that are completely inaccessible and invisible from the outside.
Battle Damage. M Chocolate.
Triangular numbers on the other hand are so seldom used that you immediately remembered exactly where you used them before. I, for one, remember them for being used to calculate the sum of all natural numbers from 1 to n; others might remember them for being the number of handshakes in a group, the number of matches played in a simplistic tournament (as above) or the number of edges in a complete graph (as you do) -- these last three are all the same thing, by the way. But the point is you wouldn't think to say anything like this regarding square numbers, specifically because of their ubiquity.