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Comment author: Thomas 20 February 2017 08:46:00AM 1 point [-]

A math problem


This one is a real one, but somewhat transformed and potentially solvable.

Comment author: Oscar_Cunningham 13 February 2017 05:12:35PM *  2 points [-]

A tetrahedron of edge length e has volume e^3/sqrt(72). So the 600-cell has surface volume 600e^3/sqrt(72). Let r be the distance from the centre of the 600-cell to the centre of one of its boundary tetrahedra. If we increase r to r+epsilon, the content increases by a shell around the 600-cell of width epsilon. This shell therefore has content epsilon*600e^3/sqrt(72) (plus terms of order epsilon^2 and smaller). Therefore we can find the content of the whole 600-cell by integrating 600e^3/sqrt(72) from r=0 to whatever r is when e=1. It remains to calculate r in terms of e.

Suppose we have a vertex of a polyhedron where n angles of size theta meet symmetrically at a point. Then the angle between adjacent faces (the so called "dihedral angle") is given by 2arcsin(cos(pi/n)/cos(theta/2)). This isn't too hard to see if you draw a little diagram. Thus the angle between two faces in a tetrahedron is 2arcsin(cos(pi/3)/cos(pi/6)). Since five tetrahedra meet at each edge of the 600-cell we can calculate the angle between two tetrahedra to be 2arcsin(cos(pi/5)/cos(theta/2)) where theta is the angle we already calculated between two triangles in a tetrahedron. Call this new angle phi. In a tetrahedron of edge length e the distance from the centre to the centre of a face can be seen to be e/sqrt(24). By considering the triangle formed by the centre of a 600-cell, the centre of one of its boundary tetrahedra, and the centre of one of the tetrahedron's faces, we can therefore see that r = (e/sqrt(24))*tan(phi/2).

After much calculation, all of the trigonometry cancels and we have r=e(sqrt(5/8)+sqrt(3/2)). Then doing the integration we get the content to be (25/4)(sqrt(5)+2)e^4.

Comment author: Thomas 13 February 2017 07:54:47PM *  0 points [-]

Well done, I think.

When I saw this site I have linked, I thought what a shame that the bulk of the cell-600 is "unknown". This would be a perfect problem for those clever guys who solved some problems already!

In a sense, I wasn't wrong.

Comment author: gjm 13 February 2017 12:43:42PM 0 points [-]

Vs lbh glcr "ibyhzr bs 600-pryy" vagb Jbysenz Nycun, gur nafjre V nffhzr lbh'er ybbxvat sbe pbzrf fgenvtug bhg.

Comment author: Thomas 13 February 2017 01:04:54PM 0 points [-]

This was quick, if true.

How does Wolfram Alpha knows that? And since when?


Those guys should update, and also should Wikipedia.

Comment author: Thomas 13 February 2017 11:22:04AM 1 point [-]
Comment author: James_Miller 09 February 2017 04:22:09PM *  6 points [-]

The podcast, part of Carlin's excellent Hardcore History series, is called "The Destroyer of Worlds". The podcast has convinced me that Truman was a horrible president. After the United States had a monopoly of atomic weapons our two sane courses of action would have been to either maintain this monopoly by threatening to go to war if another nation developed atomic weapons, or to have made an all out push for peace with the Soviet Union to avoid a future arms race. Instead, Truman used the monopoly to engage in short-term bullying of the Soviets, while doing nothing to hinder their development of atomic weapons thus guaranteeing that they would eventually have thousands of atomic weapons aimed at us. I bet in most branches of the multiverse arising out of 1953, millions of Americans die in nuclear war by 2017.

Comment author: Thomas 09 February 2017 07:00:41PM 0 points [-]

I bet in most branches of the multiverse arising out of 1953, millions of Americans die in nuclear war by 2017.

Oh, come on! This fairy tale about parallel worlds and WWIII in most of them is pretty lame.

Comment author: MrMind 09 February 2017 09:26:14AM 1 point [-]

Well, the whole point of this forum is to convince someone that the answer is most definitely not.

Comment author: Thomas 09 February 2017 09:33:26AM 0 points [-]

I know that. But the whole point of this thread is to ask stupid questions, isn't it?

And sometimes apparently the stupidest question, isn't stupid after all.

Comment author: madhatter 09 February 2017 01:35:21AM 1 point [-]

Thanks for this topic! Stupid questions are my specialty, for better or worse.

1) Isn't cryonics extremely selfish? I mean, couldn't the money spent on cryopreserving oneself be better spend on, say, AI safety research?

2) Would the human race be eradicated if there is a worst-possible-scenario nuclear incident? Or merely a lot of people?

3) Is the study linking nut consumption to longevity found in the link below convincing?


And if so, is it worth a lot of effort promoting nut consumption in moderation?

Comment author: Thomas 09 February 2017 08:45:48AM 1 point [-]

couldn't the money spent on cryopreserving oneself be better spend on, say, AI safety research?

Here comes another "stupid question" from this one.

Couldn't the money spent on AI safety research be better spend on, say, AI research?

Comment author: garabik 07 February 2017 12:08:40PM 0 points [-]

Or two ordinary small balls, one dropped from just above the geosynchronous orbit, the second one from far above the orbit. While the first one slowly drifts away to the space, the second shoots away, makes a complete (retrograde) orbit around Sun and splashes into the Atlantic while the first ball is still drifting...

Requires some careful timing, though.

Comment author: Thomas 07 February 2017 12:57:23PM *  0 points [-]

This is true, but those Moon or Sun solutions aren't my favorite. Moon, Sun, Jupiter and so on are external agents I've forgotten to explicitly forbid. Next time, I'll be even more careful when posting a problem. :-)

Comment author: garabik 06 February 2017 10:46:18PM 0 points [-]

If their gravity is significant enough, then it is incorrect to describe that they splash into the Atlantic - it's the Atlantic that splashes into them.

I'd prefer solutions that do not destroy the Earth :-)

Comment author: Thomas 07 February 2017 10:48:41AM 0 points [-]

Two giant golden balls, dropped somewhere bellow the geosynchronous orbit might do the trick of a little orbiting around each other and then splashing into the ocean, one after another.

That might cause some damage, but the Earth would survive as a planet. Rather costly and not environment friendly solution.

Comment author: Lumifer 07 February 2017 02:40:08AM *  1 point [-]

Actually, I think I've come up with a more elegant idea than altitude-triggered airbraking. There is no requirement that when released they go down into the gravity well :-D

Your two objects are leaky zeppelins.

Comment author: Thomas 07 February 2017 10:42:47AM 0 points [-]

Helium Zeppelins aren't exactly rigid bodies. But vacuum Zeppelins are. Those could be arranged to do the job.

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