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Delian problem: doubling the cube

 
 
Arjuna
 
Reply Fri 16 Jul, 2010 12:12 am
So the story goes that there was a plague in Athens. Somebody went to the Oracle and asked for the solution. The answer came: double the size of Apollo's temple at Delos. So they went out and doubled the length of each wall of the cube shaped temple.

The plague persisted. They went back to the Oracle and asked "what the heck?" The answer came back: "We told you to double the size of the temple. You didn't do it.... go do it." It was then that they realized that they hadn't doubled the volume of the temple... their efforts had actually multiplied the volume by 8.

They set out to solve the problem, only to find that they didn't know how. The answer required expanding the walls by the cube root of twice the original volume. This gave them what's called an unconstructable number. Which is to say... they couldn't use a compass and straight edge to arrive at it. The problem was given to Plato's Academy. Plato wasn't much of a mathematician, but he was known as a "maker of mathematicians." They guy who had come pretty close to calculating the circumference of the earth was a member of the Academy. And actually, he was one of the folks who worked on doubling the cube. They all failed. It's not possible with a compass, a straight-edge, and geometry. The closest they could come was a mechanical solution to the problem, which some believed irked Plato.

Descartes was pondering this problem and similarly unsolved problems from ancient Greece when he invented analytic geometry.

This story makes me wonder how much of science appears in this way... starting with a bizarre mystical agenda.
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MontereyJack
 
  2  
Reply Fri 16 Jul, 2010 12:57 am
I think you'd better add some more parameters here, since as it stands now, there's a simple solution. The way you've got it worded, all you need to do is build another cube of the same size alongside the first one and you've doubled the size--nowhere does the oracle, or the puzzle, say it has to be the same shape but bigger, only that it has to be twice the size.
engineer
 
  2  
Reply Fri 16 Jul, 2010 06:14 am
@Arjuna,
Not that it's really pertinent to the story, but Eratosthenes was the guy who calculated the circumference of the Earth (276 BC – c. 195 BC). Plato (428/427 BC – 348/347 BC) was significantly before his time.
Arjuna
 
  1  
Reply Fri 16 Jul, 2010 10:02 am
@engineer,
engineer wrote:

Not that it's really pertinent to the story, but Eratosthenes was the guy who calculated the circumference of the Earth (276 BC – c. 195 BC). Plato (428/427 BC – 348/347 BC) was significantly before his time.
You're right. The book I'm reading is wrong. Apparently Eratosthenes knew about the problem, but couldn't have known Plato. It's also possible that the story is fiction.
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Arjuna
 
  1  
Reply Fri 16 Jul, 2010 10:05 am
@MontereyJack,
MontereyJack wrote:

I think you'd better add some more parameters here, since as it stands now, there's a simple solution. The way you've got it worded, all you need to do is build another cube of the same size alongside the first one and you've doubled the size--nowhere does the oracle, or the puzzle, say it has to be the same shape but bigger, only that it has to be twice the size.
Yea, the notion is the temple had to remain cube shaped. I guess the problem isn't as much about the temple as it is a geometry problem.
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Jebediah
 
  1  
Reply Fri 16 Jul, 2010 10:36 am
Isn't it a math problem? Assuming they are doubling the volume:

4*4*4=64

x*x*x=128

x= ~5.04


Whoops, I should have read more carefully. Yeah, I don't remember the operation you have to do to find it. I just ran it through a calculator a few times to get it approximate :p

Maybe the real lesson is that approximating is ok.
Arjuna
 
  1  
Reply Fri 16 Jul, 2010 12:07 pm
@Jebediah,
Jebediah wrote:

Maybe the real lesson is that approximating is ok.
I think the idea is that approximating doesn't give you perfection. I'm thinking a temple to Apollo would especially need to be perfect because of what that god symbolized.

Even if the story is baloney, it's still poignant: because people's lives were at stake.... mathematicians to the rescue. OK, poignant and crazy at the same time.

But as we ponder exactly what kind of bridge connects the mind to the physical world... science actually is involved in saving people's lives.
DrewDad
 
  1  
Reply Fri 16 Jul, 2010 12:41 pm
@Arjuna,
http://en.wikipedia.org/wiki/Doubling_the_cube

Quote:
The answer seemed strange to the Delians and they consulted Plato, who was able to interpret the oracle as the mathematical problem of doubling the volume of a given cube, thus explaining the oracle as the advice of Apollo for the citizens of Delos to occupy themselves with the study of geometry and mathematics in order to calm down their passions.[4]
DrewDad
 
  1  
Reply Fri 16 Jul, 2010 12:44 pm
@DrewDad,
http://en.wikipedia.org/wiki/Cissoid_of_Diocles

http://en.wikipedia.org/wiki/Philo_line
Jebediah
 
  1  
Reply Fri 16 Jul, 2010 12:51 pm
@Arjuna,
Arjuna wrote:

Jebediah wrote:

Maybe the real lesson is that approximating is ok.
I think the idea is that approximating doesn't give you perfection. I'm thinking a temple to Apollo would especially need to be perfect because of what that god symbolized.


I think it does give you perfection, in the normal way we use perfection. As in, hanging a painting "left a little...right a little...perfect". Not idealist perfection but perfection none the less.
DrewDad
 
  1  
Reply Fri 16 Jul, 2010 12:56 pm
@Jebediah,
Mathematical perfection is different from hanging a picture, though.

The story of building a temple is just an attempt to illustrate the problem; they are not actually looking for a real-world, "good enough" solution.
0 Replies
 
Arjuna
 
  1  
Reply Fri 16 Jul, 2010 12:57 pm
@DrewDad,
Thanks! Cool huh?

Neither solves the constructibility problem, but you see algebra and geometry starting to come together.
Arjuna
 
  1  
Reply Fri 16 Jul, 2010 01:04 pm
@Jebediah,
Jebediah wrote:

I think it does give you perfection, in the normal way we use perfection. As in, hanging a painting "left a little...right a little...perfect". Not idealist perfection but perfection none the less.
With hanging pictures, you can "eyeball" it. Doing that you may actually make the position more perfect in terms of human perception than doing precise measurement. It's an oddity that precise measurements sometimes look wrong. I guess it has something to do with how vision works.

But yea, it occurred to me that the compass and straight edge wouldn't really create perfection. It's just the good feeling of traveling through the perfect forms of the mind on the way to constructing the temple... like that would bless the temple.
0 Replies
 
DrewDad
 
  1  
Reply Fri 16 Jul, 2010 01:27 pm
@Arjuna,
Quote:
In 1837, Pierre Wantzel showed the problem is unsolvable by compass and straightedge construction.


"Recherches sur les moyens de reconnaître si un Problème de Géométrie peut se résoudre avec la règle et le compas"
Arjuna
 
  1  
Reply Fri 16 Jul, 2010 03:53 pm
@DrewDad,
I'll take their word for it... I don't know French (though not for lack of studying it)

It's possible that the problem existed prior to Plato's time. What fascinates me is that it seems to defy necessity. Tying it to the proclamation of the Oracle doesn't explain why people would still be thinking about it centuries later.

It goes with what Democtritus said about things arising and then becoming useful. In this case, the thing was preserved for generations prior to being useful from our point of view.

And yet, all along, the problem wasn't perceived as a waste of time. It was believed that geometry and algebra were doorways to comprehension of the universe.

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