(forw) Re: paradox
- From: Neil Doane <caine@xxxxxxxxxxxxxxxx>
- To: technocracy@xxxxxxxxxxxxxxxxx
- Date: Tue, 6 Jun 2000 17:50:01 -0700
Oh, guess I could send this to the group as well...
Neil
* Jerry Hargis (CCHARGIS@xxxxxxxxxxxxxxxxxx) on [06-06-00 13:03] did utter:
> My point is that I don't think that there is anything odd going on here.
> They are two different solids, with different perimeters. When I looked
> at the pictures, I couldn't articulate what it was about them that made
> it seem odd. Does it appear that they have different areas?
>
> The best analogy I could think of is what happens when I unpack a box.
> I can never get the original contents back into the box and close the
> cover. The dimensions of the contents haven't changed, but they seem to
> be taking up more space.
>
> jwh
Well, no. It took a math Ph.D. to explain it to about six of us who were
sitting around here, so don't feel bad. :)
The quandry: look at it this way. The area of a solid triangle is (length x
width)/2, right? Let's say each of the squares are a 1'x1' square. The
total area of the pieces of the first triangle is 13'x5'/2, or 32.5sq.ft. What
then is the area of the second 'triangle'? Well, it's a full 1 square foot
less, as it is the same triangle as above with a one square foot chunk taken
from it, so 31.5 sq.ft....yet it is made up of the exact same pieces. (?!) Hmm.
Let's look at each piece in both the first and second triangles.
1st Diagram:
Red Triangle is ((8*3)/2), or 12
Green Triangle is ((5*2)/2), or 5
Orange blob is easy, just add quares, or 7
Same for Green blob, or 8
-------------------------total--------------
uh oh, we can stop right here actually-> 32
2nd Diagram:
Red Triangle is ((8*3)/2), or 12
Green Triangle is ((5*2)/2), or 5
Orange blob is easy, just add quares, or 7
Same for Green blob, or 8
--------------------------------------------
32
The pieces take up exactly the same space. But the big red flag is that the
sum of the pieces DON'T add up to the total area of the first
triangle. The error?
Red triangle's radio: 8/3, green triangle's radio: 5/2 5/2!=8/3, therefore
the "big" triangle really isn't a triangle at all, the hypotenuse isn't
straight! On the first picture, it bows downward due to the angle created
when the two distinct hypotenuses (hypoteni? I can't spell) intersect, on
the second triangle, it bows outward. The area bounded by these lines
(imagine a long, thin diamond shape sitting on the hypotenuse of the top
diagram), adds up to 1 square unit...WERD.
Neil
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