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   Heronian dissection: square
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   Author  Topic: Heronian dissection: square  (Read 1658 times)
JocK
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Heronian dissection: square  
« on: Aug 19th, 2005, 4:39pm »
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Can you dissect a square into as few as possible Heronian triangles?
 
 
 
 
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solving abstract problems is like sex: it may occasionally have some practical use, but that is not why we do it.

xy - y = x5 - y4 - y3 = 20; x>0, y>0.
SMQ
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Re: Heronian dissection: square  
« Reply #1 on: Aug 20th, 2005, 6:56am »
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I see a simple solution with eight 3-4-5 right triangles of two different sizes, so that's definitely an upper bound.
 
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Oyibo
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Re: Heronian dissection: square  
« Reply #2 on: Aug 20th, 2005, 10:30am »
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what the heck, for all practical purposes two triangles with edges 927538920, 927538921, 1311738121 will do.  That's definitely a lower bound. Roll Eyes
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Oyibo
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Re: Heronian dissection: square  
« Reply #3 on: Aug 20th, 2005, 10:48am »
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can't find an integer solution to
 
(2a)2 + (a + b)2 = c2
 
(2a)2 + (a - b)2 = d2
 
(0 < b < a)
 
if there is, you can cut a square in three heronian triangles.
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Barukh
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Re: Heronian dissection: square  
« Reply #4 on: Aug 26th, 2005, 6:14am »
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Oyibo, my sources tell that the system doesn't have non-trivial solutions.  Sad
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Sjoerd Job Postmus
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Re: Heronian dissection: square  
« Reply #5 on: Aug 29th, 2005, 1:18am »
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I'd like to state that two is impossible, because the hypotenuse would be sqrt(2) times the length of one of the other side. sqrt(2) isn't a finite number, as far as I know.
 
A rectangle? Could do... A square? Nope, not in two.
 
So, we're looking at a minimum of three.
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Oyibo
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Re: Heronian dissection: square  
« Reply #6 on: Aug 29th, 2005, 5:10am »
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on Aug 29th, 2005, 1:18am, Sjoerd Job Postmus wrote:

 
So, we're looking at a minimum of three.

 
We are looking at a minimum of four (if Barukh is right).
 
Barukh, what makes you so sure that the zero solution is the only integer solution?
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Grimbal
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Re: Heronian dissection: square  
« Reply #7 on: Aug 29th, 2005, 5:57am »
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For what it is worth, I have checked them by program.  The first "solution" it found was around a=29'000 and was due to a 32-bit integer overflow.
 
It is not a proof, but it convinces me.
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Barukh
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Re: Heronian dissection: square  
« Reply #8 on: Aug 30th, 2005, 3:50am »
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on Aug 29th, 2005, 5:10am, Oyibo wrote:
Barukh, what makes you so sure that the zero solution is the only integer solution?

In your formulation, (2a, a+b, c) and (2a, a-b, d) are Pythagorean triples. We may assume then: 2a = 2mn, a+b = m2 - n2 for some integers m, n. Plugging this into the second equation, expanding and dividing both sides by n4, one gets:
 
x4 - 2x3 + 3x2 + 2x + 1 = y2,

where x = m/n, y = d/n2 are rational numbers.
 
This looks threatening. Fortunately, there is a formal procedure to find rational solutions of the above equations using the method of Elliptic Curves. If I didn’t mess up with calculations, the only solution is the trivial (x, y) = (0, 1).
 
If there is an interest, I can elaborate on the method. There is even a software support to make things easier.
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Barukh
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Re: Heronian dissection: square  
« Reply #9 on: Sep 4th, 2005, 2:53am »
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on Aug 29th, 2005, 5:10am, Oyibo wrote:

 
We are looking at a minimum of four (if Barukh is right).

There exists a dissection into 4 parts.
 
I run out of time to present it now (JocK knows why  Grin), will make it later.
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Barukh
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Re: Heronian dissection: square  
« Reply #10 on: Sep 9th, 2005, 11:23pm »
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As attachments do not work for me, I will describe a solution in words.
 
To get rid of fractions, let ABCD be a square with side length 360. Take point E on BC, and BC = 224; F on CD, and DF = 105. Then, all 4 triangles ABE, ADF, CEF, AFE are Heronian.
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Grimbal
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Re: Heronian dissection: square   heronian.gif
« Reply #11 on: Sep 10th, 2005, 4:20am »
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Congratulations!
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JocK
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Re: Heronian dissection: square  
« Reply #12 on: Nov 1st, 2005, 9:35am »
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Excellent piece of work Barukh. Chapeau!
 
 
 
 
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solving abstract problems is like sex: it may occasionally have some practical use, but that is not why we do it.

xy - y = x5 - y4 - y3 = 20; x>0, y>0.
Barukh
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Re: Heronian dissection: square  
« Reply #13 on: Nov 1st, 2005, 11:11am »
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Thanks, JocK. And welcome back!  Cheesy
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