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Topic: The Yin of Yang (Read 6892 times) |
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rloginunix
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The Yin of Yang
« on: Aug 12th, 2016, 7:07am » |
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The Yin of Yang
Divide the yin (or the yang) portion of the entire yin-yang symbol in half square area-wise.
(ignore the small dots)
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dudiobugtron
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Re: The Yin of Yang
« Reply #1 on: Aug 12th, 2016, 9:47pm » |
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This is a cool question. 
Assuming I understand what 'square area-wise' means, here is my attempted solution:
The Yin/Yang symbol is made by first constructing a circle of radius r with centre (0,0). Then, construct two smaller circles of raidus r/2, with centres (0,r/2) and (0,-r/2). This divides the larger circle into 4 non-overlapping areas. Choose the top circle the 'Yin' circle, and the bottom one the 'Yang' circle. For the remaining two areas, call the left-most one the 'Yin' tail, and the other the 'Yang' tail.
The area of full circle is Pi*r^2
So the area of half of it is Pi*r^2 / 2
By symmetry with the Yang symbol, the total area of the Yin Circle and Yin tail must be Pi*r^2 / 2.
The area of the Yin circle is:
Pi*(r/2)^2 = Pi*r^2 / 4 = (Pi*r^2 / 2)/2
So the above construction also solves the problem. The Yin circle and Yin tail are each half of the area of the Yin symbol.
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| « Last Edit: Aug 12th, 2016, 9:49pm by dudiobugtron » |
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alien2
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Re: The Yin of Yang
« Reply #2 on: Aug 12th, 2016, 11:42pm » |
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on Aug 12th, 2016, 7:07am, rloginunix wrote:
Divide the yin (or the yang) portion of the entire yin-yang symbol in half square area-wise.
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I'm dividing nothing. Even though your riddle is well-intended I won't risk possibly offending William Wu. He is my dearest Chinese.
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rloginunix
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Re: The Yin of Yang
« Reply #3 on: Aug 13th, 2016, 9:29am » |
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You, guys, are on a problem-solving roll.
dudiobugtron - correct with a small asterisk since you suggested a, no pun intended, half of the solution. Your line that solves the problem is curved but not the only one.
PS
I hope that Mr. Wu recognizes that the only purpose of this problem is purely academic.
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dudiobugtron
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Re: The Yin of Yang
« Reply #4 on: Aug 13th, 2016, 1:41pm » |
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There are infinitely many lines that divide either symbol in half. Other than my solution, I guess that the most interesting one would be:
...a straight line that divides both Yin and Yang in half. This line would necessarily need to pass through (0,0) using my Yin/Yang symbol in the solution above.
The area of the right half of the Yin Symbol (in the first quadrant - positive x and y) is (Pi*r^2)/8. This is half of the size that we need. the rest will come from the area of the sector between the y axis, and the line we are looking for. Using the sector area formula, to get an area of (Pi*r^2)/8 this sector needs to be 1/8th of the size of the circle. So the line that passes through (0,0) must be at an angle of 45 degrees, or Pi/4 radians, anticlockwise from the y-axis. (Or, 135 degrees clockwise.)
By symmetry this line will also split the Yang symbol in half.
PS: I wasn't very rigorous in my 'construction' of the shapes in these answers - but hopefully you can see it would be easy to construct those 3 circles in my first answer, given a straight line (the y axis), a point on the line (0,0), and a length r. It's also pretty easy to construct a 45 degree angle at that point in order to draw this new straight line.
(I am definitely interested to see if there is a general solution for all possible lines!)
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| « Last Edit: Aug 13th, 2016, 1:46pm by dudiobugtron » |
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rloginunix
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Re: The Yin of Yang
« Reply #5 on: Aug 14th, 2016, 9:43am » |
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Correct.
I thought that if the question were "... with one straight line" then it would be too easy and would exclude your first solution.
(as far as a generic solution, one has to be careful - it is possible to divide the region in question with 'zig-zag' lines - the ones that do not posses even the first derivative at the breakage point, similar to f(x)=|x|)
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