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Figure this one out
- Thread starter Woody
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MikeD
Registered
It's straight forward - here's a simple explanation:-
Mitsunobu Matsuyama's "Paradox" uses four congruent quadrilaterals and a small square, which form a larger square. When the quadrilaterals are rotated about their centers they fill the space of the small square, although the total area of the figure seems unchanged. The apparent paradox is explained by the fact that the side of the new large square is a little smaller than the original one. If a is the side of the large square and θ is the angle between two opposing sides in each quadrilateral, then the quotient between the two areas is given by sec2θ − 1. For θ = 5°, this is approximately 1.00765, which corresponds to a difference of about 0.8%.
Got it?
Mitsunobu Matsuyama's "Paradox" uses four congruent quadrilaterals and a small square, which form a larger square. When the quadrilaterals are rotated about their centers they fill the space of the small square, although the total area of the figure seems unchanged. The apparent paradox is explained by the fact that the side of the new large square is a little smaller than the original one. If a is the side of the large square and θ is the angle between two opposing sides in each quadrilateral, then the quotient between the two areas is given by sec2θ − 1. For θ = 5°, this is approximately 1.00765, which corresponds to a difference of about 0.8%.
Got it?
It's straight forward - here's a simple explanation:-
Mitsunobu Matsuyama's "Paradox" uses four congruent quadrilaterals and a small square, which form a larger square. When the quadrilaterals are rotated about their centers they fill the space of the small square, although the total area of the figure seems unchanged. The apparent paradox is explained by the fact that the side of the new large square is a little smaller than the original one. If a is the side of the large square and θ is the angle between two opposing sides in each quadrilateral, then the quotient between the two areas is given by sec2θ − 1. For θ = 5°, this is approximately 1.00765, which corresponds to a difference of about 0.8%.
Got it?
Woody , I am surprised that you are not aware of this simple mathematical explanation !!....Even Mike knows how its done !!!
It's straight forward - here's a simple explanation:-
Mitsunobu Matsuyama's "Paradox" uses four congruent quadrilaterals and a small square, which form a larger square. When the quadrilaterals are rotated about their centers they fill the space of the small square, although the total area of the figure seems unchanged. The apparent paradox is explained by the fact that the side of the new large square is a little smaller than the original one. If a is the side of the large square and θ is the angle between two opposing sides in each quadrilateral, then the quotient between the two areas is given by sec2θ − 1. For θ = 5°, this is approximately 1.00765, which corresponds to a difference of about 0.8%.
Got it?
and thats straightforward?
Yeah, come on Woody I thought everyone knew about old Mitsunobu Matsuyama's "Paradox" he kept it in a box innit 
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Did that when I sat my 11+ at school !!!!!!
MikeD
Registered
What's the problem?
The government uses exactly the same principle when they balance the budget - money disappears and reappears even though nothing actually changes because it is all an optical illusion.
The Banks thought they understood the principle but used the wrong angle between the two opposing sides of each quadrilateral; MPs made the same mistake when filling in their expense claims.
Next time you are doing some segmenting remember the above formula
The government uses exactly the same principle when they balance the budget - money disappears and reappears even though nothing actually changes because it is all an optical illusion.
The Banks thought they understood the principle but used the wrong angle between the two opposing sides of each quadrilateral; MPs made the same mistake when filling in their expense claims.
Next time you are doing some segmenting remember the above formula
Well that would have seems like black magic without Mike's simple cogent explanation. Thanks Mike!
Aw ye thought that was what it was
That old Mitsunobu Matsuyama's "Paradox" put me to sleep at school........ 
Maybe thats why!!!
Maybe thats why!!!
Perhaps you're entering a Goya period of your life Brian
That or someone's painted the lenses of your glasses black
MikeD
Registered
OK, an explanation in everyday language.
You will have noticed that most of the pieces that are moved around appear to be triangles. In reality they are not, the apparent hypotenuse (the longest arm of the triangle) is not straight so the triangle is a quadrilateral (a four sided shape). Your eyes are misled because of the squares scribed on the surface.
So as you go through life be aware of bent hypotenuses they may mislead you
My previous explanation provides the same solution but in mathematical terms
You will have noticed that most of the pieces that are moved around appear to be triangles. In reality they are not, the apparent hypotenuse (the longest arm of the triangle) is not straight so the triangle is a quadrilateral (a four sided shape). Your eyes are misled because of the squares scribed on the surface.
So as you go through life be aware of bent hypotenuses they may mislead you
My previous explanation provides the same solution but in mathematical terms