That damn triangle
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I'm sorry, but I'm going to have to bring this 'optical illusion' thread up again as its driving me mad. http://www.codeproject.com/lounge.asp?msg=528308#xx528308xx[^] http://www.briandela.com/files/picture.gif[^] I read the replies, but I must be stupid or something, cause none of them made sense. This triangle problem has occupied my thoughts since this time yesterday. I've recreated versions of it in Excel and Paintshop Pro, and still can't get my head around it!!! Help!!!! :eek::eek: Basically, the surface area of both large triangles is 32.5 squares. And both triangles ARE the same, AND the hypotenuse is a straight line. Why doesn't it add up? :(( John www.silveronion.com[^]
As several people have mentioned, the hypotenuse isn't straight. The dead giveaway here comes when you look at the 6th and 9th vertical lines and note how they don't intercept equivilant horizontal lines on both composite shapes.
Shog9
drifting along with the tumbling tumbleweeds...
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John Honan wrote: And both triangles ARE the same, AND the hypotenuse is a straight line. No, the two main "shapes" are not triangles and their "hypotenuse" is not a straight line. Remember back to your days in geometry class. By definition both the red and dark green triangle MUST be similar triangles. (ie. They must have exactly equal angles). Red triangle angles: 90, x and y where tan x = 3/8 and y = 90 - x x = 20.556 degrees y = 69.444 degrees Dark green triangle angles: 90, x and y where tan x = 2/5 and y = 90 - x x = 21.801 degrees y = 68.199 degrees Let's take it a step further: Main "shape" angles: 90, x and y where tan x = 5/13 and y = 90 -x x = 21.037 y = 68.963 This "puzzle" is purely an optical illusion.
Work like you don't need the money.
Love like you've never been hurt.
Dance like nobody's watching.The problem has nothing to do with angles one way or the other. If I took two triangles - one 80 degrees and the other 20 degrees, for example, and placed the vertex of one two units above the base of the other and counted 8 units down the base of the second, I would have 16 units. If I than took and moved that same rectangle up another unit and measured out five units I would have 15 units. I could modify those angles as much as I pleased and the results will always be exactly the same. Even if the angles were exactly the same, the "hole" would still be there, and it would always be exactly the same size.
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I'm sorry, but I'm going to have to bring this 'optical illusion' thread up again as its driving me mad. http://www.codeproject.com/lounge.asp?msg=528308#xx528308xx[^] http://www.briandela.com/files/picture.gif[^] I read the replies, but I must be stupid or something, cause none of them made sense. This triangle problem has occupied my thoughts since this time yesterday. I've recreated versions of it in Excel and Paintshop Pro, and still can't get my head around it!!! Help!!!! :eek::eek: Basically, the surface area of both large triangles is 32.5 squares. And both triangles ARE the same, AND the hypotenuse is a straight line. Why doesn't it add up? :(( John www.silveronion.com[^]
http://www.knowledgeautomation.com/graphics/triangle.JPG[^] :-D Marc Help! I'm an AI running around in someone's f*cked up universe simulator.
Sensitivity and ethnic diversity means celebrating difference, not hiding from it. - Christian Graus
Every line of code is a liability - Taka Muraoka
Microsoft deliberately adds arbitrary layers of complexity to make it difficult to deliver Windows features on non-Windows platforms--Microsoft's "Halloween files" -
The problem has nothing to do with angles one way or the other. If I took two triangles - one 80 degrees and the other 20 degrees, for example, and placed the vertex of one two units above the base of the other and counted 8 units down the base of the second, I would have 16 units. If I than took and moved that same rectangle up another unit and measured out five units I would have 15 units. I could modify those angles as much as I pleased and the results will always be exactly the same. Even if the angles were exactly the same, the "hole" would still be there, and it would always be exactly the same size.
Stan Shannon wrote: The problem has nothing to do with angles one way or the other It has everything to do with the angles. The optical illusion happens because the mind wants to believe the two main "shapes" are triangles and are the same height and width yet have different areas. Since the green triangle and red triangle have only slightly different angles from each other and from the main "shape" the mind assumes they are all identical. In one case the orientation of the dissimilar angles produces a convex "hypotenuse", the other produces a concave "hypotenuse". The area difference between the convex and concave is exactly the area of the missing block.
Work like you don't need the money.
Love like you've never been hurt.
Dance like nobody's watching. -
http://www.knowledgeautomation.com/graphics/triangle.JPG[^] :-D Marc Help! I'm an AI running around in someone's f*cked up universe simulator.
Sensitivity and ethnic diversity means celebrating difference, not hiding from it. - Christian Graus
Every line of code is a liability - Taka Muraoka
Microsoft deliberately adds arbitrary layers of complexity to make it difficult to deliver Windows features on non-Windows platforms--Microsoft's "Halloween files":cool: Now I understand it as well... :-D Paul ;) I have been afraid always. When you see something insurmountable ahead of you, say to yourself: "All right! I am afraid. Now that I've been properly afraid, let's go forward." That is the whole secret. - Jeanne d'Arc
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I'm sorry, but I'm going to have to bring this 'optical illusion' thread up again as its driving me mad. http://www.codeproject.com/lounge.asp?msg=528308#xx528308xx[^] http://www.briandela.com/files/picture.gif[^] I read the replies, but I must be stupid or something, cause none of them made sense. This triangle problem has occupied my thoughts since this time yesterday. I've recreated versions of it in Excel and Paintshop Pro, and still can't get my head around it!!! Help!!!! :eek::eek: Basically, the surface area of both large triangles is 32.5 squares. And both triangles ARE the same, AND the hypotenuse is a straight line. Why doesn't it add up? :(( John www.silveronion.com[^]
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Stan Shannon wrote: The problem has nothing to do with angles one way or the other It has everything to do with the angles. The optical illusion happens because the mind wants to believe the two main "shapes" are triangles and are the same height and width yet have different areas. Since the green triangle and red triangle have only slightly different angles from each other and from the main "shape" the mind assumes they are all identical. In one case the orientation of the dissimilar angles produces a convex "hypotenuse", the other produces a concave "hypotenuse". The area difference between the convex and concave is exactly the area of the missing block.
Work like you don't need the money.
Love like you've never been hurt.
Dance like nobody's watching.Mike Mullikin wrote: The area difference between the convex and concave is exactly the area of the missing block. I suppose my problem than is that I simply see no "illusion". All I see are two rectangles, one 5 x 3 and the other 8 x 2, entirely independent of the triangles, regardless of how you draw them. It is only an illusion if you try to make an area of a triangle problem out of it, which it obviously isn't. You could change those triangles as much as you please, and I just don't see how the rectangles change any at all. Hell, you could convert both triangles into squares and nothing changes. The hole is still there.
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http://www.knowledgeautomation.com/graphics/triangle.JPG[^] :-D Marc Help! I'm an AI running around in someone's f*cked up universe simulator.
Sensitivity and ethnic diversity means celebrating difference, not hiding from it. - Christian Graus
Every line of code is a liability - Taka Muraoka
Microsoft deliberately adds arbitrary layers of complexity to make it difficult to deliver Windows features on non-Windows platforms--Microsoft's "Halloween files"Ah! Thank you! :rose: I am a 'visual' person, so that simple diagram really helped. :) The insanity is starting to ease off now. :rolleyes: John www.silveronion.com[^]
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Mike Mullikin wrote: The area difference between the convex and concave is exactly the area of the missing block. I suppose my problem than is that I simply see no "illusion". All I see are two rectangles, one 5 x 3 and the other 8 x 2, entirely independent of the triangles, regardless of how you draw them. It is only an illusion if you try to make an area of a triangle problem out of it, which it obviously isn't. You could change those triangles as much as you please, and I just don't see how the rectangles change any at all. Hell, you could convert both triangles into squares and nothing changes. The hole is still there.
Stan Shannon wrote: I suppose my problem than is that I simply see no "illusion". Having spent ages looking at it myself, My co-worker came over, took one look at it and actually saw the 'bends' in the 'pseudo-triangles'. It may have something to do with him having weird astigmatism. But I was not pleased that it took him so quickly to see the fault.:)
"..Even my comments have bugs!"
Inspired by Toni78 -
Ah! Thank you! :rose: I am a 'visual' person, so that simple diagram really helped. :) The insanity is starting to ease off now. :rolleyes: John www.silveronion.com[^]
It's funny, if you print it and cut out the pieces, you can actually see the differences in the two triangles. And you can reproduce the second drawing too. This'll make a great illusion for my kid's 7th grade class next year! Marc Help! I'm an AI running around in someone's f*cked up universe simulator.
Sensitivity and ethnic diversity means celebrating difference, not hiding from it. - Christian Graus
Every line of code is a liability - Taka Muraoka
Microsoft deliberately adds arbitrary layers of complexity to make it difficult to deliver Windows features on non-Windows platforms--Microsoft's "Halloween files" -
I'm sorry, but I'm going to have to bring this 'optical illusion' thread up again as its driving me mad. http://www.codeproject.com/lounge.asp?msg=528308#xx528308xx[^] http://www.briandela.com/files/picture.gif[^] I read the replies, but I must be stupid or something, cause none of them made sense. This triangle problem has occupied my thoughts since this time yesterday. I've recreated versions of it in Excel and Paintshop Pro, and still can't get my head around it!!! Help!!!! :eek::eek: Basically, the surface area of both large triangles is 32.5 squares. And both triangles ARE the same, AND the hypotenuse is a straight line. Why doesn't it add up? :(( John www.silveronion.com[^]
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But it was nice mind boggling (or whatever the word is) :):):) Regards, Venet. Donec eris felix, multos numerabis amicos.
The area of a triangle of angle 2:5 extended for a width of 13 units is 33.8 square units. The area of a triangle of angle 3:8 extended for a width of 13 units is 31.7 square units. for a result of 2.1 square units difference between the two, yet the "hole" is obviously one squre unit. Precisely how does that translate into a solution? You guys are on drugs. The hole is there because its there, and would be there no matter how you draw the damned trianagles. Look at the diagram for Pete's sake.
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Stan Shannon wrote: I suppose my problem than is that I simply see no "illusion". Having spent ages looking at it myself, My co-worker came over, took one look at it and actually saw the 'bends' in the 'pseudo-triangles'. It may have something to do with him having weird astigmatism. But I was not pleased that it took him so quickly to see the fault.:)
"..Even my comments have bugs!"
Inspired by Toni78The problem, though, is that I am right and the rest of you guys are wrong. There is no illusion, the math just does not work out. Am I the only one here who can calculate the area of a triangle?
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It's funny, if you print it and cut out the pieces, you can actually see the differences in the two triangles. And you can reproduce the second drawing too. This'll make a great illusion for my kid's 7th grade class next year! Marc Help! I'm an AI running around in someone's f*cked up universe simulator.
Sensitivity and ethnic diversity means celebrating difference, not hiding from it. - Christian Graus
Every line of code is a liability - Taka Muraoka
Microsoft deliberately adds arbitrary layers of complexity to make it difficult to deliver Windows features on non-Windows platforms--Microsoft's "Halloween files"You'd better hope that none of them know simple geometry.
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You'd better hope that none of them know simple geometry.
Stan Shannon wrote: You'd better hope that none of them know simple geometry. So far, so good. :-D Marc Help! I'm an AI running around in someone's f*cked up universe simulator.
Sensitivity and ethnic diversity means celebrating difference, not hiding from it. - Christian Graus
Every line of code is a liability - Taka Muraoka
Microsoft deliberately adds arbitrary layers of complexity to make it difficult to deliver Windows features on non-Windows platforms--Microsoft's "Halloween files" -
The area of a triangle of angle 2:5 extended for a width of 13 units is 33.8 square units. The area of a triangle of angle 3:8 extended for a width of 13 units is 31.7 square units. for a result of 2.1 square units difference between the two, yet the "hole" is obviously one squre unit. Precisely how does that translate into a solution? You guys are on drugs. The hole is there because its there, and would be there no matter how you draw the damned trianagles. Look at the diagram for Pete's sake.
Stan Shannon wrote: The hole is there because its there, and would be there no matter how you draw the damned trianagles. A=bh/2 The red triangle is 8*3/2, = 12 sq. u. The green triangle is 5*2/2, = 5 sq. u The blocks are 15 sq. u 12+5+15=32 sq. u The whole triangle is 13*5/2, = 32.5 The hole is actually 1/2 sq. unit, but by maintaining a constant base length, it can be made to appear to be 1 sq. unit. In the top diagram, the triangle bulges out 1/2 sq. unit, and in the bottom triangle, it caves in 1/2 sq. unit. Since we're comparing to a perfect triangle (which this is not), the caving in on the bottom drawing by 1/2 sq. unit adds the other piece, thus making a full sq. unit hole. Marc Help! I'm an AI running around in someone's f*cked up universe simulator.
Sensitivity and ethnic diversity means celebrating difference, not hiding from it. - Christian Graus
Every line of code is a liability - Taka Muraoka
Microsoft deliberately adds arbitrary layers of complexity to make it difficult to deliver Windows features on non-Windows platforms--Microsoft's "Halloween files" -
Stan Shannon wrote: The hole is there because its there, and would be there no matter how you draw the damned trianagles. A=bh/2 The red triangle is 8*3/2, = 12 sq. u. The green triangle is 5*2/2, = 5 sq. u The blocks are 15 sq. u 12+5+15=32 sq. u The whole triangle is 13*5/2, = 32.5 The hole is actually 1/2 sq. unit, but by maintaining a constant base length, it can be made to appear to be 1 sq. unit. In the top diagram, the triangle bulges out 1/2 sq. unit, and in the bottom triangle, it caves in 1/2 sq. unit. Since we're comparing to a perfect triangle (which this is not), the caving in on the bottom drawing by 1/2 sq. unit adds the other piece, thus making a full sq. unit hole. Marc Help! I'm an AI running around in someone's f*cked up universe simulator.
Sensitivity and ethnic diversity means celebrating difference, not hiding from it. - Christian Graus
Every line of code is a liability - Taka Muraoka
Microsoft deliberately adds arbitrary layers of complexity to make it difficult to deliver Windows features on non-Windows platforms--Microsoft's "Halloween files"If I take an arbitrary rectangle (X,Y) and select two arbitrary regions from it (X1,Y1) and (X2,Y2) such that X = X1 + X2 and Y = Y1 + Y2 than I get the following result: The difference between the areas of the triangles formed by the three regions is: (XY/2) - ((X1Y2)/2 + (X2Y1)/2 + X2Y2 ) this expands to: ((X1 + X2)(Y1 + Y2))/2 - ((X1Y2)/2 + (X2Y1)/2 + X2Y2 ) and: (X1Y1 + X1Y2 + X2Y1 + X2Y2)/2 - ((X1Y2)/2 + (X2Y1)/2 + X2Y2 ) or: (X1Y1)/2 + (X1Y2)/2 + (X2Y1)/2 + (X2Y2)/2 - ((X1Y2)/2 - (X2Y1)/2 - X2Y2 which reduces to: (X1Y1)/2 - (X2Y2)/2 or: (X1Y1 - X2Y2) / 2 but: X1Y1 - X2Y2 is merely the difference between the two originally selected arbitrary areas within the rectangle. This is a trivial result, and prooves that any time you take any two arbitrary points within a rectangle you get results similar to those in the "illusion". So, as I originally posted - you are merely comparing two rectangular areas, you do not need to calculate angles or anything else - the results are obvious and trivial regardless of where you select the points. In the case of the illusion, the "hole" is there merely because you subtracted a 15 square unit region from a 16 square unit region. "More capitalism, please..."
