How can this be true? Observe the picture precisely, and answer will come. Feel free to comment this illusion, but please don’t spoil other’s fun by revealing the answer.
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It’s totally as Jiggs up said the interesting part of all is that the RHOMBOID that is formed, is exactly 1 square unit. We use this examples to optimize the areas when designing in architecture, cause it’s a minimum space lost y long distances, then turned into useful square spaces. –Fernando Barragán–Mexico City.
To Do’h: first of all, it should be spelled “D’oh.” Also, as far as I’m concerned, velocity has to do with the rate at which an object moves.
**Spoiler**
Comment ten is correct. I don’t really care about it not being a 180 degree angle. What happened is that instead of putting the orange L on top of the green L, it was moved over one square. Since the red triangle is one unit longer than the dark green triangle, it was able to fit.
Paula: good luck in geometry class. I hate geometry, but if I pass this year, I don’t have to do it again in high school.
look, the triangles have different angles- since tan (red) = 3/8=0,375 and tan (blue) = 2/5=0,4
that means that both shapes aren’t triangles, but polygons with a blunt corner at the point where red and blue meet.
so how did we learn to calculate the area of a polygon in primary school? draw a rectangle around it, calculate it’s area and subtrack the area’s of the triangles and rectangles that surround the polygon. (it’s hard to explain for me in english but try to imagine/remember it)
well, the area of the rectangle is 5*13=65 and the area of the first polygon is therefore 65-33=32 [65-(5*2/5 + 3*8/2 + 8*2)]
the area of the lower polygon is 65-32=33 [65-(5*2/5 + 3*8/2 + 5*3)]
the area’s of the coloured shapes remained the same in both polygons, but this illusion makes you think that both supposed big “triangles” have the same area. but in reality they’re two polygons and the area of the last one is 1 square bigger than the first (33-32) and that is created by the empty square you see.
i didn’t really say any new things, but i’m just trying to make it a little more clear than in #10
I’m sorry, I meant #5 in the last bit, what #10 said was ridiculous… all the credits to jiggs up
ah ha! I see it now. high school trig saves the world again
Open Photoshop, then start a new project on a perfectly square canvas. Now show your grid guidelines (ctrl+’). You will need to go into the gridline preferences and make the gridlines for every 20%, with 3 subdivisions each. This will display a grid of 15×15 squares.
The Triangle is 13 squares wide, and 5 high. Draw just one triangle with these dimensions, photoshop will automatically snap to the corners of the grid boxes. You have a perfect triangle. Now realize the dimensions of the orange pattern. It’s supposed to be 5 squares across on the top row, and two squares on the bottom left, occupying 7 of the grid boxes, perfectly. Look at the triangle you drew in photoshop. The top left corner of that orange shape, will not fit cleanly between the red and green triangle, no matter which configuration you have the partitions set. The top left corner of the orange shape would have to meet exactly at a corner of one of the grid squares. But as you can see by the perfect triangle you drew, there is not one point within the hypotenuse that crosses a grid box exactly at the corner. So the Orange segment will not fit as the illusion suggests. In the illusion, the hypotenuse crosses two exact corners of two different grid boxes. This is not so when doing so on a mathematically true grid and perfect triangle. The illustration here is an illusion, the red and green triangles have to cheat in order to cross grid corners. The two triangles have slight curves.
You all saying the bend is the answer are retards, excuse me.
But you will get the same illusion if you made the figures yourselves witht the most precise accuracy to each other as possible.
Therefore, the bent line “theory” does not explain it.
OK, now, after reading almost all, and I say almost so that i dont insult someone i think you are ALL WRONG. The key here is not trigonometry, nor in the triangles. Let me explain. I think the answer is this – surface area and circumference are NOT in direct relation. This would mean that objects with the same circumference can have different surface area and the other way around. So – Leave the triangles aside, the difference in the two pictures is not their surface area. Its their circumference. Look closely. If you count the squares, their number is the same in both picures, its just that in the bottom picture the overall circumference of the figure is greater. Again to make i easier I will leave the triangles, they are there for confusion and go down to the numbers.
In the first picture, we have a rectangle with a surface are of 3×5 = 15
Now. In the second we have a figure that is not an exact rectagle (look only at the coloured part) but this figure stil has a surface area of 15 (just count the coloured squares). So where is the difference?As i said the difference is in the circumference. the first figure (the rectangle) has a circ. of 2×3 + 2×5 = 16 while the second has 2×8 + 2 = 18. (If you a re wondering why I add 2, I will explain. I calculated the circ. of the rectangle with sides – 2 and 8. then subtracted the non coloured bit, which was 1 (remember we are talking about sides, now squares here), but had to add 3, which is the number of coloured sides, all in all -1 + 3 therefore +2. I hope you got it. To sum up. The key in this is the question. And almost all you had it WRONG. the two coloured figures actualy have THE SAME SURFACE AREA. Question is why do two different circumference-s belong two figures with same surface areas? Now proving that in generalall cases is much tougher, but you can make it look easier with a simple exercise. Imagine a cube with a side =4. Now this has a circ = 16 and a surface area again = 16. now draw a rectagnle with a side A = 2, and B = 6. What do you notice. They have the same circ.-s = 16, but the surface area of the second is 2×6 = 12 which is NOT 16. There you go. All non linear conspiracy theories are refuted.
I got it in less than 3 minutes. I still don’t understand why everyone is talking about trigonometry and that stuff. It’s simple! It’s just a different arrangement of the shapes! Move the orange shape to the side and (a la “tetris”) watch it fall and form the hole. I would tell you more, but I don’t want to feed the beast.
It’s not as hard what everyone seems to think. There’s no “Line Bend” that creates enough area for the “hole”. The fact is, the “hole” is not part of the area for the triangle. you can’t use a trigonometric equation on the second “triangle” because it’s not a triangle. By moving the individual colored pieces, one can simply change the shape, and while it appears to be a triangle, it isn’t. The pices still have the same area, and this can be easily replicated: just make your pices [either on paint/photoshop, or with some paper] and rearrange them like on the picture. No “magic line bend” required.
The Hypotinuse is NOT a straight line in either image. it bows slightly outward in one and inward in the other. The gridlines distract your eye from noticing it.
I have not read all comments so forgive me if this is being restated. The two trinagle are not the same proportion. They have a slightly differing slope, but it is close enough to trick the eye.
The red traigle is 3/8 and the Greaan is 2/5 or 3/7.5. That .5 differsnce allows for the extra square to be created when the pieces are moved around. Thats all there is too it
I knew in about 5 min that the squares are not perfect squares in the grid.It’s the only way to explain how it’s possible to “gain” an extra space. Makes sense now as I looked down on hypotenuses they’re not parallel.
I know some couldn’t really wrap their minds around it. But it’s really on the fact that the hypotenuse is not a straight line. As was mentioned before by others.
Just a reminder (in case you’ve missed it), check this again http://www.scientificpsychic.com/mind/mind1.html
for a clear explanation.
just look at the slopes of the hypotenuse of the red and cyan triangles. they’re not the same, so it isnt a continuous straight line
Look at the orange shape and the lime shape. The orange shape has two squares on its head and three squares on its tail. The lime shape has three squares on it’s head and two squares on its tail.
On the top, the head of one shape lines up with the tail of the other. That makes a rectangle that is three squares high and five squares long.
On the bottom, the orange shape is moved so that the tails touch. The tails don’t match though. So, that creates the gap. It also changes the rectangle to two squares high and eight squares long.
The red triangle is one square higher and three squares longer than the green triangle. So, when you move the red one up, it makes up for the decreased height of the two non-triangles in the first image, like adding 1 and -1. It also stretches to cover their increased length as well.
The green triangle is only two high and therefore matches up with the second rectangle. Since it is three squared shorter than the red, it ends up stopping where the red triangle originally stopped even though it is attached to a rectangle that is three squares longer. It’s like adding 3 and -3.
There is no net change in the overall shape of the two triangles other than the gap.
I laugh at the people talking about curved lines and such. Draw it out with perfectly straight lines and you will get the same result. It’s pretty easy to figure out what the problem is as other people have pointed out. The red triangle has a base length of 8 squares and a height of 3. The green triangle has a length of 5 squares and is set 3 up from the bottom. That makes the space with the other two objects have an area of 15sq (length x width L is the base of the green;5, and W is the height of the red;3. 5×3=15). When you switch the triangles you are changing that space from 15sq to 16 sq. The length becomes 8 (base of the red triangle) and the width becomes 2 (height of the green triangle)so the area left to fill is 8×2=16sq. You can’t take two pieces that fit together and have an area of 15 and move them to complete an area of 16.
i dont see how you idiots dont see it
Sorry comment #118 (Nightmare) – but even the explanation you gave constitutes a curved line.
You can find a picture here to illustrate what you explained. http://www.scientificpsychic.com/mind/triangle3.gif
Explanation of the answer found here. http://www.scientificpsychic.com/mind/triangle1.html
@Anonymous comment 120
do you know the difference between curves and angles? Apparently not. Let me give you the definition of a curve.
curve (kʉrv)
adjective
Archaic curved
Etymology: L curvus, bent: see crown
noun
1. a line having no straight part; bend having no angular partI will admit that after going back and looking closer at the first 13×5 triangle drawn on graph paper, the 5×3 point on the hypotenuse is slightly below the grid line. So it’s an angle and not a curve. Either way it still has to do with the area of the object changing when you rearrange them. I’ll even quote from the site, “The top figure has an area of 32 square units. The bottom figure, including the empty square, has an area of 33 square units.” Which is because it’s not a triangle.
Even if you don’t get the math–it doesn’t matter. The point of optical illusions is to point out how your brain tricks you into “seeing” something a certain way, regardless of the physical reality of the actual thing you are seeing. That’s what is really the mystery here–the way our brains work!
Take a blank sheet of paper and line up the edge with hypotenuse of the red triangle. Watch as the hypotenuse of the green triangle magically jumps off of the edge of the paper!
point.
IF ANYONE OF YOU KNOWS HOW TO USE AUTOCAD, YOULL FIGURE IT OUT, JUST DRAW THE EXACT SAME FIGURE AND YOULL FIND A CURIOUS SURPRISE
The big triangle isnt a triangle!!!
i cut it out of paper. it still gives an extra square, so it is not an illusion and it is not because of bended lines. i can’t explain it, but i am really bad at geometry
This is a very interesting puzzle. I remember finding it and figuring it out when I was in 9th grade, but it took me a good while. Very intriguing. I think people are getting hung up on terminology here. There are no “bent” lines per se. All individual shapes in the top and bottom figure are identical and drawn with straight lines. The grid is not flawed either. If you add up the areas of the individual shapes (in either picture) you get 32. If you take the area of the top figure as a triangle you get 32.5 and for the bottom figure 31.5. Neither is correct, so the obvious answer, the large figures are not triangles, but then you muse find out why. The slope of the small red triangle is not identical to the slope of the small green triangle. Therefore, the total figure is not a triangle, but a quadrilateral because it’s “hypotenuse” has a slight angle change at the meeting of the red and green triangles. I don’t want to say a bend because of terminology again. All lines are straight. But the “hypotenuse” of the large figures is really two separate lines that are convex in one figure and concave in the other, accounting for the “missing” unit.
If this does not make sense to you, may I propose something. Get a piece of graph paper and draw the red (8×3) triangle and green (5×2) triangle on top of each other. To make it even more pronounced you can double or triple all distances, as that does not change the angles. Draw an 24×9 triangle on top of a 15×6 triangle using a ruler. You should very clearly see that their hypotenuses are at different angles.
Warning, here goes a rant. Now what I find amusing is the people who laugh and say how this problem is so easy, how can everyone not see it? You amuse me. Especially when you bring up, my little 13 year old mind can see this easily! Oh you. If your solution is that the shapes were moved, all I can say is, REALLY?! The rest of us didn’t notice the shapes were moved around! Thank you for enlightening us so!! Now please go back to your tinker toys and lego while the rest of us move forward with our lives by using our brains. If your answer says anything about the two triangles in the first figure producing a 5×3 cavity (15 area) while the two in the second figure they produce an 8×2 cavity (16 area) or some mention of tetris, probably followed by a comment about how you don’t believe everyone can’t see this right away, let me explain something to you. Not only are you too dense to be able to solve the problem, you are too slow to even understand that there is a problem. And that, my friends, is far worse. In the illusion, a unit of area appears to have disappeared. No matter how much you try, you cannot explain a change in area by moving things around. Its like I shuffled a deck of 52 cards and ended up with a deck of 51 cards. And your answer is, well you shuffled it. Ok. Yes I did. But that’s not the problem. Or I moved a table across the room, and it turned into a desk. Your answer is, well you moved it. That’s not the problem. There is an illusion here, and you are not even seeing the illusion, much less the explanation to it. So if you cannot even understand the problem, don’t try to act like everyone else is stupid for not seeing the answer. Understand the problem before you even try to solve it, please.
Thanks.
the two triangles shown simply arent similar triangles.
good illusion
this is not an exercise in plane geometry as much as an exercise in being able to explain the answer coherently. one person didn’t know the difference between circumference and perimeter, and another the difference between bent and curved. the illusion is that the large “hypotenuse” is a straight line, which it is not. the two small angles of the triangles aren’t equal because 8:3 and 5:2 aren’t equal ratios; therefore when you reverse the triangles the “hypotenuse” bends outward instead of inward. the increase in total volume of the large triangle is enough to accomodate the extra square. this can be proven by simply adding the four individual areas.
oops – i spelled accommodate wrong!
I don’t get it.:s
You can buy a similar problem in the form of a wooden puzzle: http://www.thinkgeek.com/geektoys/games/be62/
somthing u dont notis is the tryangles are bent diffrently but 1 thing id like to say is that i cut out the shapes out of a peice of paper and re aranged them and got the same illusions. savy? (savy means ok)
READ THIS FOR THE TRUTH
oh and 1 more thing the bend is just making people think the wrong thing (only the tryangles are bent) if u cut out your own shapes like i did u can see for yourself
If you take the smaller triangles and do a little trigonometry on them, you will see they are not similar. The smallest angle on the small triangle is 21.8 degrees, the big (red) triangle has an angle of 20.6 degrees. I tried roughly sketching this out on graph paper, and they appeared similar, but a difference of 1.2 degrees is imperceptible.
If you take the top shape and kind of fold it over on the bottom shape, so that the green triangles and the red triangles make a 2×5 rectangle and a 3×8 rectangle, respectively, then you can see this more clearly. You get a 5×13 rectangle with one piece missing, because the area of 5×13 is 65 and the areas of all the colored shapes only adds up to 64.
There is no bending, just dissimilar triangles.
The “whole” triangle is not REAL triangle, it’s just a look-alike.
The red&green triangles appear to be similiar (to have equal angles), but actually their angles differ. If you assume all the sizes of the shapes (which form the whole picture) are determined exactly by the number of squares they occupy, you can easily calculate the angles of the red&green triangles, using trigonometry.
This is a mathematical proof that the hypotenuse of the top (whole) triangle is not actually a straight line, which means that this object is actually not a triangle, which means that its area cannot be calculated (accurately) with the “right-triangle-formula”, but rather “the-sum-of-its-parts” approach should be used.
The conundrum here is that, the point where the red&green triangles meet appears to be exactly on top of the grid, which gives the erroneous impression that their sides are well-defined. In reallity, this picture is ambiguous. If the grid represents the sizes accurately, then the sides of the red&green triangles can’t be measured with whole numbers (integers), but irrational numbers should be used instead.
So… my conclusion is that:
1) The “whole” triangle is not REALLY triangle (its hypotenuse isn’t straight line)
2) If you get a grid of squares and draw a STRAIGHT line in such a way that it forms the hypotenuse of a right triangle with sides 5 and 13, you’ll notice that this line doesn’t go through any of the grid intersection-points (it will come close, but not exactly over where the grid lines intersect)
3)This is one EXCELLENT visual illusion.
[i]Anonymous says:
February 23, 2009 at 10:36 pm
OK, now, after reading almost all, and I say almost so that i dont insult someone i think you are ALL WRONG. The key here is not trigonometry, nor in the triangles. Let me explain. I think the answer is this – surface area and circumference are NOT in direct relation. This would mean that objects with the same circumference can have different surface area and the other way around. So – Leave the triangles aside, the difference in the two pictures is not their surface area. Its their circumference. Look closely. If you count the squares, their number is the same in both picures, its just that in the bottom picture the overall circumference of the figure is greater. Again to make i easier I will leave the triangles, they are there for confusion and go down to the numbers.
In the first picture, we have a rectangle with a surface are of 3×5 = 15
Now. In the second we have a figure that is not an exact rectagle (look only at the coloured part) but this figure stil has a surface area of 15 (just count the coloured squares). So where is the difference?As i said the difference is in the circumference. the first figure (the rectangle) has a circ. of 2×3 + 2×5 = 16 while the second has 2×8 + 2 = 18. (If you a re wondering why I add 2, I will explain. I calculated the circ. of the rectangle with sides – 2 and 8. then subtracted the non coloured bit, which was 1 (remember we are talking about sides, now squares here), but had to add 3, which is the number of coloured sides, all in all -1 + 3 therefore +2. I hope you got it. To sum up. The key in this is the question. And almost all you had it WRONG. the two coloured figures actualy have THE SAME SURFACE AREA. Question is why do two different circumference-s belong two figures with same surface areas? Now proving that in generalall cases is much tougher, but you can make it look easier with a simple exercise. Imagine a cube with a side =4. Now this has a circ = 16 and a surface area again = 16. now draw a rectagnle with a side A = 2, and B = 6. What do you notice. They have the same circ.-s = 16, but the surface area of the second is 2×6 = 12 which is NOT 16. There you go. All non linear conspiracy theories are refuted.[/i]
So, dude… if you can’t figure out why the apparently straight line isn’t really straight, you can pick your favorite ruler and see this for yourself. Or maybe they’re not that straight anymore? It’s a fact, not a theory. With trigonometry, you can (on theory) see exactly which portions of the elements form the missing square.
long story short, anyone smart enough to recognise the problem already knows the answer. two of the same shapes cant have a different area… so whatever cant be true, isn’t. the shapes are either different, or the area is the same. in this case the shapes are different. END OF STORY.
but ill keep talking anyways
if simple theory doesnt work on you you can do as one person did somewhere above and prove it with simple geometry. he figured it out by calculating all dimensions looking for the inconsistency, and found it.. a different angle on the two triangle.
you cannot argue that you built the pieces and it still works, because that is 100% correct, you will accomplish the exact same effect. the illusion here isnt how it is drawn, its just taking a small area, hiding it over a long distance or focusing it in a small space, the small angle is just real hard to see with the eye, and will be only hidden better by trying to tape little pieces of cut out paper together. now if you cut these out v ery accurately, and large enough to work with, and very accurately traces the perimiter, rearanged them as shown, you would see a gap totaling the same area develop, spread out along what is a bent line. basically a waste of time thought, all you did was copy what was already shown on your screen.
some of you are making this way way too complicated, others are too stupid to understand the problem. the trick to this isnt math, geometry, trig, its simple theory. the area an object cannot dissapear by rearranging pieces of it, but the resulting shape can change. so the shape of the two triangles MUST be different. and it is.
the missing area is hidden along the length of the hypotenuse by the different angle created.
This is quite simple to figure out. The slope of the hypotenuse of the two triangles are not equal. The smaller green triangle’s hypotenuse has a slope of 2/5 while the larger red triangle has a slope of 3/8. Since these slopes are not equal, then the hypotenuse of the assembled triangle is not a straight line.
In the top figure the long hypotenuse is actually concave (bowing inward) while in the bottom figure the hypotenuse is convex (bowing outward) making the contained area of the lower triangle greater than that of the top.