January 4, 2007 by Vurdlak
I was so thrilled when I found this optical illusion. It’s ancestor (see here), was one of the hardest optical illusions for me to understand. I had really much trouble with the original, but after few days I finally got it! When you understand the principle, you understand all the variations, like the one you see posted below! It’s pretty cool, don’t you think? Try figuring out the solution. Share it with your friends… it’ll give them some interesting moments. I love math as well, so it’s kinda “emotional” for me, if you know what I mean ;) You can also try and solve another, rather easy math optical illusion, I posted it in my early days. Enjoy!
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Hey it’s so cool! Love your site and widget!
=D
Coqui
In fact this aren’t the same tiles,
thats the only explanatiion
very cool.
quite simple tho. the shapes in the second diagram are not exactly the same as in the first.
if you take the shapes from the first digram and rearrange them as shown in the second diagram you will find a very narrow gap between the two sets of triangles, in a kind of ellipse shape.
Either cut and paste, draw ur own or just use trig :-)
i’m lost on this
you just have to count the left edge of the purple tile and the top edge of the green tile and you see how it adds up to 13…thus the 5*13
The Illusions Is… HEY! There’s My Hershy Kiss From Last Weekend! But..It’s Vanilla? Sombody Tell me Why…
they are the same shapes the extra “square” comes from a sliver down the middle diagonal of the second figure that you can’t see due to the the thickness of the black lines.
if you don’t believe me…see for yourself. take graph paper and cut out the first figure. then put the pieces together to make it look like figure 2. you’ll see that there is some space down the middle diagonal that causes the area of the rectangle itself to increase while the total area of the peices will of course remain the same.
i’m too mathy for my own good sometimes.
Yeah, yeah, like Spoon said..
It’s the rearrangement of the diagonal lines that creates the extra area.
The small diagonal part in the middle of the 2nd drawing is 3 blocks by 1 block, so the diagonal parts of the purple and the pink figures are 2 blocks, so they should be 6 blocks long. So in real the total diagonal cannot be a straight line.
Chuck Norris can make each of them add up to 66.
At first guess:
the area of the yellow+blue triangle =5*13/2=32.5
The area of the yellow triangle=8*3/2=12
The area of the pink stuff=(3+5)*5/2=20.
Therefore 12+20=32.5
Therefore 32=32.5
Now we all know that is not true (thickness of the line has no bearing on this).
Since yellow and pink are basic forms and yellow+pink is composed of the two, the latter is not a geometric form. Therefore there is no yellow+pink triangle (the “diagonal line” is not really a diagonal nor is it a line, but two rather~180 degrees arranged lines).
To test this use sinus. You are supposed to get the same value for the left yellow angle and, after splitting the pink stuff into a 3*5 rectangle and a triangle, the left pinky angle. They are different 0.35 versus 0.37.
There. I always hated wasting my time with math. Still do :)
Adi.
the line down the center of the rectangle is curved. Take a piece of paper and try to match the two corners without covering the line. Its impossible.
i actually took the time to figure this one out, i think, i cut it out on graph paper and it works, its true, but i noticed when i measured that the rectangle has one less block ( i measured in blocks) on the edges on the inside than the square does, im 12 so this is my logic it might be wrong tho
No more math!!!!!!!!!!!!! I had to do this at school It scares me. Math not the illusion
I see folks have figured it out, but just so it can be easily seen with just this diagram, look at the slopes. The slope between the blue and red shapes is 5/2, while the one between the yellow and green ones is -3/8. Turn the first one and it becomes -2/5, which is close but not the same. The last slope is -5/13, different from both the others. Maybe it won’t help visualize, but it adds up.
Like Spoon said, it is along the diagonal. I’ve done this with grid paper, and if you line up the pieces, there is the slightest bit on the edge wchich would line up to make the extra square.
-Ashton
yea the second one doesn’t have the same exact shapes. In order for the two big triangles (blue + green and red + yellow) to make a rectangle, they have to be similar to the small yellow and green triangles. but since 5/13 (sides of big triangle) isn’t equal to 3/8, they aren’t similar.
I counted the blocks in the square and there are 65 blocks. In the rectangle I counted 65 blocks also.
You don’t have to be very mathy to figure this one out. Just look at the shapes. They’re the same as in the first picture, just rearranged, yet you get a different volume. Pretty cool.
Hooray, trig! 3×1 overlap of yellow and green not similar to 8×3 whole, so they don’t fit together. Now my brain doesn’t hurt anymore!
a square has greater area than any other shape. thus the difference in number of squares.
a square has a greater area than any other shape thus the difference in the number of squares.
Yeah, I agree with MED. That makes sense because it is making the triangle longer. But at first glance this seemed impossible!
Comment No. 9 is right: imagine the edge to edge cut of a tile of 8 by 3 - green and yellow: factor 2.6666 and a cut of a tile of 5 by 2 - cutting segment between purple and violet: factor 2.5. The angle definetely cannot be the same -> the tiles (e.g. red and green) cannot fit, the gap is hidden by a thicker line that bends a bit.
The slopes of the triangle and the “trapazoid” are different
OMG. I aint no scientist but here are some of the daftest comments I have yet to see posted on this site….
No.23 who cut them out but somehow managed to get the new shape to be only 12 squares long :-S
No.28 who thinks 8×8 = 65… d-oh!
but the top prize goes too….
No.31/2 who thinks that shapes can magically change their area without changing their sizes…. hmmm
Well done to everyone else tho, this was a bit of a stinker lol.
fun one. you can use the slopes to find the extra space. first image slope 3/5 does not equal 5/13 in the second. but it is close which is why it still looks right.
dats kew
the diagonal on the right rectangle is not a true diagonal (i.e. it is not a straight line segment).
It’s very simple. The two diagonal lines are not exactly the same and don’t have the same ratio. If you look closely, you can see the difference.
hi! love ur site, but aren’t these just moved around? u can move them back correctly!
I think it’s because, in the places where a diagonal line cuts through a square, the squares are not cut exactly in half.
all you have to do is this, take tyhe yellow triangle on the left and slide up 3 squares so its the same on the right, now notice the lefis 3X6 while the right is 3X5 thats the easiest way to see the slight difference in slope angle without going into trig.
its all to do with the parallel postulate…derrrr!
triangles.
The surface area of the second one is more that the first. by rearranging the blocks you can make change the surface area to be greater using the same blocks.
noooooooooo math it burns it burns!!!
use PEDMAS kids…