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enter image description here

Can anyone tell me please where does the gap come from?

Thanks and sorry if the question is not exactly relevant, I just didn't know where else to ask.

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    $\begingroup$ The slope angle of the red and blue pieces is different. $\endgroup$
    – almagest
    Apr 26, 2016 at 19:12
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    $\begingroup$ Or, watch carefully the points $(5,2)$ and $(8,3)$ on both pictures. $\endgroup$
    – Berci
    Apr 26, 2016 at 19:27
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    $\begingroup$ Martin Gardner has an informative chapter on this in his book Mathematics, Magic, and Mystery. $\endgroup$
    – Brian Tung
    Apr 27, 2016 at 4:26
  • $\begingroup$ This question gave me a huge craving for infinite chocolate! $\endgroup$ Apr 27, 2016 at 18:38
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    $\begingroup$ Some people would say this is an optical illusion. Your brain seeing what it expects to see. I say take a lesson from wood shop, pick up the paper, and sight down the lines. You'll see they aren't really straight without doing any math. $\endgroup$ Apr 27, 2016 at 20:35

3 Answers 3

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If I did this

enter image description here

Would you ask where the (white) hole came from?

Addeudum

enter image description here

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    $\begingroup$ Yes, actually I probably would... to my eyes that hole looks way more massive than the added area! $\endgroup$
    – user541686
    Apr 27, 2016 at 9:10
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    $\begingroup$ @Mehrdad - Than what added area? All I did was move four objects around. The only difference is that I exagerrated the dimensions of the two triangles. $\endgroup$ Apr 27, 2016 at 10:25
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    $\begingroup$ @StevenGregory In the lower figure, you also added a line creating a new white rectangle, and thus increased (compared to the upper figure) the area surrounded by lines. $\endgroup$
    – JiK
    Apr 27, 2016 at 10:35
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    $\begingroup$ @StevenGregory: No, I'm just looking at the overall shape, not the individual shapes. I'm saying the difference of the two shapes' areas seems to be way less than that of the white rectangle. $\endgroup$
    – user541686
    Apr 27, 2016 at 11:25
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    $\begingroup$ @Mehrdad - I added another picture. Does that help? Note that the difference in area will be 1 if, for example, b,h,B,H are consecutive Fibonacci numbers. $\endgroup$ Apr 28, 2016 at 16:29
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Figure out the area of the little gap in the middle:

enter image description here

(you can probably guess it, but it's easy to calculate).

Try it for yourself before peeking at the explanation at the bottom.

From there it's easy to see where the area for the square comes from.


Total area of the 13 × 5 region = 65 squares.

Total area of the two unmarked rectangular parts = (5 × 3) × 2 = 30

Total area of blue triangles = (5 × 2 × ½) × 2 = 10

Total area of red triangles = (8 × 3 × ½) × 2 = 24

Remaining area = 65 - 30 - 10 - 24 = 1 square.

If it's still not clear, rotate the upper two triangles each about the center of its own diagonal (which leaves the upper edge of the gap unchanged), you get the triangles from your two diagrams overlaid on each other in the correct positions:

enter image description here

It takes the extra square to make up for the area of that skinny parallelogram-shaped gap between the regions covered by the two arrangements of shapes.

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The line over the red part is not quite as steep as that over the blue part. Just compute the two slopes and you'll see that.

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