Timeline for Can you fold a square into a square of one-fifth the area?
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Aug 19, 2020 at 14:31 | comment | added | Mitsuko | I was amazed by its beauty, and then the idea came to my mind to make an origami problem out of it, asking the other way around - i.e., asking not to calculate the area of the small square, but to find how to make a square of one-fifth the size of the original square. That way the problem is made especially hard. I didn't really expect anyone to solve it, but you and @PaulPanzer solved it within just an hour or so, with your visual proof exactly coinciding with what had inspired me. I'm really impressed :) | |
| Aug 19, 2020 at 14:30 | comment | added | Mitsuko | I know I shouldn't post such a comment, but I can't help typing I'm really impressed. I was sitting at a very boring meeting and tried to entertain myself by folding a small square piece of paper. I folded it along the same lines as shown in your picture, and got curious as to what is the area of the resulting square. I did calculations and derived 1/5. Curious whether such a simple number can't be obtained in a simpler way, I gave it a thought and found the visual solution. | |
| Aug 18, 2020 at 12:09 | comment | added | TenMinJoe | @Chris We can see that e.g the large teal triangle is similar, but double the size of, the small teal triangle, because we know the small triangle's base is half as long. Given that, we can see that the red square runs along exactly half of the large teal triangle's side, because it ends where the small triangle begins. | |
| Aug 18, 2020 at 10:03 | comment | added | Chris | @Deusovi I'm not clear on the reasoning of why the line from the corner of the big square to the red square is the same length as the side of the red square... What have I missed? | |
| Aug 18, 2020 at 1:16 | comment | added | Deusovi♦ | @BlueRaja-DannyPflughoeft The proof I had in mind was easier (for me at least): You can reassemble the two shapes of each color into a square the same size as the red. (This is a square because (1) all four angles are right angles; (2) two adjacent sides are the same, because each of the four lines touching the corners are the same by symmetry.) | |
| Aug 17, 2020 at 23:43 | comment | added | BlueRaja - Danny Pflughoeft | Proof: consider the triangle consisting of both yellow tiles + the small teal triangle. Assuming a 1x1 square, the total area is (1/2)(1/2)(1) = 1/4. The small triangles are each 1/4th the area of the larger colored triangles (since they're similar triangles with half the base); thus the large yellow triangle takes up 4/5th of that area, or (4/5)(1/4) = 1/5 total area. So all triangles together = 4/5 area, leaving 1/5 for the red square. | |
| Aug 17, 2020 at 4:26 | comment | added | rrauenza | @nasch thank you - that's totally an optical illusion that the opposite intersections are not at the same height! | |
| Aug 17, 2020 at 2:52 | comment | added | nasch | In case anyone else is wondering, it took me a minute to figure this out. You can make those diagonal folds because you have the horizontal and vertical centerlines marked with creases. Fold between (for example) the upper right corner and the bottom center. | |
| Aug 16, 2020 at 20:23 | vote | accept | Mitsuko | ||
| Aug 16, 2020 at 18:04 | comment | added | Paul Panzer | Kind of a visual proof. Neat! | |
| Aug 16, 2020 at 17:55 | history | answered | Deusovi♦ | CC BY-SA 4.0 |