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    • Folding fractions

      Rachel Thomas
      2 March, 2015
      16 comments

      Can you fold a piece of paper in half? Of course you can, it’s easy, you just match the two corners along one side. But can you fold it in thirds? You might be able to fold it into thirds with a bit of fiddling and guessing, but what about into fifths? Or sevenths? Or thirteenths? Here is a simple way you can fold a piece of paper into any fraction you would like – exactly – no guessing or fiddling needed!

      Folding a third

      To start, take a square piece of paper and mark half way along top side with a small crease. Now fold the bottom left corner to meet this halfway mark and crease the paper.

      Mark half way
      Fold bottom corner to middle of top side

      The first thing to notice is an interesting relationship between the three triangles you've created, one on the top-left, one on the top-right, and one on the bottom-left overhanging the side of the paper.

      We've folded three similar triangles

      At the centre of the top edge three angles fit together to make 180 degrees. One of these is a right angle (marked with a square corner in the picture), which means that the pair of other angles (marked with the single and double lines in the picture) add to 90 degrees. Each of the angles of this pair also form part of a right-angled triangle (the top-right and top-left triangles), which means the other angle of the pair also appears in each of these right-angled triangles. And the bottom-right triangle also shares an angle with the top-right triangle. And, as it is also a right-angled triangle, it too contains the same pair of angles.

      This all means that the three triangles are similar – they have the same shape. That is to say, they all have the same angles, and so, have the same ratios of lengths of their sides. We can use these similar triangles, along with Pythagoras' theorem, to fold the paper in thirds.

      We've folded three similar triangles

      If we take the length of the side of our square paper to be 1, our top left triangle has one side of length 1/2, one side of unknown length which we'll call x, giving the other side a length of 1−x. Then, by Pythagoras' theorem, we know that x2+(12)2=(1−x)2. Expanding this out: x2+14=1−2x+x2 which, by rearranging, gives us x=3/8.

      Finding the length of the sides of the top-right triangle

      We can now calculate the length of the sides in the top-right triangle. This triangle has one side of length 1/2 and another side of unknown length y. Because the triangles are similar we know that the ratio of the lengths of their sides must be the same. So y12=12x. And as x=3/8, we find that y=2/3. So we can construct 1/3 by folding this length y in half.

      Folding a third

      Folding any fraction

      Kazuo Haga, a retired professor of biology from Japan, came up with this ingenious method. Although a biologist, he was very interested in using origami to explore mathematics (you can find our more in his fascinating book). In fact, Haga realised that this method was even more useful.

      Suppose instead of folding your bottom left corner of the paper to some point halfway along the top edge, you instead fold the bottom left corner to a point a distance of k along the top edge.

      Folding any fraction

      Then, the top-left triangle has sides of length k, x and 1−x. Then, as above, Pythagoras' theorem tells us that: x2+k2=(1−x)2 which rearranged gives x=(1−k2)/2. And by the similar triangles argument we have that y1−k=kx. If we put these two equations together we find y1−k=k(1−k2)/2 which can be rearranged as y=2(1−k)k1−k2. And (since 1−k2=(1−k)(1+k)) this can be simplified to y2=k1+k. This is known as Haga’s Theorem and it allows us to fold any fraction we would like from a square piece of paper.

      Folding a half
      Folding a third

      We've already seen k=1/2 allowed us to fold 1/3. What if we rotate the square paper and fold the bottom left corner up to a point 1/3 along the top edge? Then we have k=1/3 in Haga's Theorem which means that y2=1/34/3=1/4.

      Starting from a third
      Folding a quarter

      So far, starting from 1/2, we've used Haga's method to fold 1/3 and then 1/4. And if we carry on repeating this method we can fold any fraction. If we start with k being some number 1/N, then y2=1/N1+1/N=1N+1. So by repeating Haga’s method over and over, we can construct every unit fraction (one with 1 in the numerator): folding 1/5....

      Starting from 1/4
      Folding 1/5

      ... folding 1/6....

      Starting from 1/5
      Folding a 1/6

      ...folding 1/7...

      Starting from 1/6
      Folding 1/7

      ...and so on. And then we can use these to create any multiple of these unit fractions, and so fold any rational number. Hooray for Haga!


      About the author

      Rachel Thomas is Editor of Plus.

      About this article

      This article was inspired by content on our sister site Wild Maths, which encourages students to explore maths beyond the classroom and designed to nurture mathematical creativity. The site is aimed at 7 to 16 year-olds, but open to all. It provides games, investigations, stories and spaces to explore, where discoveries are to be made. Some have starting points, some a big question and others offer you a free space to investigate.

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      Comments

      Anonymous

      4 March 2015

      Permalink

      Very cool. Interesting that you can trisect a side, but this does *not* allow you to trisect an angle (i.e., fold a 30 degree angle at the corner) since this would require you to divide a side by 1/sqrt{3} which, alas, is *not* rational.

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      Anonymous

      9 March 2015

      In reply to Still can't trisect an angle. by Anonymous

      Permalink

      You can trisect an angle with folding. But you can't do so with a compass and ruler. See trisect angle

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      Anonymous

      9 March 2015

      In reply to Still can't trisect an angle. by Anonymous

      Permalink

      It's true you can't trisect an angle with a compass and ruler. But you can trisect an angle with folding/origami.

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      Anonymous

      5 March 2015

      Permalink

      The link for Haga's book is broken

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      Marianne

      9 March 2015

      In reply to The link for Haga's book is by Anonymous

      Permalink

      Thanks for spotting that, we have fixed it.

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      Anonymous

      2 April 2015

      Permalink

      Lovely idea .. just that in practice it's usually an A4 piece of paper that needs folding into thirds to fit in an envelope ... can anyone improve on this idea for this context?

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      Jon

      5 September 2016

      In reply to Practically by Anonymous

      Permalink

      Haga's method is great for squares, but you're right, it won't work very well for an A4 (or any rectangle) sheet. You will need to look at the crossing diagonals method for that. Fold one crease diagonally across the A4 sheet, and one line diagonally from one corner to halfway down the edge, and they will intersect at the 1/3 point.

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      Fu Wei

      18 July 2019

      In reply to Haga's method is great for by Jon

      Permalink

      I have summerized 16 methods for trisecting A4 Paper:
      https://www.flickr.com/photos/119967028@N08/48308136022/in/dateposted-p…

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      Anonymous

      2 April 2015

      Permalink

      It is a nice article.

      Haga's theorem are there,
      http://www.origami.gr.jp/Archives/People/CAGE_/divide/02-e.html
      Mathematics of paper folding,
      http://en.wikipedia.org/wiki/Mathematics_of_paper_folding#cite_note-12

      Haga's book, we can get by Amazon,
      ORIGAMICS: MATHEMATICAL EXPOLORATIONS THROUGH PAPER FOLDING

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      Anonymous

      28 April 2015

      Permalink

      It is not as elegant as the intercept theorem, thousands of years ago.

      https://www.flickr.com/photos/129045076@N07/16394585596/in/set-72157650…

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      Anonymous

      10 June 2015

      Permalink

      Please provide more information for the sections of which you work the equations out. I am still stuck on how you converted X^2+1/2^2=(1-X)^2 into X^2+1/4=1-2X+X^2. Please provide the answer to that equation in a comment or edit the page to include more information. Thanks :)

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      Marianne

      10 June 2015

      In reply to Please more info by Anonymous

      Permalink

      You simply multiply out the brackets on each side of the equation: (1/2)^2=1/4 and (1-x)^2 = 1-2x+x^2.

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      Anonymous

      31 May 2016

      Permalink

      This was really easy to follow for someone who knows how to do stuff with equations. It was really helpful and it looks like it took a while to make so thanks for putting in the effort.

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      Hiroshi Okumura

      25 December 2017

      Permalink

      A generalization of Haga's fold and several new theorems can be seen in the paper "Haga's theorems in paper folding and related theorems in Wasan geometry Part 1". The paper can be dowload at http://sangaku-journal.eu/.

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      Hiroshi Okumura

      27 December 2017

      Permalink

      The relationship between the two fractions can be obtained from
      Theorem 3.1 in the paper. Thank you very much.

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      George Plousos

      7 June 2020

      Permalink

      The same folding of the square has two more interesting properties. I have made their interactive constructions below (are activated by double clicking).

      A famous result from Japan (1893)
      https://www.geogebra.org/m/P8JF3mHB

      Solution to the problem of doubling the cube
      https://www.geogebra.org/m/hdEMmBWY

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      Read more about...

      pythagoras' theorem
      origami
      creativity
      Haga's theorem
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