February Moment of Science:
Fold & Cut Theorem

February Moment of Science – Fold and Cut Heart

This Moment of Science is all about math.  Did you know that you can cut any shape made of all flat edges by folding a single piece of paper and cutting once?  The earliest known description of the so called “Fold and Cut Problem” comes from Japan in 1721.  The first proof of this theorem was published in Canada in 1999…that’s 278 years later!  Although this complex mathematical theorem may seem like it is only useful for creating advanced origami and impressing your family and friends, it has actually turned out to be scientifically important for figuring out how to fit roofs onto irregularly shaped buildings. Follow these instructs to fold and cut a heart for Valentine’s Day.

Start with a square piece of paper (such as origami paper).  If you do not have any, take a piece of letter copy paper (8.5” x 11”) and fold and cut it into a square:

Begin by taking the top right corner (*) and folding it down until the top edge overlaps the left edge of the paper.

1.1

Next take the top left corner (*) and fold it down so the left edge touches the new right edge, forming the top of a triangle.

1.2

Make sure to run your finger along the crease so it is a smooth and flat fold.

1.3

Fold and cut off the bottom, excess paper.  Unfold the triangle – this will be your square piece of paper.

Now that you have a square piece of paper, you can begin!

1. Fold the paper in half so the left edge touches the right edge.

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2. Fold the bottom left corner up so the bottom edge overlaps the right edge.

2.1

3. Your paper should look like this.

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4. Turn the paper over so it looks like this.

3.2

5. Fold the right edge so it is about a third (1/3) of the way across the paper – make sure the top edge is still straight.

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6. Flip the paper around so the triangular point is pointing away from you.

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7. Take the top triangular point and fold it down and to the right. As you start to fold it, you will notice a point on the other side of the paper. You want to line up the two points so that it looks like this:

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8. Turn the paper upside down and rotate so the long edge of the hidden triangle is facing you (the point of the hidden triangle is pointing away from you)

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9. Take the top point and fold it down so that a 90 degree (or right angle) is formed in the top right. Your paper should now look something like this:

10. Fold the left edge so it is on top of the fold in the middle of the paper. Your paper should now look like this:

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11. It is time for your single cut! Unfold your heart.

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heart

*Hint: If the shape does not turn out how you want it to look, try folding a new piece of paper, making sure to angle the straight cut more toward the right.

Several modern mathematicians have challenged themselves to use the Fold and Cut Theorem to create their own alphabet.  Can you figure out how to fold and cut the letters of your name?  Here are some helpful hints for alphabets others have created (as well as a few more designs for fold-and-cut hearts): http://www.takayaiwamoto.com/Onecut_Origami/onecut_origami.html.