Make an Impossible Triangle

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Vietnam era veteran (USAF), former air traffic controller, former entrepreneur, former clergy, fo...

Intro: Make an Impossible Triangle

Here's a little paper mystery that will drive your mathematically inclined friends - and quite possibly your geometry teacher - completely bonkers. Using a simple visual aid, you can prove that a triangle you have constructed from card stock - right before their eyes if you like - is larger on one side than it is on the other. I learned this little oddity many moons ago from Mr. Martin Gardner, who wrote the column "Mathematical Games" for Scientific American magazine for twenty-five years and regularly came up with fascinating and mind bending curiosities.

Step 1: Print and Cut the Triangle

All you need is an 8 1/2 by 11 inch piece of card stock of other fairly stiff paper, a pair of scissors and the graph grid with triangle in Figure 1. Copy and save the graph grid to your computer, open it in the graphics program of your choice, then print it onto your card stock, selecting "Fit to Available Space" before you print it.

Once you've printed it out you will notice that you have an Isosceles triangle - a triangle with two equal length edges. The triangle is printed on a grid of squares that is 10 squares wide by 12 squares tall, which means that the rectangular grid has an area of 120 squares. If you remember your geometry, you'll know that an Isosceles triangle constructed in a 10 by 12 grid of 120 squares will have an area equal to exactly one half of the area of the rectangular grid from which it was constructed, that is, an area of 60 squares in this case. If you don't remember you'll have an opportunity to prove it to yourself in just a few seconds.

As carefully as you can, cut the rectangular grid from your paper, then cut the triangle from the rectangular grid along the black lines. Now you can demonstrate to yourself (or your audience) that the resulting triangle is exactly half the area of the original rectangle by taking the two leftover right triangles and arranging them on top of your Isosceles triangle to show that they exactly cover the Isosceles triangle - the Isosceles triangle has an area of 60 squares and the two right triangles you cut away each have an area of 30 squares. After this demonstration just throw away the two leftover right triangles. Now carefully cut the Isosceles triangle into six pieces along the red lines. Your six pieces, reassembled, will look like those in Figure 2.

Step 2: Assemble the Triangle With the Back Side Up

Turn the six pieces upside down and arrange them as shown in Figure 3 back into an Isosceles triangle. You will notice that when you're putting the triangle back together the pieces are not in the same positions they were originally...but they still form the Isosceles triangle. If you're demonstrating this to someone and they don't notice it there's no reason to point it out. If they do notice it, simply explain it as I just did - the pieces still form the Isosceles triangle. But here's where it gets weird. As you can see, you now have a two square hole in your triangle. This means that either (a) the back side of your triangle is two squares larger than the front side or (b) your paper is two squares smaller on the back side than it is on the front!

Step 3: Arrange the Triangle Pieces Into a Rectangle

But you're not done yet. Now turn one of each of the shapes back over so the grid side is up and arrange them as shown in Figure 4. As you can see, they form a nice rectangle, but it is missing four squares. There are two ways to interpret this phenomenon as well. First, you could deduce that your paper has shrunk again, as you are now missing four squares. However, some wise guy might look at the rectangle you've constructed and notice that it is seven squares by nine squares, for a total area of 63 squares. But since there are clearly four squares missing it means that the area of your triangle pieces is now actually 59 squares...and that means that you've only lost one square. You can explain this by telling your audience that it makes perfect sense: if the back side of your triangle was two squares smaller than the front side and now you're using half of the pieces facing forward and half of them facing backward it only stands to reason that you'd only lose half as much area as you did when they were all facing backward. If you're demonstrating this to your geometry teacher this is the point at which he or she might run out of the room screaming!

This is a lot of fun to mess with and all you need is your imagination to come up with ways to present this or stories to tell as you demonstrate it. And this is where the unicorn comes in: for instance - and especially if you're demonstrating this to a younger audience - you might tell them it is said that if you make a triangle large enough so that when you assemble the pieces backward you wind up with with a large enough hole in the center of the triangle a unicorn will come running through the hole. I cannot personally attest to the veracity of this claim as I have never tried to construct an impossible triangle large enough to attract the attention of a unicorn. You could just as easily make a large version out of poster board and do a little presentation for a geometry or science class...or just be satisfied with driving everyone you know crazy! I'm not going to try to explain why or how it works - part of the fun is the mystery. Just know that there is absolutely no sleight of hand involved; the grid squares are just that - perfectly square; there is no fancy or deceptive cutting of the pieces; just cut them as carefully and precisely as you can. Let me know how many people you drive bananas with this little curiosity, and if you find this Instructable intriguing I would truly appreciate your vote. Thanks, and...

Peace,

Radical Geezer

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    2 Discussions

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    HolopupenkoS

    7 months ago on Step 3

    It is not difficult at all to explain the apparent paradox. In fact, there is no paradox. Why? Because the larger and the smaller that form a pair, when compared, are not similar... and so the second version of the isosceles triangle (the one with a "hole" in the middle) is, in fact, NOT an isosceles triangle. It's close (hence easily missed), but not quite there. The easiest way to show this is to compare one of the larger triangles with one of the smaller ones. If, indeed, the second "hole" triangle was actually isosceles, then the larger and smaller triangles would be similar. In fact, they are not. If you compare the ratios of the lengths of the two sides, you get the expression 3:7 :: 2:5 which, of course, doesn't hold. A slightly more advanced way of understanding this is that the ratio of the straight sides of both the triangles are different, hence the slope (related to the derivative of the line) of the inclined sides are different. That is to say, in the second case of the "hole in the middle" triangle, the sides are ever so slightly kinked, i.e., not a straight line... and hence on a triangle in the first place. QED.

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    Radical GeezerHolopupenkoS

    Reply 7 months ago

    That is, indeed, one possible explanation. I personally prefer the alternate set of facts that says that the unicorns themselves devised this challenge in order to find those imaginative enough to construct something that looks like an Isosceles triangle but isn't in order to free something that looks like a horse but isn't because they knew that only such imaginative people would be properly equipped to care for the unicorns freed in the process. But I'll check with my unicorns again to see what they think. WTF ;-)