Geometrical Vanishes

Look at the rectangle below:

There seems to be nothing strange about it.

No, not really. Now, if you cut the square along the lines and then rearrange it as shown,you will get another rectangle of the same size, but with a hole in it! It is nearly taken as an axiom in Euclidean geometry that the area of anything in 2-dimensional space is a constant regardless of how it is placed or arranged.

However, the square above leaves a hole when the pieces of the square are rearranged to give another square with the same dimension! Where has the area gone?

Indeed there are many examples that give similar peculiar results.

Let¡¦s look at another example:

Look at the triangle

If we rearrange the two triangles at the bottom and the 2 pieces at the middle, we would again get another triangle exactly the same, yet with 2 holes inside. What¡¦s more, you are able to control the number of holes inside it by arranging the pieces to different positions.

The following square will verify what I just said.

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The squares and triangles I just mentioned are called the Curry Paradox. Actually what you see is an illusion because we should have more faith in Euclid Geometry. The area is not going to diminish!

So how all these things happened?

Let¡¦s look at the rectangle I first mentioned again. If point X is located exactly 5 units from the side and 3 units from the base, then the diagonal line will not be perfectly straight, though the deviation will be so slight as to be almost undetectable. When triangles B and C are switched, there will be a slight overlapping along the diagonal in the second figure. The missing square unit spreads from corner to corner, forming the overlap along the diagonal.

On the other hand, if the diagonal in the first figure is ruled accurately from corner to corner, then line XW will be a trifle longer than 3 units. As a consequence, the second rectangle will be slightly higher than it appears to be. The missing square is distributed along the width of the rectangle, only that the discrepancy is too minute to be detected.

In short, the secret of all these paradoxes lies in the diagonal in a similar manner as the example just shown.

Another interesting example goes like this.

Cut the checkerboard above along that diagonal. Then shift part B downward as shown and snip off the projecting triangle at the upper right corner and fit it into the triangular space at the lower left, a rectangle of 7 by 9, which is a square unit less than the original area of the square! This is the Checkerboard Paradox.

The answer lies in the fact that the diagonal line passes slightly below the lower left corner of the square at the upper right corner of the checkerboard. This gives the snipped-off triangle an altitude of 1 1/7 rather than 1, and gives the entire rectangle a height of 9 1/7 units. The addition of 1/7 of a unit to the height is not noticeable, but when it is taken into account, the rectangle will have the expected area of 64 square units. The paradox is even more puzzling to the uninitiated if the smaller squares are not ruled on the figure. When the square units are shown, a close inspection will reveal their inaccurate fitting along the diagonal cut.

There are other examples of disappearing area of different forms but actually applying the same principle.

Let's look at the following example:

All these puzzles may not be familiar to you. However, all of them originates from one of the most famous paradox.

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Remember the "vanishing face"? If not, look at the picture below.

(We can see that if we move the part under the broken line leftward in the upper picture, there will be a face missing as shown by the lower picture.

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To give a simple explanation, we can first look at the following figure.

Cut the rectangle along the diagonal and slide the lower part downward and leftward to the position shown and count the number of vertical lines and you will find that there are only nine, with one missing. Slide the lower part back to its former position and the missing line returns. Which is the line that has returned and where does it come from? This is the Line Paradox.

The secret, like the previous example, lies in the fact that there is a progressive decrease in the length of the segments above the diagonal and a corresponding increase in the length of segments below. What happens is that eight of the ten lines are broken into two segments, then these sixteen segments are redistributed to form nine lines, each a trifle longer than before. Because the increase in the length of each line is slight, it is not immediately noticeable. In fact, the total of all these small increases exactly equals the length of one of the original lines. Therefore, there is actually not a line which vanishes.

Many puzzles operates by the same principle, including the Checkerboard Paradox.

Here are more interesting examples:

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The American flag of fifteen stripes is cut into 2 pieces and is fitted together to form a flag of thirteen stripes. A puzzle devised by Sam Loyd.

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Another Puzzle, called "Get off the Earth", is again invented by Sam Loyd. The thirteen warriors becomes twelve when the circle is cut and rotated clockwise until the arrow points to N.E.. What is interesting is that Sam ingeniously bend the line paradox into a circular form and replaced the lines with figures of Chinese warriors.

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The DeLand Paradox is also interesting.

Cut the card along horizontal line AB and vertical line CD to form three pieces. Exchange the positions of the lower part and the same result as the line paradox is achieved. One of the playing card vanishes. The two paradoxes actually work following the same principle. To shadow the arrangement still further, DeLand added some extra cards which play no part in the paradox.