Curvilinear and three-dimensional paradoxes. Effective trick and its solution

EFFECTIVE FOCUSES AND THEIR CLUES
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Curvilinear and three-dimensional paradoxes. Focus secret

Spectacular tricks and their clues

Directory / Spectacular tricks and their clues

Focus Description:

The field of paradoxes with area variation is just beginning to be developed. Are there any curvilinear shapes, such as circles or ellipses, that can be cut into pieces and then reassembled in such a way that, without noticeable distortion of the figure, internal holes are obtained? Are there three-dimensional figures that are specific to three dimensions, i.e., are not a trivial consequence of two-dimensional figures? For it is clear that any plane figure we have encountered in this chapter can be "added a dimension" by simply cutting it out of fairly thick cardboard, the height of which is equal to the "length of the third dimension"*. Is it possible to cut a cube or, say, a pyramid in a not very complicated way into parts so that, by composing them in a new way, we get noticeable voids inside?

The same non-triviality requirement should also be applied to the problem of cutting curvilinear plane figures. It is clear that if cutting and composing in a new way admits a square, then a plane figure with any boundary, containing the given square inside itself, also admits cutting and composing.

The answer will be this: if you do not limit the number of parts, then it is not difficult at all to indicate such spatial figures. This is clear enough in the case of a cube. Here an internal void can be obtained, but the question of the smallest number of parts with which this can be achieved is more complicated. It can certainly be made from six parts; it is possible that this can be achieved with a smaller number.

Such a cube can be effectively demonstrated as follows: take it out of a box made exactly according to the cube, disassemble it into parts, revealing a ball inside, put the parts back into a solid cube and show that it (without the ball) still fills the box densely. We will conjecture that there must be many such figures, both flat and spatial, and also distinguished by simplicity and elegance of form. Future explorers of this curious field will have the pleasure of discovering them.

Author: M.Gardner

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