Is it possible to cut a triangle into such a number of parts so that one can put a square together?
An affirmative answer to this question was given back in 1807. In a more general form, it sounded like this: "Any two polygons of the total area should have a common cut." This is the Boyle-Gervin theorem, proved in 1807. If we have a triangle and a square, and we know that their surfaces are the same, cutting the triangle into several polygons, we can fold the square like a mosaic.
But here is a more difficult question. Is it possible to cut so that all parts are connected in a continuous chain?
Dudeny dissection (by the name of the author), made in the form of animation, shows us how the triangle is converted into a square, and then into a hexagon and back into a triangle (Wikipedia animation movie was used).
Initially, the task of cutting a triangle was proposed by Henry Dudeny in the form of a puzzle and published in the Daily Mail newspaper (issues of February 1 and 8, 1905). Later this puzzle was included in the book "Canterbury puzzles" and to this day is included in the hundred best puzzles of "all time."
The original text is as follows:
Many attempts were made to induce the Haberdasher, who was of the party, to propound a puzzle of some kind, but for a long time without success. At last, at one of the Pilgrims' stopping-places, he said that he would show them something that would "put their brains into a twist like unto a bell-rope." As a matter of fact, he was really playing off a practical joke on the company, for he was quite ignorant of any answer to the puzzle that he set them. He produced a piece of cloth in the shape of a perfect equilateral triangle, as shown in the illustration, and said, "Be there any among ye full wise in the true cutting of cloth? I trow not. Every man to his trade, and the scholar may learn from the varlet and the wise man from the fool. Show me, then, if ye can, in what manner this piece of cloth may be cut into four several pieces that may be put together to make a perfect square."
Now some of the more learned of the company found a way of doing it in five pieces, but not in four. But when they pressed the Haberdasher for the correct answer he was forced to admit, after much beating about the bush, that he knew no way of doing it in any number of pieces. "By Saint Francis," saith he, "any knave can make a riddle methinks, but it is for them that may to rede it aright." For this he narrowly escaped a sound beating. But the curious point of the puzzle is that I have found that the feat may really be performed in so few as four pieces, and without turning over any piece when placing them together. The method of doing this is subtle, but I think the reader will find the problem a most interesting one.
The figures show how you can cut a triangular piece of matter into 4 parts. Then, how to put the resulting pieces into a square.
9. Put the segment JK equal to BE.
10. From point D, lower the perpendicular to EJ with base at L.
11. From point K, we drop the perpendicular to EJ with base at M.
You can make such an entertaining puzzle, which allows you to convert a regular triangle into a square by an unbroken chain of polygons, independently from paper. But in order to make it movable, it is necessary that it be not a flat, but a three-dimensional construction. Therefore, each of the polygons receives the height and is transformed into a prism.
We suggest you download the net parts in the form of the 4th prism.
For each "piece" of the puzzle, we glue together a separate prism, and then we connect the individual prisms together in the form of hinges.
One can specify the following mathematical characteristics in each of the five Platonic solids: 1....