Is it possible to cut a triangle into such several 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 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 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 Dudeney 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."
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 that he knew no way of doing it in any number of pieces after much beating about the bush. "By Saint Francis," saith he, "any knave can make a riddle methinks, but it is for them that may 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.

1. Divide AB in half at point D
2. BC split in half at E.

 

 

4. Build an arc from point E, with radius EB. At the intersection, with line AE we get point F.
5. Divide AF in half in G.
6. Draw an arc AF with center at point G.
7. Continue the EB to the intersection with the arc at point N.

 

8. From E as from the center with a radius of EH, we describe the arc HJ.

 

After this operation, we managed to get the first cut of the triangle - the segment EJ.

 

9. Put the segment JK equal to BE.

10. From point D, lower the perpendicular to EJ with the base at L.

11. From point K, we drop the perpendicular to EJ with the base at M.

 

We obtain segments along which cuts should be made.

 

 

The author of the problem, the English mathematician Henry Ernest Dyu-Deni (1857–1930), is known as one of the eminent creators of mathematical puzzles. Speaking with this task, set in a more general form, in front of the Royal Society at Burlington House and at the Royal Institute, he added another figure, in which the solution of the problem is shown in a more interesting and practical form. All parts of the model can be made of mahogany, fastened with bronze hinges so that it was convenient to show to the audience. It is easy to see that all four parts form a kind of chain. If you twist this chain in one direction, you get a triangle, and if you twist it in the opposite direction, you get a square. Now such a division of the triangle is called “Dudeny dissection” in honor of the author or “hinged cutting” since there should be hinges at the junction points of the polygons, which make it easy to turn the structure and go from triangle to square and back.

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 to make it movable, it must 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 in the form of hinges.

Net part 1

Net part 2

Net part 3

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In addition to the "classic" cutting off the triangle, you can also make other beautiful articulated cuttings of geometric shapes. 1. Articulated cutting of a square, which allows us to convert it into a regular hexagon.
The task is as follows: how to cut a square so that a regular hexagon can be assembled from its parts?

 

Download net 1 cutting square

Download net 2 cutting square

Download net 3 cutting square

Download net 4 cutting square

Download net 5 cutting square

Download net 6 cutting square
 

 

2. Articulated cutting of a regular hexagon, which allows us to convert it into a triangle.
The task is: how is a regular hexagon so that a triangle can be assembled from its parts?

 

Download net 1 of cutting a regular hexagon

Download net 2 of cutting a regular hexagon

Download net 3 of cutting a regular hexagon

Download net 4 of cutting a regular hexagon

Download net 5 of cutting a regular hexagon

 

3. Articulated cutting of the cross, formed by five squares, allows converting it into a single square.
The task is as follows: how to cut a cross formed by five squares so that a single square can be assembled from its parts?

 

 

Download net 1 cut of cross

Download net 2 cut of cross

Download net 3 cut of cross

Download net 4 cut of cross

 

 

Canadian designer Emmanuel Peluchon repeated the puzzle Dyudeni, making it out of wood. And placed the clock inside the part of the puzzle:

The design is made of wooden prisms, neatly fastened with conventional hinges.
The cost of such a designer puzzle is 380 Canadian dollars.

 

 

The picture shows all the puzzles illustrating the hinge cut, collected from the proposed shape nets.