What Do You Know About Hinged Dissections?

The pieces of a hinged dissection are hinged at the vertices. This means that the points where the individual pieces connect and rotate are located at the vertices of the shapes involved in the dissection.

A twist hinged dissection is different from a regular hinged dissection because it is three-dimensional. While a regular hinged dissection occurs in a plane, a twist hinged dissection exists in three dimensions. This means that the objects being dissected can be manipulated and reconfigured in three-dimensional space, allowing for more complex and intricate transformations.

A hinged dissection is a geometric puzzle where a shape can be divided into smaller pieces that are connected by hinges. These hinges allow the pieces to move and be rearranged into different configurations. Therefore, the correct answer is "hinges" as they are the connecting element in a hinged dissection.

Henry Ernest Dudeney is the correct answer for introducing the idea of hinged dissections. Dudeney was a famous English mathematician and author who specialized in mathematical puzzles and recreations. He is known for his contributions to recreational mathematics, particularly his work on dissection puzzles. Dudeney's interest in hinged dissections led him to explore and create various puzzles and solutions involving the rearrangement of polygons by hinging them along certain edges. His work in this area has had a significant impact on the field of recreational mathematics.

In a twist hinge, the hinges are placed on the edges of the pieces. This allows the pieces to rotate or twist around the edge, providing flexibility and movement. Placing the hinges on the edges ensures that the pieces can pivot smoothly and maintain stability while twisting. Therefore, the correct answer is "Edge."

Polyominoes are formed by joining unit squares at their edges. A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. The term "polyomino" is derived from the Greek words "poly" meaning "many" and "omino" meaning "square". Each individual square in a polyomino is called a unit square. Therefore, polyominoes are the correct answer to the question.

Adding a new piece is not a way to form a new shape with a hinged dissection. Hinged dissection involves rearranging and repositioning the existing pieces of a shape to form a new shape, without adding any new pieces. This process relies on the flexibility of the hinged connections between the pieces. Adding a new piece would not be considered a hinged dissection, as it involves introducing a completely new element to the shape transformation.

If two polygons have a common hinged dissection, it means that they can be dissected into the same set of smaller polygons that can be rearranged to form either of the original polygons. In order for this to be possible, both polygons must have the same area. If one polygon was twice the size of the other, it would not be possible to rearrange the smaller polygons to form both of them. Similarly, if one polygon was half the size of the other, the smaller polygons would not be able to form both of them. Therefore, the correct answer is that both polygons must have the same area.

When forming a new shape in a dissection, the chain will be broken. This means that the original arrangement or sequence of connected components will be disrupted or disconnected. This is because dissection involves rearranging or separating the parts of a shape to create a new configuration or arrangement. The other options state that the original shape will be changed, the original shape will remain intact, and the chain will not be broken, but these are incorrect as they contradict the nature of dissection.

The first hinged dissection refers to the first known transformation of an equilateral triangle into a square using only hinged movements. This means that the triangle can be rearranged by folding along certain points without any cuts or additions. The equilateral triangle to square transformation is considered the first hinged dissection because it was the earliest discovered and documented example of this type of geometric transformation.