What Do You Know About Moser's Worm Problem?

1) Which mathematician formulated Moser's worm problem?

Adam Logan

Newt Moser

Leo Moser

Jones Moser

Leo Moser is the correct answer because he was the mathematician who formulated Moser's worm problem.

Explanation

Leo Moser is the correct answer because he was the mathematician who formulated Moser's worm problem.

2) In the convex case, from whose selection theorem is the existence of a solution acquired?

Pythagoras

Avogadro

Blaschke

Venn

Blaschke's selection theorem is used to prove the existence of a solution in the convex case. This theorem states that for any sequence of convex sets in a compact metric space, if the diameter of the sets tends to zero, then there exists a point that belongs to all the sets. Therefore, Blaschke's theorem is the appropriate theorem to establish the existence of a solution in the convex case.

Explanation

Blaschke's selection theorem is used to prove the existence of a solution in the convex case. This theorem states that for any sequence of convex sets in a compact metric space, if the diameter of the sets tends to zero, then there exists a point that belongs to all the sets. Therefore, Blaschke's theorem is the appropriate theorem to establish the existence of a solution in the convex case.

3) What does Moser problem relate to?

The problem asks for the region of smallest area that can accommodate every plane curve of length.

A mathematical problem that involves a hexagonal lattice.

The Moser problem relates to finding the region of smallest area that can accommodate every plane curve of length. This means that no matter what shape or length the curve is, it can fit within this region. The problem is asking whether every problem that can be quickly verified can also be solved quickly. This relates to computational complexity and the ability to solve problems efficiently. The answer provided is a concise summary of the problem statement.

Explanation

The Moser problem relates to finding the region of smallest area that can accommodate every plane curve of length. This means that no matter what shape or length the curve is, it can fit within this region. The problem is asking whether every problem that can be quickly verified can also be solved quickly. This relates to computational complexity and the ability to solve problems efficiently. The answer provided is a concise summary of the problem statement.

4) What does the term accommodate mean in Moser's worm problem?

It means that the curve may be rotated and translated to fit inside the region.

The existence of a solution follows from the Blaschke selection theorem.

It is not trivial to determine whether a given shape forms a solution.

It is not completely trivial that a solution exists.

The term "accommodate" in Moser's worm problem refers to the ability to rotate and translate the curve in order to fit it inside the given region. This implies that the curve can be adjusted in position and orientation to satisfy the problem's requirements.

Explanation

The term "accommodate" in Moser's worm problem refers to the ability to rotate and translate the curve in order to fit it inside the given region. This implies that the curve can be adjusted in position and orientation to satisfy the problem's requirements.

5) In which year was it formulated?

1965

1966

1967

1968

1969

The correct answer is 1966. This means that the formulation being referred to was created in the year 1966.

Explanation

The correct answer is 1966. This means that the formulation being referred to was created in the year 1966.

6) Which part of mathematics does it deal with?

Logic

Mathematical physics

Calculus and analysis

Geometry and topology

Dynamical systems and differential equations

Geometry and topology deal with the study of shapes, sizes, and properties of objects. They both involve the study of space and how objects are related to each other. Geometry focuses on the properties of specific shapes and figures, while topology studies the properties of spaces that are preserved under continuous transformations. Both fields are important in various areas of mathematics, including physics, engineering, and computer science.

Explanation

Geometry and topology deal with the study of shapes, sizes, and properties of objects. They both involve the study of space and how objects are related to each other. Geometry focuses on the properties of specific shapes and figures, while topology studies the properties of spaces that are preserved under continuous transformations. Both fields are important in various areas of mathematics, including physics, engineering, and computer science.

7) What is the minimum area of a shape that can cover every unit-length curve?

The problem asks for the region of smallest area that can accommodate every plane curve of length 1

The problem asks for the region of highest area that can accommodate every plane curve of length 1

The given answer explains that the minimum area of a shape that can cover every unit-length curve is 0.232239. This is achieved by using a min-max strategy for the area of a convex set containing a segment, a triangle, and a rectangle. The answer suggests that a possible solution is a rhombus with vertex angles of 60 and 120 degrees, and a long diagonal of unit length. This solution provides a lower bound for the convex cover, ensuring that it can accommodate every plane curve of length 1.

Explanation

The given answer explains that the minimum area of a shape that can cover every unit-length curve is 0.232239. This is achieved by using a min-max strategy for the area of a convex set containing a segment, a triangle, and a rectangle. The answer suggests that a possible solution is a rhombus with vertex angles of 60 and 120 degrees, and a long diagonal of unit length. This solution provides a lower bound for the convex cover, ensuring that it can accommodate every plane curve of length 1.

8) Which group of scientists showed that no finite bound on the number of segments in a polychain would suffice in a solution?

Panraksa, Wetzel, Wichiramala

Gerriets, Poole, Wetzel

Leo Moser, Wichiramala, Panraksa

Blaschke, Panraksa, Wetzel

Panraksa, Wetzel, and Wichiramala are the group of scientists who showed that no finite bound on the number of segments in a polychain would suffice in a solution. The other options do not include all three of these scientists, so they are not the correct answer.

Explanation

Panraksa, Wetzel, and Wichiramala are the group of scientists who showed that no finite bound on the number of segments in a polychain would suffice in a solution. The other options do not include all three of these scientists, so they are not the correct answer.

9) Which of the following shapes gives the optimal solution with larger areas?

Rectangles

Squares

Rhombi

Circles

Rectangles give the optimal solution with larger areas compared to squares, rhombi, and circles. Rectangles have four sides with opposite sides equal in length, allowing for more flexibility in adjusting the dimensions to maximize the area. Squares have equal sides, limiting the potential for increasing the area. Rhombi have equal sides but their angles are not right angles, further limiting the area. Circles have a fixed area formula based on the radius, making it difficult to increase the area beyond a certain point. Therefore, rectangles offer the best solution for achieving larger areas.

Explanation

Rectangles give the optimal solution with larger areas compared to squares, rhombi, and circles. Rectangles have four sides with opposite sides equal in length, allowing for more flexibility in adjusting the dimensions to maximize the area. Squares have equal sides, limiting the potential for increasing the area. Rhombi have equal sides but their angles are not right angles, further limiting the area. Circles have a fixed area formula based on the radius, making it difficult to increase the area beyond a certain point. Therefore, rectangles offer the best solution for achieving larger areas.

10) What is the region of the smallest area which will cover every planar arc of length 1?

This means that any solution region must have area at least (0.43893)/2 _> 0.21946.

The area of a solution region to the problem lies between 0.21946 and 0.27524.

The problem asks for the region of smallest area that can accommodate every plane curve of length 1.

The question asks for the region of smallest area that can cover every planar arc of length 1. The explanation states that it is easy to show that the circular disk with a diameter of 1 will cover every planar arc of length 1. The area of the disk is approximately 0.78539. Therefore, the circular disk with a diameter of 1 is the region of smallest area that can accommodate every plane curve of length 1.

Explanation

The question asks for the region of smallest area that can cover every planar arc of length 1. The explanation states that it is easy to show that the circular disk with a diameter of 1 will cover every planar arc of length 1. The area of the disk is approximately 0.78539. Therefore, the circular disk with a diameter of 1 is the region of smallest area that can accommodate every plane curve of length 1.