Question

Which of the following is / are TRUE for the strip of paper shown in the image?
A. It has two edges.
B. An ant can walk all over the surface of the strip without having to cross an edge.
C. A rotating belt that is similar to the strip of paper will have a uniform wear and tear.
D. This is an optical illusion that can be created as a 2D image but is not possible in 3D.

• A. A
• B. B
• C. C
• D. D

Hint:

The Möbius strip is a classic example in topology. Remember its two main counter-intuitive properties: one side and one edge. These properties lead to practical applications, like the conveyor belt example in option C.

Answer 1

The Correct Option is B, C

Step 1: Understanding the Question:
The image shows a Möbius strip, a surface with special mathematical properties. We need to identify the true statements about this object from the given options.
Step 2: Analyzing the Properties of a Möbius Strip:
A Möbius strip is a non-orientable surface created by taking a strip of paper, giving it a half-twist (180 degrees), and then joining the ends.

Sides: It has only one continuous surface. If you start drawing a line down the middle, you will end up back at your starting point having covered the entire strip, without lifting your pen.
Edges: It has only one continuous edge. If you trace the edge with your finger, you will return to the starting point having traced the entire boundary. Step 3: Detailed Explanation (Evaluating the Options):

A. It has two edges. This is FALSE. A key property of the Möbius strip is that it has only one edge.
B. An ant can walk all over the surface of the strip without having to cross an edge. This is TRUE. Because the strip has only one side, an ant can access the entire surface area without ever crossing the single boundary edge.
C. A rotating belt that is similar to the strip of paper will have a uniform wear and tear. This is TRUE. A normal belt loop has an "inner" and "outer" surface, and only one surface experiences wear. A Möbius belt uses its entire single surface area, distributing the wear and tear uniformly and effectively doubling its lifespan compared to a simple loop.
D. This is an optical illusion that can be created as a 2D image but is not possible in 3D. This is FALSE. A Möbius strip is a well-known and easily constructible 3D object. Step 4: Final Answer:
The statements that are true for a Möbius strip are B and C.