Shown on the left is a woven mat using threads of two colours. The mat is flipped at the edge Q to reveal the back side. Identify the back side of the mat from the given options.
Show Hint
For weaving and flipping problems, break it down into two steps: first, determine the pattern of the back side (often a color inverse), and then perform the geometric flip/rotation.
Step 1: Understanding the Concept:
The question involves two main concepts: the nature of a simple woven pattern and the geometric transformation of flipping an object.
1. Weaving Pattern: In a simple one-over-one-under weave (plain weave), the back side is the negative of the front side. Where one color is on top on the front, the other color will be on top on the back.
2. Flipping: Flipping the mat at the edge Q is a reflection across the vertical axis Q.
Step 2: Key Formula or Approach:
We must perform two transformations on the original image to get the final view:
1. Color Inversion: Swap the orange and blue colors to represent the back side of the weave.
2. Horizontal Reflection: Flip the color-inverted image horizontally across the axis Q.
Step 3: Detailed Explanation:
Step 2.1: Invert the colors.
First, let's visualize the back side of the mat before it's flipped. We swap orange for blue and blue for orange.
The original mat has a large blue diamond with orange shapes inside. The color-inverted back side will have a large orange diamond with blue shapes inside.
Step 2.2: Flip the color-inverted image horizontally.
Now, we take this color-inverted image and reflect it across the vertical line Q.
The original image has a central four-pointed star shape made of orange triangles. In the color-inverted image, this star will be blue.
Let's look at the asymmetry. The four triangular elements around the center are not identical. The top and bottom ones are larger, and the left and right ones are smaller. When flipped horizontally, the left and right elements will swap positions, but since they are symmetrical across the vertical axis, they will look the same. The overall pattern is symmetrical left-to-right, so a horizontal flip will not change its appearance.
So, the final result should be the color-inverted version of the original mat. This means we are looking for an option with:
A large orange diamond as the background.
A blue four-pointed star in the center.
Blue triangles at the four corners.
Now, let's check the options:
(A) Has the original colors, incorrect.
(B) Has the colors inverted, but the central star is oriented incorrectly (rotated 45 degrees).
(C) Has the original colors, incorrect.
(D) Has the correct inverted colors: an orange background diamond and blue foreground shapes. The shapes and their orientations are preserved from the original.
Step 4: Final Answer:
The back side of the woven mat is a color-negative of the front. Flipping it horizontally does not change the pattern as it is symmetric. Therefore, the correct representation is the color-inverted image, which is option (D).
