Tile X was used to create a pattern which is seamless (without gaps) when arranged as shown in Y. Which tile(s) from the options will create a seamless pattern?
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A quick way to check if a complex tile can tessellate is to look for "interlocking" features. If a "bump" on one side has a corresponding "dent" on the opposite side, it's very likely the tile will tessellate by translation.
Step 1: Understanding the Concept:
The question is about tessellation, which is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. For a single, non-regular tile to tessellate by translation, there's a simple rule: any modification made to one side of a basic repeating shape (like a square or hexagon) must be compensated by an identical modification on the opposite side.
Step 2: Key Formula or Approach:
We will examine each tile to see if it follows the principle of translational tessellation. This is often called the "cut and paste" method. We check if the shape of each edge on one side matches the shape of the opposite edge.
A convex "bump" on one side should fit perfectly into a concave "dent" on the opposite side.
Step 3: Detailed Explanation: Tile X is a clear example of this principle. It appears to be based on a square or a rhombus. A piece is cut from two adjacent sides and pasted onto the other two sides, allowing them to interlock perfectly.
Now let's analyze the options:
Tile (A): This tile appears to be constructed from a rectangle. The convex curve on the top edge seems to be identical in shape to the concave curve on the bottom edge. Similarly, the wavy curve on the left edge appears to match the one on the right edge. Therefore, these tiles can be shifted horizontally or vertically to fit into each other seamlessly. This tile will create a seamless pattern.
Tile (B): The boundaries of this tile do not show corresponding pairs of opposite sides. The curved top-left edge does not match the curved bottom-right edge, and the straight top-right edge does not match the curved bottom-left edge. This tile will not tessellate.
Tile (C): This tile is another example of the "cut and paste" method on a rectangle. The curved piece removed from the bottom edge appears to have been added to the top edge. The piece removed from the right edge seems to have been added to the left edge. The opposite sides are complementary, so this tile will create a seamless pattern.
Tile (D): This tile also follows the principle. It seems to be based on a square. A piece is removed from the top side and added to the bottom side. Another piece is removed from the right side and added to the left side. Thus, it will interlock with copies of itself and create a seamless pattern.
Step 4: Final Answer:
Tiles A, C, and D are all constructed in a way that their opposite sides are complementary, allowing them to tessellate the plane without gaps. Tile B does not have this property. Therefore, options A, C, and D will create a seamless pattern.
