Can the figure below tesselate a plane? Explain your answer.
The shape is an arch (like a rainbow)
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The top of the first rainbow will not fit into the bottom of the one above it.
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However, an arch-shaped figure like a rainbow cannot create tessellation because it cannot be repeated in a way that covers a plane without leaving gaps or overlaps. The curved shape of the arch makes it impossible to fit perfectly next to each other without leaving spaces or overlaps.
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A tessellation is created when a shape is repeated over and over again covering a plane without leaving any gaps or overlaps. Tessellation is also known as tiling. Triangles, squares and hexagon are perfect examples of figures that can create tessellation.
Look at the shaded triangle in the drawing below. Describe in detail the transformations needed to create the rest of the square design. You can use a combination of translations, reflections, and rotations of the shaded triangle (Figure I) to create the square design.
A square is drawn on a coordinate plane with the center of the square on the origin. Two diagonals are drawn through the origin forming four congruent triangles on the inside of the square. The triangle that overlaps the positive y-axis is shaded.
To create the rest of the square design, we can use a combination of translations, reflections, and rotations of the shaded triangle. Here are the transformations needed:
1. Translation: We can translate the shaded triangle to the right by the length of one side of the square. This will create another triangle that is congruent to the shaded triangle and adjacent to it.
2. Reflection: We can reflect the two triangles we have now across the y-axis or x-axis to create two more triangles. This will fill up two more positions in the square.
3. Rotation: We can rotate one of the triangles we have now by 90, 180, or 270 degrees to fill up another position in the square.
Using these three transformations, we can repeat the process to fill up the entire square with the shaded triangle. We can perform the following transformations:
- Translate the shaded triangle to the right to fill the second position in the square.
- Reflect the two triangles across the y-axis to fill the third and fourth positions in the square.
- Rotate one of the triangles by 90 degrees clockwise to fill the fifth position in the square.
- Reflect the two triangles across the x-axis to fill the sixth and seventh positions in the square.
- Rotate one of the triangles by 180 degrees to fill the eighth position in the square.
- Reflect the two triangles across the y-axis to fill the ninth and tenth positions in the square.
- Rotate one of the triangles by 270 degrees clockwise to fill the eleventh position in the square.
- Reflect the two triangles across the x-axis to fill the twelfth and final position in the square.
After performing these transformations, the shaded triangle will fill the entire square design.