Visualize a cube that is held with one of the four body diagonals aligned to the vertical axis. Rotate the cube about this axis such that its view remains unchanged. The magnitude... Visualize a cube that is held with one of the four body diagonals aligned to the vertical axis. Rotate the cube about this axis such that its view remains unchanged. The magnitude of the minimum angle of rotation is?
Understand the Problem
The question is asking to visualize the rotation of a cube around one of its body diagonals and determine the minimum angle of rotation required to maintain its view. This involves understanding the geometry of the cube and how rotation affects its orientation.
Answer
$60^\circ$
Answer for screen readers
The minimum angle of rotation is $60^\circ$.
Steps to Solve
- Identify the body diagonal of the cube
The body diagonals of a cube connect opposite vertices. We will denote the diagonal that is aligned with the vertical axis as the rotation axis.
- Visualize the rotation of the cube
When the cube is rotated around its body diagonal, the view remains unchanged after certain angles of rotation due to the symmetry of the cube. We need to determine the minimum angle for the view to be unchanged.
- Determine the geometry of rotation
For a cube, rotating it about the body diagonal results in a view that looks unchanged at specific angles: $60^\circ$, $120^\circ$, and $180^\circ$.
- Analyze angle preservation
Since each face of the cube can rotate into the position of an adjacent face after $90^\circ$ or $180^\circ$ rotations, we need to find the smallest angle that will achieve an unchanged view.
- Confirm the minimum angle for unchanged view
After analyzing the rotations, $60^\circ$ is found to be the smallest angle that allows for a return to an unchanged view, followed by a $120^\circ$ rotation for an adjacent position.
The minimum angle of rotation is $60^\circ$.
More Information
The symmetry of the cube indicates that certain rotations will cause the cube to appear as though it has not moved. The unique relationship of its geometry means a minimum rotation of $60^\circ$ preserves its visual orientation.
Tips
- Mistaking the vertical angle with the angles at which cubes can maintain an unchanged view. Each angle (e.g., $90^\circ$, $120^\circ$, and $180^\circ$) behaves differently based on cube symmetry.
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