In how many ways can a cube with 8 different vertices be oriented?

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The discussion confirms that a cube with 8 different vertices can be oriented in exactly 24 distinct ways. This conclusion is derived by fixing one vertex at a point and selecting one of its three adjacent vertices, leading to the calculation of 8 (choices for the first vertex) multiplied by 3 (choices for the adjacent vertex), resulting in 24 orientations. The concept of "physical symmetries" is also highlighted as a key term in understanding these orientations.

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alexmahone
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My solution:

Imagine that we have 8 points in space which must be occupied by the vertices of the cube.
Let A be one of the points. Fix anyone of the vertices at A. This can be done in 8 ways.
We have 3 different choices for an adjacent point (say B) because each vertex of a cube is connected to 3 other vertices.
Once A and B (being adjacent points) are determined, the other points are automatically determined. (Is this true? If so, how would I prove it?)
So we have 8*3 = 24 orientations.

Is this correct?
 
Physics news on Phys.org
Yes, there are 24 cube rotations, or "physical symmetries", as they are called in this article.
 

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