How many rotations are there in $\mathbb R^3$ which take $C$ to itself?

Question

Let $C$ denotes the cube $[-1,1]^3\subset\mathbb R^3.$ How many rotations are there in $\mathbb R^3$ which take $C$ to itself?

A. $6$

B. $12$

C. $18$

D. $24$

Please add your approach to the problem and we will be glad to provide hints. — gt6989b, Dec 10 '13 at 13:07
http://math.stackexchange.com/questions/326625/automorphism-group-and-congruences-of-the-cube/326680#326680 Also maybe this: https://en.wikipedia.org/wiki/Octahedral_symmetry — Jean-Claude Arbaut, Dec 10 '13 at 13:13

mathematics2x2life · Answer 1 · 2013-12-10T13:23:57.077

4

Let $H$ be the set of vertices of the cube, i.e. $H=\{1,2,\cdots,8\}$. Label the vertices of the cube $H$ such that $1$ is adjacent to $2$ and so forth. Consider the rigid motions (that is rotations) of vertex $1$.

There are how many different vertices to which $1$ can be sent, not including the motion which fixes the cube?

There are $7$.

So including the one that fixes the cube, there are .... total. Then there are how many possibilities for the placement of vertex $2$ as it must be adjacent to vertex $1$.

There are $3$.

This yields a total of how many total possible rigid motions of the cube?

$3\cdot 8=24$ total ways.

Since a rigid motion must fix the edge connecting vertex $1$ and vertex $2$, this is sufficient to determine the cube. Hence, there are .... total rigid motions (rotations) which fix the cube.

edited Dec 10 '13 at 13:23

answered Dec 10 '13 at 13:07

mathematics2x2life

13,322

@PraphullaKoushik I have adjusted my answer to be a more 'appropriate' hint. Think about each step carefully. – mathematics2x2life Dec 10 '13 at 13:17
yes yes.. I would try.. Thank you :) – Dec 10 '13 at 13:19
Will you please say why fixing $1$ and $2$ fixes the whole cube – Learnmore Dec 05 '16 at 05:20

How many rotations are there in $\mathbb R^3$ which take $C$ to itself?

1 Answers1

Linked