# Is there a cube with quaternionic (Q8) symmetry?



## caveman (Mar 13, 2013)

I'm trying understand the group Q8 better: I want to build a puzzle (at least a java applet or something) with exactly these symmetries. My first idea was that Q8 might be a subgroup of the 2x2x2 cube so coloring some faces the same way would give a puzzle with Q8 symmetry.. I am not sure if that would work, any thoughts on this or references to similar things?

My next idea was to arrange the quaternion numbers into a pattern and look at how the group moves them around, it twists them in a very puzzling way! Here's the diagram I made:







The problem is you can't twist just one thing at a time, for the puzzle to have Q8 symmetry you have to twist pairs together in the specific relative directions (which you can see in my diagram).. the biggest problem is that vertices of the cube pass through each other when you do this: So it wouldn't be possible to build one. To fix that I thought about using the dual, an octohedra instead of a cube - I don't think that works out either..

So I'm pretty stuck, I really want to get my hands on this group so I'm hoping someone knows how to get a puzzle which inherently has this symmetry! Welcoming any input!

(Also I know there's a rubiks tube which can encode any group what-so-ever but I feel that generic constructions like that doesn't give any insight into the group).


----------



## jaap (Mar 15, 2013)

caveman said:


> I'm trying understand the group Q8 better: I want to build a puzzle (at least a java applet or something) with exactly these symmetries. My first idea was that Q8 might be a subgroup of the 2x2x2 cube so coloring some faces the same way would give a puzzle with Q8 symmetry.. I am not sure if that would work, any thoughts on this or references to similar things?



Q8 occurs as a subgroup of the 3x3x3 cube however, using the following three move sequences to represent i, j, and k:

i = (UR,UF)+(UL,UB)+ = R2 UR'U'R F R2 URU' F' UR2
j = (UB,UF)+(UR,UL)+ = L'U'B' U2 BLU FU2F'
k = (UL,UF)+(UB,UR)+ = B'U'R U2 FRF'R' U' BRU'R'

I don't know where I first encountered this fact, but it is in Christoph Bandelow's book "Inside Rubik's Cube and Beyond".


----------



## TMOY (Mar 23, 2013)

Q8 also occurs as a subgroup of the 2^3 cube:

i = U' D = y'
j = ki =z2 U2 y'
k = z2 U2


----------

