# How is this cube symmetric?

I am having a hard time with this, but I am reading in a published journal that this cube is symmetric when distributing +1 and -1 in the following corresponding points.

Positions 1-12: $$[+,+,+,-,-,+,-,-,-,+,-,+]$$

So how is this ordering symmetric? They say this cube has a center of symmetry given this ordering. Any help greatly appreciated!

Paper I am referencing: https://jcheminf.biomedcentral.com/articles/10.1186/s13321-018-0268-9

• With that arrangement, I only see a plane of symmetry - the diagonal plane through the edges marked 1 and 12 (which reflects 2 to 6 and 3 to 10. – Jaap Scherphuis Feb 20 at 15:32
• @JaapScherphuis please see my edit, position 12 is a +, and tell me if that is your answer still. As well as why is it symmetric? Is it because the halves that we draw will both sum to $0$? – Nick Pavini Feb 20 at 15:45
• I don't know what halves are you talking about, but that doesn't matter. No, it is just a coincidence. – Ivan Neretin Feb 20 at 15:56
• With that plane I mentioned, one half contains 2,3,4,7,8 (charges ++---), which mirrors to 6,10,5,11,9 respectively (also ++--- of course), with 1 and 12 (both +) in the mirror plane itself. – Jaap Scherphuis Feb 20 at 15:59
• Yes, that is the correct plane of symmetry for that arrangement of charges. – Jaap Scherphuis Feb 20 at 16:39

### The Original Article (Chirality vs Symmetry)

I found the article (doi: 10.1186/s13321-018-0268-9) via reverse image search. In the future, please consider including such details in the question in order to provide context; in this case it was absolutely essential for understanding the problem. As an aside, I don't recall coming across Spectrophores before and they seem like a really neat technique.

It looks to me like the authors actually meant to refer to an improper axis of rotation (aka rotation-reflection axis) as opposed to a center of symmetry. Then again, chemical chirality is always a bit of a mind bender for me so hopefully I'm not making a complete fool of myself here.

Initially, the article speaks of symmetrical vs asymmetrical

...it is possible to construct either 12 or 18 unique cages, depending on whether the + 1 and − 1 values are distributed in either a symmetrical or asymmetrical manner along the cage.

and then later refers to "a center of symmetry" (and I believe conflates this with an improper axis of rotation) in the caption for table 1.

There are 12 cages with a center of symmetry (hence non-stereospecific cages), and 18 cages without a center of symmetry (stereospecific cages).

Table 1 itself is split into the categories stereospecific and non-stereospecific, which refers to the concept of chirality from chemistry. Paraphrasing from the linked page, a chiral arrangement is one which does not contain an improper axis of rotation (typically denoted Sn). This implies that such an arrangement cannot contain either a plane of symmetry or an inversion center (aka center of symmetry). However, note that while a chiral arrangement will be dissymmetric (ie lack Sn), it will not necessarily be (though often is) asymmetric (ie lack all symmetry elements except the trivial identity).

Importantly, asymmetric implies chiral but chiral does not imply asymmetric. The authors aren't actually differentiating between symmetric and asymmetric, but rather between dissymmetric (chiral, stereospecific) and non-dissymmetric (achiral, non-stereospecific). The first quoted passage ought to read "values are distributed in either a chiral or achiral manner", and the table caption ought to refer to improper axes of rotation instead of centers of symmetry.

### The Original Question

As to your original question, as noted by @Jaap in the comments I do see a plane of symmetry in non-stereospecific cage number 12 from table 1 but do not see a center of symmetry. Specifically, if we place the origin at the center of the cube then the plane of symmetry would contain the points (-1, -1, 1), (1, -1, 1), and (-1, 1, -1). I believe this corresponds to Miller index (011). As such, that cage is indeed non-stereospecific as claimed.

### An Alternative Approach

A different (perhaps more intuitive) way of looking at this is to ask, for a given first row of values, does there exist a second row of values such that:

• The second row is different from the first row.
• The second row describes a cube that is related to the first cube by either a mirror or inversion operation.
• The second cube, following a rotation operation, is the same as the first cube.

In the case of the cube from your question, it is achiral because the row +-+-, ---+, ++-+ describes a cube that is a mirror image and which is superimposable after a rotation of 180° about the X axis.

On the other hand, stereospecific cage number 1 from table 1 (++--, +-+-, +--+) is an example of an arrangement that is chiral but not asymmetric. It appears to have rotational symmetry about the X axis by 180°, but there doesn't appear to be any alternative row of values which describes a mirror image that, when rotated, would produce the original row of values.

In contrast to both of the above, stereospecific cage number 3 from table 1 (+++-, -+--, --++) appears to be chiral as well as asymmetric.

• My apologies, didnt think you all wanted the full craziness. Thank you, yes this is research I am conducting with the DOE in the USA. Trying to come up with a way to measure Domain of a Neural Network applied to Computational Drug Design... For some reason I could not visualize the plane. – Nick Pavini Feb 23 at 14:10