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Examples of axial symmetry abound, but I could not find an example of pure central symmetry (that is, without axial symmetry)! Do you know of any? A butterfly shows axial symmetry, what shows point/central symmetry in nature?

So, many examples resemble even functions, but I am looking for something that resembles an odd function. The best I can come up with is the image of a wave. But natural waves tend to be complicated and simple waves look too mathy.

Edit and to clarify: The question and terminology were vague. I left it as it was so that the connection to discussions below is not lost. Here is another attempt at explaining the question.

Odd and even functions are brought up early in college algebra. y=x^2 is even and exhibits symmetry with respect to the y-axis. Symmetry with respect to an axis is quite common in nature and one can find many examples. y=x^3 is odd and it exhibits symmetry with respect to the origin, if (x,y) is on the graph so is (-x,-y). I am looking for natural examples of symmetry with respect to origin, without the presence of symmetry with respect to an axis.

Thanks to the answers below I found the following (Figure 1b is what I was looking for) taken from this paper.

If you have other examples please post a picture or give a link.

Trivia: After this post I asked ChatGPT the same question. After two obviously incorrect answers it said it cannot find any!

enter image description here

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    $\begingroup$ What about a swastika? But beware: in India they'll love you for this example, in Europe they'll hate you for it :-) $\endgroup$
    – Dominique
    Commented Jan 31, 2023 at 8:51
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    $\begingroup$ It's not clear what you are looking for. A butterfly's symmetry is reflexive about the left/right plane; it's not axial symmetry. A starfish might have 5-fold rotational symmetry about an axis (in addition to reflexive symmetry about each of 5 planes). $\endgroup$
    – Rosie F
    Commented Jan 31, 2023 at 11:03
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    $\begingroup$ First, it would be very helpful if you could define your terms. It is possible that the definitions you have in mind are different from the definitions that others know, hence giving your definitions will ensure that you get answers that are relevant to you. Second, this is Math Educators SE. Questions asked here should be about mathematics education. What is the educational question you are looking to answer? If you are just looking for examples (even if they are for a class), you are probably better of asking at Mathematics. $\endgroup$
    – Xander Henderson
    Commented Jan 31, 2023 at 12:32
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    $\begingroup$ I would say that a butterfly shows plane symmetry, not axial symmetry. The world is in 3d; axial symmetry in 3d would be a 180° rotation around the axis. $\endgroup$
    – Stef
    Commented Jan 31, 2023 at 17:31
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    $\begingroup$ What about a spiral symmetry? It could be considered a central symmetry, but not an axial: commons.wikimedia.org/wiki/File:Artichokes_(3473638245).jpg $\endgroup$
    – flawr
    Commented Feb 1, 2023 at 14:33

3 Answers 3

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It is easy to have axial symmetry, without an inversion center (e.g. the picture Opal showed).

It is more difficult to have the converse. Many molecules with inversion centers also have rotational symmetry, at least a C2. However, it is possible (in 3D). What you want is the "Ci" point group.

https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Inorganic_Coordination_Chemistry_(Landskron)/02%3A_Symmetry_and_Group_Theory/2.02%3A_Point_Groups [third point group]

enter image description here

As you can see, there's not a C3 along the C-C bond because of the three different subsittuents. You might think there's a C2, perpendicular to the C-C bond, but there isn't either. You can rotate and make the hydrogens work, but then the Cl and Br mess things up and don't coincide.

[Note that if this were staggered ethane (CH3-CH3) instead of this halogenated ethane beastie, you'd actually still have an inversion center, but you would have a C3 axis along the C-C long axis and three C2s perpendicular to it.]

P.s. I do not believe there is a 2D analogue of Ci though.

P.s.s The butterfly does not have axial symmetry. It's not a C2, because the legs are on the bottom. It has a mirror plane. This is the case with most animals (at least at a gross level) and is called "bilateral symmetry". Maybe a starfish is "C5". But most higher animals are "Cs" (no rotation, but a single mirror plane).

Here is a molecule:

enter image description here

If you think about that molecule like an animal, it would have a purple head, and two dark green legs and be dragging a light green tail/body. Sort of like an airplane, maybe.

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    $\begingroup$ Is that molecule supposed to look like a 2-headed dog? $\endgroup$
    – Barmar
    Commented Jan 31, 2023 at 16:21
  • $\begingroup$ @Barmar with only one front leg? It does though. Anyway, cool, now I can make balloon molecules. $\endgroup$
    – Chris H
    Commented Feb 1, 2023 at 11:09
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Does this example of a flower with rotational, but not reflective, symmetry hit what you are looking for? (Name: Pinwheel Flower or Tabernaemontana divaricata)

pinwheel flower

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  • $\begingroup$ Interesting. It comes very close! If it had 4 petals it would be perfect. With 5 it just has pentagonal symmetry. The fact that individual leaves are not symmetric makes it interesting. $\endgroup$
    – Maesumi
    Commented Jan 31, 2023 at 3:15
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    $\begingroup$ What is your definition of "central symmetry" that excludes this "pentagonal symmetry"? $\endgroup$ Commented Jan 31, 2023 at 6:57
  • $\begingroup$ @GregMartin in 2D if (x,y) is on graph then (-x,-y) is also on the graph. $\endgroup$
    – Maesumi
    Commented Jan 31, 2023 at 10:57
  • $\begingroup$ The flower in the Figure 1b of the following paper seems to fit the bill, am I right? mdpi.com/2223-7747/11/15/1987?type=check_update&version=2 $\endgroup$
    – Maesumi
    Commented Jan 31, 2023 at 11:14
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    $\begingroup$ @Maesumi that sounds like you want specifically 2-fold rotational symmetry (180° symmetry). In your question you only mentioned that as an example $\endgroup$
    – Chris H
    Commented Jan 31, 2023 at 13:44
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Just to clarify:

There are zero two dimensional "examples of central symmetry [(x,y) -> (-x,-y)], without axial symmetry[Cn], in nature". This is because it is impossible mathematically. In two dimensions an inversion center is equivalent to a 180 degree rotation around the origin.

However, in three dimensions, you can have an inversion center [(x,y,z) -> (-x,-y,-z)] without any axial symmetry. This is the Ci point group.

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  • $\begingroup$ The question is more about $Rot_\alpha (Fig) = Fig$, I guess. $\endgroup$
    – Dominique
    Commented Jan 31, 2023 at 16:16
  • $\begingroup$ I don't understand this answer at all. In 2d it is very possible to have a central symmetry without an axial symmetry. In 3d it is both very possible to have an axial symmetry without a central symmetry, and it is very possible to have a plane symmetry without an axial symmetry. $\endgroup$
    – Stef
    Commented Jan 31, 2023 at 17:35
  • $\begingroup$ You are saying that, in the plane, the composition of the symmetries about the two coordinate axes is a rotation about the origin of 180 degrees. This implies that if a figure is symmetric about the two coordinate axes, then it is invariant by a rotation of 180 degrees about the origin. You're right, of course, but the converse is false: consider for example the letters Z, S, and N (What is true is that, if a figure is invariant by rotation of 180 degrees about the origin, then it has to be symmetric about the two coordinate axes, or none of them) $\endgroup$
    – Taladris
    Commented Feb 1, 2023 at 1:18
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    $\begingroup$ @Taladris But Z, S and N are centrally symmetric, inversion through the center does not change the shape. It doesn't matter if the shape has both reflection symmetries independent of each other, like with H, I or X, just that their composition is a symmetry. That's why we even have the concept. $\endgroup$
    – No Name
    Commented Feb 1, 2023 at 6:48
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    $\begingroup$ @EricDuminil: I understood "axial symmetry" in the plane (the context of the first sentence of this answer) as "symmetry about an [coordinate] axis", that is a reflection about the $x$-axis or the $y$-axis. AFAIK, axial symmetry is defined in space (wikiwand.com/en/Axial_symmetry). (The edit of the OP speaks about even functions, whose graphs are invariant under reflection about the $y$-axis) $\endgroup$
    – Taladris
    Commented Feb 1, 2023 at 13:39

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