How the Pentagon Has Just One Line of Symmetry

The homework for 3rd Graders was to draw as many lines of symmetry as they could for a regular pentagon. I thought they knew their way around this stuff — they’d had a lot of success working with a cut-out hexagon the week before, and had been getting a bunch of practice —  but I was surprised by the huge variety of responses in their homework, so I decided to figure out what was going on.

I figured we could settle this by having a whole-group conversation. Probably, the kids who found less than five lines stopped looking too early. If I shared a variety of work, I bet that kids would quickly recognize that these other lines split the pentagon into two congruent pieces. (In 3rd Grade speak, this is criterion is usually expressed as “you could fold it over perfectly.”)

It didn’t go down like that. I noticed that some of the kids who saw 2 or 3 lines switched positions during the early part of the discussion, but the 1s and 5s weren’t budging.

F and I offered (what I thought was) a pretty convincing argument against the possibility of there being only one line of symmetry. After all, they argued, if you think that the vertical line of symmetry works, just rotate the pentagon a bit and you have a new “top” of the shape to split in half.

It was around now — especially when T shared — that I started to form a theory about what the “one line” camp was thinking. Their responses to the arguments made me think that they didn’t see the regular pentagon as actually having this rotational symmetry. They saw it as irregular, in some way, so that these other lines wouldn’t actually fold over perfectly.

“Why don’t we cut them out?” R asked. Perfect, I said. I was in for a surprise!


With the cut out pentagons, you could just see that the rotations returned the pentagon to its original shape, and made room for a new “vertical” line of symmetry. And, for sure, this helped some kids.

But Bo and Annie [pseudonyms] called me over, because they were caught in argument and couldn’t make progress.

Annie: “I know that you can rotate it, but there’s still only one line.”


Annie: “I’m saying that you can draw that line if you rotate it, but it’s the same line?”

Do you mean that it wouldn’t be a new line?

Annie: “No, it would be a different line on the page, but I’m saying that it would still be the same line.”

Hold on, I think I’ve got you. Are you saying, look, you could draw a new line if you rotate it, but that’s not actually a different line, since it’s still basically that same, vertical line that you started with?

Annie: “Yes!”

Huh. So if you have all these different lines, you wouldn’t say that there are five lines of symmetry or anything?

Annie: “No, because they’re all the same.”

At the moment that I understood her, I no longer knew what to say. I’m used to thinking of all sorts of mathematical similarity and difference — congruence, modular congruence, similarity, equivalence, symmetry — but I hadn’t ever thought of mathematical sameness quite in the way that Annie did.

She’s saying (what is she saying, exactly?) that these lines aren’t different because they’re intellectually similar. There is, for her, just one line of symmetry because there’s basically only one idea of a line of symmetry. Sure, you could repeat that process five times, but that’s not the point. The point is that you get a line of symmetry by connecting a point and its opposite side.

And maybe she’s also saying that when you rotate the shape, you’re really making a new shape. That the regular pentagon is a brand new regular pentagon when you turn it in the air, and you have a new opportunity to draw that vertical line and divide it into parts.

I wonder what she’d say if she compared her pentagon thoughts to her hexagon results. And earlier in the week she had drawn a ton of lines of symmetry in a circle.

Maybe this has something to do with the unloveliness of the pentagon? Maybe pentagons feel different to her than these other shapes?



6 thoughts on “How the Pentagon Has Just One Line of Symmetry

  1. This seems to me that Annie is thinking topologically in some way. She’s essentially defined an equivalency class for lines of symmetry around how they change a geometric figure and so all five lines of symmetry in her mind are the same.

    Maybe a question to ask is, how many lines could you draw that are the same in this way?


  2. I think it has more to do with the “unloveliness” of the pentagon. The fact that the symmetries occur at 72 degrees makes them less obvious than symmetries which occur at 45, 60 or 90 degrees. She doesn’t see them until the shape is rotated in the pentagon, but she sees them just fine in the hexagon, octagon and circle. Another interesting aspect is the odd number of sides – the symmetry goes from a vertex to the middle of the opposite side. I wonder how many lines of symmetry she would see in an equilateral triangle.

    Blog with a sense of mystery – ask and you shall produce!


  3. My reaction: Along the lines of what you’re saying, the student may be saying that you can create a line of symmetry by connecting a vertex with the midpoint of its opposite side. By that logic, all regular n-gons with an even n have two lines of symmetry (connecting opposite vertices and connecting opposite side-midpoints), while those with an odd n have one line of symmetry (connecting a vertex with its opposite side-midpoint).

    I’m more confused as to why she would see more than two lines of symmetry in a hexagon, given her reasoning here. There are no vertices in a circle and student experience with circles commonly have multiple radii (e.g., bicycle wheels have spokes), so it’s not necessarily the same mentation.

    I would test her on a variety of regular polygons and see if a pattern emerges. 🙂


  4. I think it’s partly to do with the fact that we tend to always draw 2D shapes in the horizontal base orientation – especially simple ones like rectangles, squares, triangles – so students tend to only ‘see’ shapes this way.
    But anyway, I’m wondering if this particular issue could be fixed by getting students to fold a cut-out regular pentagon so they get the 5 symmetry fold lines anyway they like, but then glue the pentagon down by putting glue on just one of the small resulting triangles. Then they should be able to see that even if the pentagon is fixed in one orientation, they can still fold along their 5 fold lines to see the symmetry.


  5. First of all, I want to say that I frickin love this kid’s thinking, and I appreciate you for sticking with her long enough to grok it.

    Second, my take: I mean, a propos of David’s, Paul’s, and Dan’s comments, it sure sounds to me like she’s coming from an intuition rooted in the utterly mathematically valid point of view that “how many lines?” means “how many equivalence classes of lines under the symmetries of the figure?” By this count, 1 for a regular n-gon with n odd, and 2 if n even, as Paul observed. And 1 for a circle. This is every bit as coherent as actually counting the lines. (And in some contexts, perhaps more interesting – for example in the tessellation in this Escher print, counting the actual centers of rotation is not that compelling – clearly there are infinitely many – but counting the classes of centers of rotation under symmetries of the tessellation is a real question and the answer is a small finite number.)

    Then the question becomes why she didn’t apply the same standard to the hexagon and circle. I mean she’s a kid, nobody expects her to be 100% logically consistent all the time, that’s not what I mean. I just mean, why did the intuition about the several lines being “different… on the page” but yet really the same only first come up for her with the pentagon and not with the earlier figures? Here I think maybe Spencer’s and Kay’s analyses sheds light: if the visibility of the line of symmetry depended on its vertical orientation, then the other lines of symmetry are sort of invisible until the figure has been rotated – but the rotation calls attention to (in fact, is the essence of) the sense in which they are “the same.” If it was possible for Annie to see the various lines of symmetry of the hexagon and circle without the rotation, then it might not have come onto her screen that this sense of sameness would unify some of them. This makes me wonder (like Spencer) what she would have done with an equilateral triangle, but even more so also wonder if her views on the hexagon would have changed after her encounter with the pentagon (now that she has developed this sameness idea).


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