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The imagined students are in elementary school, say around 9-13 years old.

I want to use rather precise terminology when talking to my students. However, it seems like we typically use the same terminology for a geometric figure and for its boundary.

Does the word "triangle" refer to the $2$-dimensional geometric figure or the $1$-dimensional geometric figure? It seems to me like we kind of treat "triangle" to mean either, or even both at the same time.

two things called triangle

There are similar issues with other terms, like "square", "circle" and "cylinder". Here I can imagine the word "cylinder" also being used for cylinders open at the bottom, or a cylinder open in both ends. There seems to be some distinction when talking about "balls" and "spheres".

I fear that when the terminology is not precise enough that this will lead to unexpected problems when defining and working with other concepts later on. For instance, if one defines a triangle as some union of edges and vertices, then one needs to be a bit careful when defining the area of the triangle.

Many of the geometric figures are so elementary that they are deeply rooted in daily language, and there seems to be no great solution.

I have played with the idea of calling things something like "1D-triangle" and "2D-triangle", but this seems to make things worse and I have not come up with good alternative terminology.

My current go-to would be to tell my students something like "We call this a triangle, but we also call this a triangle. Sometimes the distinction matters, but most often it is clear from context." Another possibility would be to just let the students realize the ambiguity themselves. I don't really have any evidence that suggests that this actually is an issue.

Question

What can (and should) an educator do about ambiguous terms like "triangle", "square", etc?

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    $\begingroup$ Calling it "1D" will confuse them, since it is referring to a figure's intrinsic property not "where it lives". (I speak from experience discussing topology to math majors.) I'd steal terminology from 3 dimensions and call the filled-in figures "solid". $\endgroup$ – Adam Jun 2 at 0:16
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    $\begingroup$ Is there research showing that this causes, or does not cause, misunderstandings? $\endgroup$ – Tommi Jun 2 at 6:32
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    $\begingroup$ Going down this rabbit hole leads to the sort of obfuscation I have seen in USA high school geometry, where the text pedantically differentiates between "an angle" and "the measure of an angle," whereas everyone else (both in real life and in STEM subjects) knows exactly what is meant by "the angle is 35 degrees" and never says "the measure of the angle is 35 degrees." $\endgroup$ – alephzero Jun 2 at 12:10
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    $\begingroup$ Surely, a "1D Triangle" is a degenerate triangle in which all 3 points lie on the same line, in which case it ceases to be a triangle, and is called a "line segment". TLDR - There is no such thing as a 1D triangle! $\endgroup$ – Glen Yates Jun 2 at 14:49
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    $\begingroup$ For what it's worth, as an engineer working in the field, I recently tried to disambiguate a similar circumstance, defining the difference between a circle and a disk, where "circle" meant the 1d edge and "disk" meant the area contains by the circle. To the best of my understanding, those were the "correct" mathematical terms. The result actually caused substantially more confusion than was previously caused by letting context dictate, much less the subsequent confusion between a "square" and a "square surface." We have since reverted to calling them circles and squares. $\endgroup$ – Cort Ammon Jun 3 at 5:20

11 Answers 11

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I'm confused. Are you really going to try to make this sort of distinction when teaching geometric figures to students "around 9-13 years old"? Students that age (and engineers my age -- much, much older) think that a triangle is a triangle. It's a polygon formed by three non-colinear points.

A triangle has many ways you can think about it. Three points, and the connecting line segments. The three angles. The area of the triangle. The three ways you can measure the height of a triangle (used in calculating that area). Whether the triangle is equilateral, isosceles, acute, scalene. Etc. But, it's a triangle.

I finished at or near the top of my math classes all the way through K-12 and junior college (the top got a lot more crowded in my engineering math classes) I've never bothered to think of it any differently, and I can't imagine anything but confusion if you tried to make a distinction.

Take a circle. If I cared about the 1-dimensional aspect of it (your term, not mine), I'd speak of the circumference of the circle. If I cared about the region it bounds, I might talk of the area. If I care about it's trigonometric features, I'd discuss them separately. But, a circle is a circle is a circle.

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    $\begingroup$ Interesting point: Natural language is fuzzy and we should not make rules that unnaturally force one meaning and exclude all the others. Instead, we should specify what we mean with additions (e.g. the surface area of a triangle or its perimeter). In a way you are close to Dan's opinion, except that he has a pronounced preference (the circumference). $\endgroup$ – Peter - Reinstate Monica Jun 2 at 19:46
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    $\begingroup$ The circle is an interesting example in that it's not like the triangle: mathematicians, at least, pretty universally mean the boundary curve when they say "circle", and the bounded region when they say "disk". So we see that the distinction is important enough to warrant different terminology in this closely related case. $\endgroup$ – Nick Matteo Jun 2 at 22:28
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    $\begingroup$ Really? For students ... in elementary school, say around 9-13 years old? I went to school a long time ago (graduating from high school in 1974). I never had any issues in math - I was in the minority who "just got it". It seems like math education has been working hard to increase the proportion of students "who don't get math". I've lived my whole life comfortable that a circle is a circle is a circle. It has a radius, a diameter, a circumference, an area. If it's a unit circle, trigonometry makes more sense. Disks make me think of Frisbees. $\endgroup$ – Flydog57 Jun 2 at 22:33
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    $\begingroup$ I agree with everything, except the last paragraph. That is just false. A circle is the set of point whose distance to its center is equal to the radius. The circumference of a circle is a number, not a curve. The area of a disc is a number, not a region. A circle doesn't have an area (or it has area 0). $\endgroup$ – Stef Jun 3 at 21:00
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    $\begingroup$ @Stef: You might only use the term "circumference" to refer to a length, but others disagree with that view. $\endgroup$ – psmears Jun 4 at 10:19
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One encounters exactly the same issue teaching multivariable calculus when one treats integrals over three-dimensional regions and integrals over the surfaces that are their boundaries. In particular the word sphere is particularly confusing in this context. (Mathematicians use sphere to mean the the two-dimensional surface; colloquial speech and some physicists and engineers use the same word to describe the three-dimensional ball.)

Except when teaching students studying mathematics per se, my sense is that the best practice is simply to say clearly what interpretation is intended. So one can say the triangle formed by the edges connecting vertices A, B, and C or the triangular region with vertices A, B, and C. Writing this I betray that I tend to use triangle to mean the one-dimensional polygonal curve and say something like triangular region to refer to the region it bounds and would usually simply speak of the the triangle with vertices A, B, and C and the triangular region that it bounds. In calculus I might speak of a the surface of the sphere and the spherical solid. In my private usage I would regard the first phrase as redundant (a sphere is a surface), but in the classroom I am talking to students and (hopefully) not to myself (I think of area as two-dimensional volume but I know better than to ever call an area a volume in calculus class).

At any rate, in the classroom what matters is being clear and consistent. One does not want to make (fussy) fine distinctions with an audience unprepared to hear fine distinctions, and one does not want to use the same word(s) to refer to a region and its boundary. The second goal is more fundamental, but meeting it should not entail cumbersome and/or subtle terminology, particularly with students, such as typical high school students, who are simply unprepared to hear mathematical and terminological subtleties. We want to help them learn to listen more carefully, but this process is gradual, and should not come at the cost of more basic matters.

One should also try to be aware of what other teachers the students might have might say in regards to the same figures. So when I teach multivariable calculus I try to figure out what the professor teaching the same students electromagnetism says in regards to spheres/balls. I don't necessarily follows his/her practice, but if I can I make clear where mine differs from it.

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  • $\begingroup$ I would say that in high school geometry, I sometimes find it important to distinguish between the interior and the boundary of a circle. It doesn't seem to be necessary to be bogged down in terminology, though; students seem to be able to differentiate between a point being on a circle vs. in a circle. $\endgroup$ – Matthew Daly Jun 3 at 17:20
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    $\begingroup$ @MatthewDaly: This amounts to the same issues as for sphere. A professional mathematician says circle for the curve and disk for the region it bounds, but colloquial speech does not make the same distinction and uses circle for both. It's worth being careful to distinguish the two when the distinction matters, and probably best to not fuss much about it when it could be distracting, though if it's costless to make the distinction, then one should do so. What matters for calculus students might not matter for fourth graders. $\endgroup$ – Dan Fox Jun 3 at 17:28
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I think the distinction you are raising is not natural to students at this age. I teach undergraduates and graduate students, not elementary schoolers, but I find that it is not natural for undergraduates who have not had a theoretical math course. In my experience, students do not naturally think of geometric figures as sets of points.

If $P = (-1,-1)$, $Q = (1,0)$ and $R = (0,1)$, then students understand just fine the idea that $(0,0)$ is in the interior of the triangle $PQR$, and is not on the boundary of the triangle $PQR$, but they don't think of the triangle $PQR$ as a set of points, so the question of whether or not $(0,0) \in PQR$ is not a natural one to them. (They also don't distinguish naturally between the open triangle, which does not contain its boundary, and the closed triangle which does.)

When you get to point set toplogy or to multivariate integration, you need to raise this issue. Until then, though, I think it is just fine to use the words "interior of the triangle" or "boundary of the triangle" when you really need to specify, and ignore the issue otherwise. (Again, I'm not an elementary teacher, though.)


If you do want to teach this distinction, you might want this cartoon:

enter image description here

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  • $\begingroup$ +1 for first sentence. There will always be a student or two that will debate definitions, but for most, “a polygon with 3 sides” suffices. $\endgroup$ – JTP - Apologise to Monica Jul 3 at 18:09
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I confess, when you start talking about 1-D triangles, my own first thought is "how can you have non-colinear points in 1-D?". So, I imagine most students that age will have a far more difficult time with that.

Keep in mind age appropriateness. For 9-12 year old children, you are generally looking at a level of psychological development characterized by concrete reasoning (Piaget as the primary theorist here). For that level, a triangle is commonly imagined as something physical, having the attribute of three edges/corners. As a shape, it is understood (once the concepts are introduced) to have the additional properties of area and perimeter. The formal definition of these from an advanced math perspective is probably beyond what most students in that age range will understand.

Instead, stick with simple physical properties they should be familiar with, commonly described as the "enclosed area", "filled area", or simply "area", versus the "perimeter", "boundary", or "outline", as needed. Save the advanced definitions for advanced students - eg, those taking various Euclidian and non-Euclidian geometry classes, multivariate calculus, etc. If it is easier for you, you can suggest that, unless you say otherwise, when you name a shape, you mean the boundary and not the enclosed area, but keep in mind that the majority of your students will need the reminder whenever the question comes up.

You could also give them a heads up that there is a formal definition, but for your purposes you will be going with the common usage as above. (I like this approach when discussing color, for example, since you will get art students talking about primary and secondary color, which is useful for some purposes, but not the same as specific frequencies of light as discussed in science and physics classes.) A brief heads up of "these work for reasons we aren't going to get into this year", with a later "offline" discussion for those who are interested enough to ask, is probably the best approach.

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Educator here who has worked with many students in the aforementioned age range (9-13) on triangles and squares. In my experience, it has never come up that a confusion between the boundary and the interior of a plane region was relevant to problem solving at that grade level. For these types of elementary shapes, the boundary and the interior completely determine one another anyway, so which one you pick to be the True Meaning Of A Triangle wouldn't make much of a difference.

It seems to me that the precision gained by saying "The triangle is the set of boundary points" is a small payoff compared to the drawback of having to pedantically reword every statement to be unambiguous (and the resulting increase in prolixity/verbosity). Is a ten-year-old going to understand the area of a triangle formula better if I tell him, "Here is the formula for the area of the interior region of a triangle", vs. "Here's the formula for area of a triangle"? They're going to know what I mean either way. Having worked with students that struggle on math (and in some cases, may be apathetic towards the concept of gaining a deeper understanding), none of them would have benefited from splitting hairs over the distinction. Conversely, there's significant gain in conciseness using a word like "triangle" as an umbrella concept that can refer to one or more of:

  • any three non-collinear points
  • the line segments connecting same
  • the 2D region inside those line segments in the plane passing through all three

and letting the context clarify which one is meant. The umbrella concept also reinforces for the student that these are all interrelated, rather than balkanizing them artificially into "this one is about the 2D region, this one is about the 1D perimeter, and this one is about the 3 zero-dimensional corner points".

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Many of the geometric figures are so elementary that they are deeply rooted in daily language, and there seems to be no great solution.

I agree with you here, and I think this is the key point. To me they are clearly well-defined: "Triangle", "square", and polygons in general, are bounded regions on the Euclidean plane, i.e., 2D figures.

In your pictures, the left one is a triangle. When we say "the triangle ABC", it is a (very) economic way to describe a 2d region enclosed by a "three-non-colinear-points-and-three-lines" system.

On the other hand, the intention of your right picture is to show the edges of a triangle. A more technical definition would be a "closed polygonal chain", roughly speaking, just a set of points connected by lines. (https://en.wikipedia.org/wiki/Polygonal_chain)

Maybe the most remarkable difference between those objects is the area. In Euclidean Geometry, the area of DEF is 0 (the area of 1 line is zero, so the area of 3 lines is just 0+0+0). Nevertheless, the area enclosed by DEF is the same as the area of the "triangle ABC", and it is nonzero.

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  • $\begingroup$ From your link: "Often the term "polygon" is used in the meaning of "closed polygonal chain", but in some cases it is important to draw a distinction between a polygonal area and a polygonal chain. " $\endgroup$ – Improve Jun 2 at 7:35
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    $\begingroup$ But instead of "closed polygonal chain" just say "outline" or "perimeter". $\endgroup$ – Ben Voigt Jun 2 at 15:05
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Programmers consider the naming of things to be one of the three leading problems in our field.

For the cases you describe, we do already have a well-established and widely-used set of terms that even non-computer users should recognize and be familiar with.

A circle can be called a solid or filled circle, contrasted with wireframe or outlined circle.

Generally "filled/outlined" suggest 2D, while "solid/wireframe" suggest 3D, but that's not a hard and fast rule. You could pick one pair and use it universally, if you wanted. Or even use both pairs synonymously.

In the third dimension, there's an ambiguity again: what is the term for a shape formed from planes? Maybe box or container?

For the cylinder problem, "open-ended" feels a little ambiguous: I'd tend to be more explicit about which specific faces are missing, so "open-topped" or "open both ends".

  • Filled or solid cube, cylinder, prism, sphere, tetrahedron.
  • Open-topped cubic box.
  • Open-topped square outline.
  • Cylindrical box, open both ends; aka tube/pipe.
  • Wireframe cube.
  • A filled circle; aka disk.

There are areas of uncertainty, still: geometrically speaking, is it meaningful to speak of a wireframe sphere, torus, or other curved shape? I'd argue they'd be ambiguous, as you can't say how the curves are subdivided. And in theory, a wireframe cylinder should just be the two wireframe circles of the ends, but that's not what people would imagine: they'd picture connecting lines subdividing the curve, too. But these are edge cases unlikely to be visited by your target audience.

So for kids and teens, and indeed for almost all cases, I think you'd do well with wireframe/outlined, box/container, filled/solid, with explicit specification of any missing edges or faces.

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    $\begingroup$ I think this is a very interesting answer. It reminds me of this discussion: matheducators.stackexchange.com/questions/13700/… Implicit teaching of hierarchical thinking by careful, but not pedantic, mathematical vocabulary is valuable. Not sure I would use vocabulary like "filled circle" in a geometry class, but your answer is a good reminder of the importance of teaching to think hierarchically. $\endgroup$ – user52817 Jun 4 at 22:18
  • $\begingroup$ In most cases dealt with in school, I think I might agree with those who argue that they're distinctions without a difference. It'd be fine to consider a plate, planet and orbit all as circular things. I remember only once being confused by geometric naming ambiguity in school, by the use of "cylinder". I was accustomed to that meaning pipe/tube (cylindrical box, open both ends), rather than circular-cross-section prism (solid cylinder), so an exercise question at first made no sense to me. But I got what they meant from context of the question, so even that time wasn't a big deal. However... $\endgroup$ – Dewi Morgan Jun 5 at 4:10
  • $\begingroup$ ...however, I'd rather eat my own spleen than advocate imprecise naming in an answer :P $\endgroup$ – Dewi Morgan Jun 5 at 4:14
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I agree with the point that a lot of the answers here are making - some distinctions, while correct and important, are not accessible to the age group you're talking about - I also want to point out an important benefit of not making the distinction for them. While it is crucial in higher mathematics to be able to be extremely precise, it's also important to be able to understand the "sense" of what's going on, even when it's not made explicit.

For example, consider the well-known fact that "the volume of a sphere is $\frac43\pi r^3$". I doubt anyone here would disagree with that, but a student who's highly aware of all of the important differences in math might raise any or all of the following concerns:

  • "Sphere" may refer to a figure in any number of dimensions, so the statement claims that spheres have a fixed volume no matter the dimension!
  • The only sphere that actually has volume (volume being explicitly three-dimensional) ought to be the one in four dimensions, but then this formula is incorrect.
  • The English term "a" is ambiguous; do we mean that there is a sphere with this volume, or that every sphere has this volume?
  • Strictly speaking, the term "sphere" makes sense in any metric space, but the term "volume" does not.

To make that statement really, truly, mathematically precise, in a way that will be sure to never set a student up for misunderstanding in the future, we would have to say the following:

"The three-dimensional Lebesgue measure of every ball in $\mathbb{R}^3$ of radius $r$, under the standard Euclidean metric, is $\frac43\pi r^3$."

Now, certainly, a student who prefers this second formulation over the first knows more math; they must, at minimum, be aware of the terminology used in measure theory, metric spaces, and higher-dimensional geometry. But the student who prefers the first formulation understands the concept better, because the formula really isn't about $n$-dimensional geometry or Lebesgue measures!

An important skill for students to build is navigating the connection between the "fuzzy but intuitive" nature of natural language and the "precise but confusing" nature of mathematical language. If you jump directly to the fully precise version, they lose the opportunity to develop that skill!

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A triangle is born from three non-collinear points and the axiom that two points determine a line. In the context of neutral geometry, a triangle has no structure other than three lines and three points. In particular, there is no notion of the interior of a triangle without more axioms.

In the real projective plane, one cannot define the "interior" of a triangle. Similarly, the notion of the interior of a triangle is problematic in finite geometries.

In Euclidean geometry, one uses Hilbert's betweenness axiom (plane separation) and Pasch's theorem to make sense of the interior of a triangle.

As educators, when teaching Euclidean geometry, we should try to use vocabulary that makes a distinction between a triangle and its interior.

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    $\begingroup$ Really? For students "around 9-13 years old"? A triangle is a triangle. Do you really want to introduce Hilbert's betweenness axiom (plane separation) and Pasch's theorem to make sense of the interior of a triangle or talk about projective planes or finite geometries? I don't remember any of this from my high school Euclidean geometry class (which was only only Euclidean geometry - start with the axioms and prove your way up from there - it was an advanced students only class). A triangle was still a triangle (correction, I do remember betweeness, but I don't think I met Hilbert) $\endgroup$ – Flydog57 Jun 2 at 22:17
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    $\begingroup$ @Flydog57 The original question was framed as "students of any age" but it was edited (after I answered) to 9-13 years old. But in any case, I am not advocating that educators should introduce Hibert/Pasch etc. I am just saying that educators do have a broader and deeper understanding of the subject matter than the students they are teaching. So with this perspective, we should be attentive and say "triangle interior" when we are referring to the interior of a triangle. $\endgroup$ – user52817 Jun 3 at 1:20
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    $\begingroup$ @user52817 Just because you have a broader/deeper understanding of the subject matter than the students, doesn't mean you need to conspicuously display it at every opportunity. There's a lot of value in meeting the students where they're at, and if you're working with 12-year-olds who are learning formulas for perimeter, area, sum of the angles, etc. for a triangle, pedantically insisting on reminding them of the distinction between boundary and interior every chance you get (a) borders on hairsplitting and (b) doesn't add anything to their understanding of those formulas. $\endgroup$ – Rivers McForge Jun 3 at 16:57
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I don't really know what's the big problem. Terms need not be ambiguous; it's up to you to define and use them in clear and unambiguous manner. For example you can use "triangular region" or "cylindrical volume" to clearly differentiate from "triangle" and "cylindrical surface", and of course you have to define whether your region/volume includes the boundary. I also don't see why you say you have to be careful when defining area of a triangle. Just define the area of a polygon via the shoelace formula and you're done (it is better than unsigned area).

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    $\begingroup$ "I also don't see why you say you have to be careful when defining area of a triangle.": Because if the triangle is a collection of line segments, its area would always be zero. $\endgroup$ – Nick Matteo Jun 2 at 22:32
  • $\begingroup$ At first you say "use "triangular region" [...] to clearly differentiate from "triangle"," which implies that you mean a triangle is not a triangular region. But then you go on to say "defining area of a triangle," which implies that you mean that a triangle is a triangular region. $\endgroup$ – Joel Reyes Noche Jun 3 at 5:49
  • $\begingroup$ @JoelReyesNoche: No. Just because we have a measure-theoretic definition of area of regions does not mean we must also define area for synthetic geometric objects in the same manner. $\endgroup$ – user21820 Jun 4 at 19:15
  • $\begingroup$ @NickMatteo: Like Joel, you are thinking based on other mathematical notions of area. You can't assume that the area of a polygon is zero until you define area. There is no inconsistency if I define area for synthetic geometric figures in a manner different from how I would define it in analytic geometry. $\endgroup$ – user21820 Jun 4 at 19:18
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    $\begingroup$ Okay, I understand. Thanks for the clarification. $\endgroup$ – Joel Reyes Noche Jun 5 at 1:14
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What you are trying to do is way too complicated for this age group. You have been exposed to years of math but all this stuff is new for the students. What might seem easy/trivial for you might be very complicated for them. If you try to explain it this way you run the risk of confusing your students thoroughly and possibly even creating an aversion for math in general.

The solution is simple. If there's ever a risk of confusion just spend one or two sentences to explain exactly what you mean. Maybe add even a picture.

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