Can two shapes occupy the exact same area on a plane?

Suppose there are two triangles on a plane. The coordinates for each point of both triangles are the same.

It seems to me that there is nothing to differentiate these two triangles, and as such, they must be identical. Therefore, there can't be more than one such triangle on the plane.

I might describe two congruent triangles positioned such that either can rotate to overlap the other one perfectly. In that case, it seems that I could say that the two triangles are identical after the rotation.

I'm in first year, so I can only speculate about how set theory represents geometry, but I have a hunch that something would disappear from a set after the rotation, which would decrease the cardinality of the set. However, if there were only one triangle, I suspect that you could rotate it all you want without affecting the cardinality of a set. It would seem very weird if rotation could end the existence of a triangle in some circumstances, but not in others. Then again, perhaps that's just a consequence of the rules.

Nevertheless, I can imagine an instructor asking a student to 'rotate the triangle so that it exactly overlaps the other one, record the angle of rotation, dilate it by a factor of x, and then sketch the result. In this case, the instructor seems to talk about the plane as though it includes two identical triangles, even after the rotation; it also seems that the correct sketch would include two triangles, one inside the other.

Can a plane include two indiscernible triangles?

• One can have arbitrarily large family $(T_i)_{i\in I}$ of identical triangles, just set each $T_i$ to be a given $T$. – user2345215 Dec 27 '14 at 22:15

Let me first address how set theory represents geometry.

The same way the CPU in the device you are using to write this question interprets the HTML data from the website. Not interesting in details whatsoever. Set theory allows you to interpret the real numbers, and so on and so forth. It's really quite standard, and you can find it on this website on several threads. The point is that set theory interprets geometry in a way that ensures that whatever we prove "naively" without regarding the set theoretic underlying structure will be true in the interpretation. The same way that basic HTML is displayed correctly (modulo available fonts) on my Linux desktop boasting [an old] QuadCore processor, and my Nexus 4 with its ARM processor.

Now you have to ask yourself. What is a triangle? Is it a subset of $\Bbb R^2$? Is it a more abstract object? In the former case, congruent triangles are not the same, exactly in the same way that $[0,1]$ and $[2,3]$ are not the same, despite having the same length as line segments.

This means that when moving and rotating a triangle you consider a different triangle which has different elements as a set, but not different geometric properties.

If you consider triangles to be more abstract objects, not represented by sets on the plane, then the answer will depend on how you consider them. Are two congruent triangles the same? Is moving a triangle from one point on the plane to another gives you the same triangle? If the answer is yes, then there is only one triangle of each "type". If the answer is no, then it depends on your definition of a triangle.

• I think you did not quite answer the question. It is not just about congruent triangles, but about congruent triangles that are at the same place (they are related by an identity translation so to speak). In that case set theory does say they are the same, hence "the two" are really one triangle. – Marc van Leeuwen Dec 28 '14 at 9:15

You can speak about triangles being similar up to different levels. For example, if the set consists of triangles that are equal up to rotation, then it would only contain one triangle of all the ones that are only different by a rotation. Otherwise, one could consider triangles up to translation by a vector, in which case two triangles that are the same up to rotation would be different. Or you could consider all triangles different unless they occupy the exact same space in the plane. Or you could consider a family of labeled triangles, all the same except they have different labels. The only rule with sets is that they allow no repetition, but you can define what is meant by repetition.

Yes, mathematically "two" objects that have nothing to distinguish them are in fact just one object, mentioned twice. If $x$, $y$ designate any kind of objects of the same kind (natural numbers for instance, but it applies to triangles just as well), then the set $\{x,y\}$ has cardinality (number of elements) equal either to $1$ or to $2$, depending on whether $x=y$ or $x\neq y$ holds. But mathematically one cannot modify objects, even though informal language makes it seem so. If one subtracts $1$ from $3$, then one is not modifying $3$ so as to make $3$ equal to$~2$ (which would be absurd), but one in performing a (subtraction) procedure whose outcome is a "new" value,$~2$ (although of course that value always existed, independently of our procedure). Similarly if one translates a triangle $T_1$, this is producing a triangle that (usually) differs from$~T_1$. When the translation is such that the result is equal to another given triangle$~T_2$, then set containing the translated $T_1$ and the triangle$~T_2$ indeed has only one element. But the translated$~T_1$ is not the same as $T_1$, so one did not change the cardinality of $\{T_1,T_2\}$ (and no operation could possibly do that). If one continues to scale the translated $T_1$, then this produces yet another triangle, this times different from $T_2$. In fact it would be just the same if "instead" one continued to scale $T_2$; this too would produce that same other triangle that differs from $T_2$.

But one must distinguish between values (to which the above applies) and methods used to designate or obtain values. Even if $x=y$, it is still the case that $x$ and $y$ are different variables, which differ in some properties not deduced from their values (they have different places in the alphabet, they may have been introduced at different points in the text). Similarly one may say that the vector $(0,0,0)\in\Bbb R^3$ has three components, all of which are equal to the number$~0$. This is really a manner of speech: a "component" here is a method (a projection) to obtain a number from an element of $\Bbb R^3$; the first, second and third components are distinct such methods, but applied to $(0,0,0)$ they happen to produce the same value.

Each one decide, to some extent, what a triangle is. Traditionally a triangle ABC is something determined by the tree points A, B and C, but if you consider triangles in motion you'll have an other way to identify triangles.

Identity is associated with substitution: whenever x=y it is always possible to substitute x with y. The context decides what the objects are and when they are identical/substitutable.

Yes, they can. There is a difference between "identity" and the state of being "identical". You can separately identify two identical triangles in math (e.g. triangle A and triangle B), just as you can identify two electrons in physics.

• One of several reasons why two electrons cannot be said to be a single electron is that two electrons have twice the charge of one electron. Triangles do not have that property in the usual mathematical model. You can separately identify two congruent triangles in math; if "identical" means the same as "congruent" to you, then of course you can distinguish identical triangles; if "identical" means the exact same triangle in the same place, then it's not so clear that you can distinguish the triangles. – David K Jan 2 '15 at 15:50
• @David K. I am not comparing one electron to two. I am comparing one electron to another single electron. To clarify, I intended one to understand that electrons are identical in every property that we can measure, but one electron can be identified from another. You yourself used T1 and T2 as names to identify two identical triangles. I used analogy as a tool for explanation, just like some other answers on this same question. You can disagree, but it may still be helpful to see it this way – Jonny D Jan 2 '15 at 19:09
• $T_1$ (in its original location--there is no other triangle I call $T_1$) and $T_2$ are distinct (not identical in the sense I would say "identical") because they are in different locations and one triangle does not contain the same points as the other. So yes, the original $T_1$ and $T_2$ are as distinct as two electrons, which nobody questions. The question is whether the image of $T_1$ is distinct from $T_2$. That situation is not analogous to two electrons because the things that guarantee you always have two electrons are not even analogous to any properties of triangles. – David K Jan 2 '15 at 19:53
• In your example, if R is a rotation of 0 degrees and D scales by unity, T1 and T2 could indeed be identical (triple bar, math identical) yet have two unique identities. The actual question is simply whether two shapes can occupy the exact same area. It sounds perhaps like semantics and the question is meaninglessly subjective and certainly not practical. It's not really properties of triangles that are the matter. It is properties and representations of math systems. Not worth spilling more ink over, I'd say. – Jonny D Jan 2 '15 at 22:32
• The question is mostly semantics. One view is that if the transformation is the identity, $T_1$ and $T_2$ were never two triangles in the first place, only one. But even then we've given the triangle two names. It's not something I'll lose sleep over. – David K Jan 3 '15 at 12:47

I would interpret the instructions, "rotate the triangle so that it exactly overlaps the other one, record the angle of rotation, dilate it by a factor of x, and then sketch the result," as a paraphrased form of the following instructions:

Given triangles $T_1$ and $T_2,$ find a rotation $R$ under which the image of $T_1$ is $T_2.$ Record the angle of the rotation $R.$ Let $D$ be a dilation by a factor of $x$; sketch the image under $D$ of the image of $T_1$ under $R,$ that is, sketch the image of $T_2$ under $D.$

Since merely drawing the final image gives no sense of scale, my sketch would also show at least the original triangle, and probably the second as well.

This is from a point of view in which the plane is a set of points, and a triangle is a subset of the plane (and therefore also a set of points). The image of the first triangle under the rotation is the second triangle, and the set of points "after the rotation" is a strict subset of the set of points "before the rotation," so yes, in a sense you do "lose" something when this figure replaces the figure consisting of the original two triangles. (Whether the cardinality is less depends on how many points a triangle contains: if you count only vertices, then you may go from as many as $6$ to $3,$ but if you consider all the points on the triangle's perimeter as part of the triangle, the cardinality of both sets is the same infinity.)