During a rather lively debate on SSM, the thread, somehow, ended discussing the reality of geometric shapes.
Do triangles exist? Are triangular-shaped objects really triangles?
I undertand that, for shorthand and convenience, many things are called triangles. But are they geometric triangles?
From Wikipedia "A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ABC.
In Euclidean geometry any three non-collinear points determine a unique triangle and a unique plane (i.e. a two-dimensional Euclidean space)."
So if I draw a “triangle”, it isn’t really a triangle, but a series if ink-dyes applied in uniform lines to a pulp-based recepticle for demarkation?
It all sounds incredibly nit-picky to me. We’ve got to call things that look like triangles something. And I’ll go out on a limb here and suggest, to hell with it, let’s call them “triangles.”
Since it’s impossible to draw a perfect line, and given the infinitely small nature of a point on said line, every attempt to draw a line in the real world will necessarily be a series of line segments(with vertices) before they come to the intended “vertex” of the “triangle”. So each “line segment” between the vertices of a “triangle” would, in actuality, have multiple vertices along its length. Thus no real-world “triangle” can fit the definition of “having only three vertices.”
The glaring redundancy of this sentence should be a pretty big clue what the answer is. What’s a triangle? It’s a shape. What, then, is a “triangular-shaped object”? Obviously, a triangle.
Triangles exist objectively, as norms. By this, I mean that there are certain purposes which objects serve better the closer they (the objects) are to a triangular ideal. This is an objective fact about those purposes, and that is the sense in which I think triangles exist objectively.
This question is as answerable as “Does blue exist?” There is a spectrum of light that our eyes perceive as blue, but does it exist independently? The answer for both is that they are descriptor words. Some thing triangle shaped is not a triangle (unless it is), it is a triangle shaped object.
This also applies to numbers. Is there such a thing as two or zero? Does division really exist? They are not physical objects but they are real concepts that have real world, concrete applications.
We use the word “triangle” to refer to both the mathematical object and the real-world shape. Triangles in the first sense do not exist anywhere in the material world, but they do in the Platonic realm. Triangles in the second sense can be readily found in the real world.
Only a few three-sided polygons actually have the will to become triangles. The rest are mired in a slave-like mentality, subject to the masterful dominations of polygons with more sides.
I think if you were going to be picky like that, you couldn’t really say such objects are triangle-shaped, but would rather have to describe them as triangle-like. If it’s triangle-shaped, it’s a triangle. If it’s triangle-like, it’s merely similar.
I heard one guy say that a triangle can prove the universe is flat. Another said it only seems Euclidean because we can’t detect a tiny arc on a plane that size.
As a concept, a “triangle” exists, in the same sense that a “chair” exists, or a “man.” They are abstractions that represent x number of entities, which have certain properties in common, while differing in their particulars. There are as many possible triangles as there are possible chairs or men, but the concept remains constant.
Eh; a non-Euclidean universe has triangles too: any three points determine one in the usual way. It may not have all the properties of Euclidean triangles (angles adding up to 180 degrees, Pythagorean theorem/Law of Sines, etc.), but it’s still an instance of the basic notion of “three vertices and the line segments which join them”.
Sure, according to relativity, it is true that our universe is non-Euclidean and only seems otherwise on small scales where the deviation is negligible. But it doesn’t mean triangles in our universe are actually pseudo-triangles with bending arcs for edges; their edges are straight lines, just in an intrinsically curved universe.
Right, and this is why a triangle can prove whether the universe is flat or not. The three spatial dimensions of our universe constitute an enormous three-manifold. It is possible that this manifold is Euclidean, and it is possible that it is not. One way to tell is to draw a triangle in space and measure the angles. If those angles add up to anything but 180 degrees, the universe is not Euclidean. If those angles add up to 180 degrees well, then you don’t know anything.
The man Lib probably heard was Jeff Weeks, an unaffiliated mathematician who travels about and, among other things, has an incredibly entertaining lecture on the shape of space and what exactly that means. You can look at his website here.
Good one!