Triangles in the 3d world.

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JulianMathHelp

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When someone says "Pick up the triangle." Isn't the "triangle" actually a triangular prism as triangles itself can't exist in the 3d dimension?
 
JulianMathHelp said:
When someone says "Pick up the triangle." Isn't the "triangle" actually a triangular prism as triangles itself can't exist in the 3d dimension?
Click to expand...
Please provide an example of usage. I've never heard "Pick up the triangle." before.
 
Sorry for not copying the initial question word for word, but
Albert has a neat trick. Given any triangle, he can place it on the tip of his pencil and it balances on his first try! The whole class wonders, “How does he do it?”
 
I also have another question if you don't mind.
Why is it possible to put a 2-dimensional shape in a 3 dimensional plane when proving formulas and such? Isn't it not possible as 2 dimensional shapes itself can't exist in the 3 dimensional plane?

www.khanacademy.org

Triangle medians & centroids (video) | Khan Academy

Seeing that the centroid is 2/3 of the way along every median
www.khanacademy.org www.khanacademy.org
 
JulianMathHelp said:
Sorry for not copying the initial question word for word, but
Albert has a neat trick. Given any triangle, he can place it on the tip of his pencil and it balances on his first try! The whole class wonders, “How does he do it?”
Click to expand...
Don't understand. Is there more? Place a triangle on the tip of a pencil? No idea what this means.
 
I’m guessing they’re referring to a triangular prism with a super small thickness but I’m not sure, that’s all I have.
 
JulianMathHelp said:
I also have another question if you don't mind.
Why is it possible to put a 2-dimensional shape in a 3 dimensional plane when proving formulas and such? Isn't it not possible as 2 dimensional shapes itself can't exist in the 3 dimensional plane?

www.khanacademy.org

Triangle medians & centroids (video) | Khan Academy

Seeing that the centroid is 2/3 of the way along every median
www.khanacademy.org www.khanacademy.org
Click to expand...
A plane is 2-dimensional. It's a perfectly suitable place for 2d shapes to inhabit. Why do you think it's 3-dimensional?
Even if it were, why can't 2d shapes be in 3d space?
 
JulianMathHelp said:
When someone says "Pick up the triangle." Isn't the "triangle" actually a triangular prism as triangles itself can't exist in the 3d dimension?
Click to expand...
The issue here is the distinction between technical terms and their definitions on one hand, and everyday usage in the real world on the other. A mathematical triangle is a plane figure, which has no thickness. But when we describe a physical object as a triangle (or any other plane figure), we mean that it is an approximation to the plane figure. A "triangle" drawn on paper (or cardboard, or sheet metal, ...) and cut out, which is what your question is about, is such an approximation. It just has to be thin enough that we are willing to ignore the thickness.

In fact, any mathematics applied to the real world is an approximation -- a model. We don't quibble over such things, because we ourselves live in the physical world.
 
Ok, so I understand that when people say “triangles” they are referring to triangular prisms so flat that they model it to be a triangle. I thank you for that. However, I still don’t understand why you can put a 2 dimensional shape on a 3 dimensional plane when proving stuff (In khan academy video)
 
What does "3 dimensional plane" mean? I didn't take the time to watch your video, but I get the impression that on one hand he is talking about ideal geometrical planes embedded in space, so that the plane itself is 2-dimensional (and, in fact, flat); why shouldn't a triangle be embedded in a plane, regardless of what the plane is in turn embedded in? Can I not put a flat piece of paper (modeled by a rectangle) on a flat table (modeled by a plane) that is surrounded by air (i.e. in space)? You'll have to explain your thinking.

Then, on the other hand, I see a reference in the transcript to an "iron triangle"; that would be a physical model, and would not be part of a literal, geometrical plane. I don't think that's where you're objecting, but you'll have to state what your objection is.
 
My objection is: Screen Shot 2020-04-23 at 4.38.49 PM.png
How can you put a triangle with each vertex on each axis? Is there another plane that the triangle is laying on?

Dr.Peterson said:
Can I not put a flat piece of paper (modeled by a rectangle) on a flat table (modeled by a plane) that is surrounded by air (i.e. in space)?
Click to expand...

I mean of course you can model a rectangle on a plane by using a flat piece of paper on a flat table as how else could you represent it without literally drawing on the paper, but I think my objection is different (not sure though).
 
JulianMathHelp said:
So there’s another plane, it’s just embedded on the other “visible” planes?
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Another plane? In addition to which plane? We have 3d space with a coordinate system. We pick 3 points on its axes. These 3 points define a plane. They also define a triangle in the video. The triangle lies in that plane.
 
JulianMathHelp said:
So the plane that the triangle is in is like “slanted” upwards?
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Draw a 2d coordinate system. Pick a point on x and on y axes. Draw a line through the points. The points define this line. Similarly, in 3d case the 3 points define a plane. You can call it "slanted", not sure what it means exactly. Theoretically it can be vertical or horizontal too, if one of the points is at the origin.
 
By slanted I mean the plane that the triangle is contained in is like laying on the other planes (the other planes are the Xz plane, yz plane, and xy plane.)
 
JulianMathHelp said:
By slanted I mean the plane that the triangle is contained in is like laying on the other planes (the other planes are the Xz plane, yz plane, and xy plane.)
Click to expand...
Julian, it appears to me that you have very confused ideas of \(3D\) geometry. Here is a very readable textbook. I have the paperback version.
I often use it for reference.
 
Okay, I did some reading, and here is what I know.

Triangle (2D) CAN exit in the third dimension (xyz axis). However, it will still have no thickness. Because of this, even though it can exist in the third dimension, in our physical world, it does not exist. This is because even though our world IS third dimensional, it requires thickness.

However, I'm not super clear on why it can exist in the third dimension even though it has no thickness. Is it just because it does? Or is there an explanation on why?

Thanks!
 
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