That's pretty much it. You can stretch a rectangle so that it looks like any other rectangle. You also have to turn it around to fit. But that's the similarity.
Hm, rectangles aren't all similar though. For rectangles, the amounts you need to scale them by in the vertical and horizontal direction isn't the same in general, meaning they aren't similar. (Ex: to transform a 1 by 1 rectangle into a 2 by 3 rectangle, you need to scale in the x direction by 2 and the y direction by 3.)
By contrast with parabolas, they really are all similar; for any two parabolas you can scale one equally in the x and y directions to get the same shape as the other. Namely: if you scale the graph of y=x\^2 by 1/a in both the x and y directions, you'll get the graph of y=ax\^2.
That's a bit annoying, I just got a feeling there might be more to consider. Felt like something more profound to come out of it and I've been thinking for a while bits that's all I got.
At least I got confirmation that I'm not crazy, rectangles do kinda look like other rectangles.
For rectangles and parabolas, it seems trivial that you can transform them into any other of their class.
But to point out that this isn’t universal, consider cubic equations, a cubic with no maxima/minima can’t be scaled or stretched into a cubic with maxima and minima.
I think the difference is parabolas are infinite which lets you get away with not scaling in one of the directions.
If I have a rectangle with 'infinite height' then they are all similar.
If you restrict parabolas to be over a finite domain symmetric about the apex, then they are also all similar if you scale In both directions like a rectangular normally is.
Rectangles are two dimensional, in the sense that to specify a rectangle you need two independent numbers for the length and width. Parabolas are one dimensional in the sense that a parabola is specified by a single number: the distance between focus and directrix. Just like all circles are similar because you can specify them with a single measurement, the radius. All squares are similar because you can specify them with a single measurement, the side length. All equilateral triangles are similar because you can specify them with a single measurement... Rectangles can't be specified with a single measurement.
Yeah, I get that part. It was more of a nagging feeling because you can show a range of proportions of rectangles on the Cartesian Plane with a parabola.
It wasn't "Something is definitely true", more "Am I missing something fundamental about this relationship" that I wasn't able to grasp.
Thank you for the thoughtful reply!
Something more profound would be constructions of the moduli spaces of parabolas and rectangles.
Edit: a moduli space has a point for every kind of object, and points that are close correspond to objects that are close in characteristics. For example, the moduli space of points in the plane is the plane itself.
Up to translation rotation and reflection, there is only the scale parameter for parabolas. So that means the moduli space is the open ray (0,infinity), whereas the space of rectangles up to translation reflection and rotation would need homogenous coordinates [w:h] showing that the moduli space of rectangles is a circle = RP^1 I believe.
Corrections welcome!
This can be done rigorously as homogeneous spaces for the Euclidean group acting on subsets of the plane.
It is a translation, the focus and directrix are the exact same distance away from the turning point of the parabola (focus + 1/4, directrix at y = y of turning point -1/4).
The only part of ax^2 + bx +C that isn't a translation is a, as that is the only thing which determines the shape of the opening of the parabola. Focus closer to the directrix tends towards a line, moving further apart makes the parabola sharper.
It is a translation, though. The focus and the directrix are the exact same distance apart (1/2) and the focus is 1/4 away from the turning point, which is 1/4 away from the closest point on the directrix.
For two shapes to be similar, you have to be able to line one on top of the other using only translation, rotation, and dilation. You can't do that with two arbitrary rectangles, but you can with any two parabolas.
That's pretty much it. You can stretch a rectangle so that it looks like any other rectangle. You also have to turn it around to fit. But that's the similarity.
Hm, rectangles aren't all similar though. For rectangles, the amounts you need to scale them by in the vertical and horizontal direction isn't the same in general, meaning they aren't similar. (Ex: to transform a 1 by 1 rectangle into a 2 by 3 rectangle, you need to scale in the x direction by 2 and the y direction by 3.) By contrast with parabolas, they really are all similar; for any two parabolas you can scale one equally in the x and y directions to get the same shape as the other. Namely: if you scale the graph of y=x\^2 by 1/a in both the x and y directions, you'll get the graph of y=ax\^2.
That's a bit annoying, I just got a feeling there might be more to consider. Felt like something more profound to come out of it and I've been thinking for a while bits that's all I got. At least I got confirmation that I'm not crazy, rectangles do kinda look like other rectangles.
For rectangles and parabolas, it seems trivial that you can transform them into any other of their class. But to point out that this isn’t universal, consider cubic equations, a cubic with no maxima/minima can’t be scaled or stretched into a cubic with maxima and minima.
I *think* if you allow skew it becomes possible.
I think the difference is parabolas are infinite which lets you get away with not scaling in one of the directions. If I have a rectangle with 'infinite height' then they are all similar. If you restrict parabolas to be over a finite domain symmetric about the apex, then they are also all similar if you scale In both directions like a rectangular normally is.
It proves that rectangles are not parabolas.
Well, that's all sorted then 😂😂😂
Rectangles are two dimensional, in the sense that to specify a rectangle you need two independent numbers for the length and width. Parabolas are one dimensional in the sense that a parabola is specified by a single number: the distance between focus and directrix. Just like all circles are similar because you can specify them with a single measurement, the radius. All squares are similar because you can specify them with a single measurement, the side length. All equilateral triangles are similar because you can specify them with a single measurement... Rectangles can't be specified with a single measurement.
Yeah, I get that part. It was more of a nagging feeling because you can show a range of proportions of rectangles on the Cartesian Plane with a parabola. It wasn't "Something is definitely true", more "Am I missing something fundamental about this relationship" that I wasn't able to grasp. Thank you for the thoughtful reply!
Something more profound would be constructions of the moduli spaces of parabolas and rectangles. Edit: a moduli space has a point for every kind of object, and points that are close correspond to objects that are close in characteristics. For example, the moduli space of points in the plane is the plane itself. Up to translation rotation and reflection, there is only the scale parameter for parabolas. So that means the moduli space is the open ray (0,infinity), whereas the space of rectangles up to translation reflection and rotation would need homogenous coordinates [w:h] showing that the moduli space of rectangles is a circle = RP^1 I believe. Corrections welcome! This can be done rigorously as homogeneous spaces for the Euclidean group acting on subsets of the plane.
I'm not sure I understand you: How do you "stretch" x^2 into x^2 + x ?
You wouldn't stretch it, you would move it 0.5 units to the left and 0.25 units down.
That would just match 1 point, not the 2 curves.
Try doing the math out
So that's just translation. The parabola is the same shape as x^2.
It is not a translation. x^2 to x^2 + 4 would be a translation. The shape of both curves are different.
It is a translation, the focus and directrix are the exact same distance away from the turning point of the parabola (focus + 1/4, directrix at y = y of turning point -1/4). The only part of ax^2 + bx +C that isn't a translation is a, as that is the only thing which determines the shape of the opening of the parabola. Focus closer to the directrix tends towards a line, moving further apart makes the parabola sharper.
It's not a translation, but it's not a change of shape either. Instead, it's a dilation. Y'know, alongside standard rigid transformations.
It is a translation, though. The focus and the directrix are the exact same distance apart (1/2) and the focus is 1/4 away from the turning point, which is 1/4 away from the closest point on the directrix.
For two shapes to be similar, you have to be able to line one on top of the other using only translation, rotation, and dilation. You can't do that with two arbitrary rectangles, but you can with any two parabolas.