Yes! Even a rhombus!! https://comb.io/Sks1NZ
Have you ever seen a square? Congrats, it’s a type of rhombus.
A square is also a rectangle.
Indeed. And square is, by its definition, the only shape that is both.
We learn to learn how to learn.
Rhombusoh they don’t appear again, but its generalization, the parallelepiped, does:
it’s used to calculate volumes of curved objects. basically you chop down the object into a lot of small parallelepipeds (mentally), and then calculate the volume of each of them small things and sum over them. Done.
to calculate the volume of a parallelepipede, there’s a surprisingly simple mathematical formula. If you have the vectors for the three sides a, b, c, then the volume V = (a × b) · c, where × is the cross product and · is the dot product. it’s very simple and an effective way to calculate volumes of curved / deformed objects.
Links:
- (german) https://de.wikipedia.org/wiki/Transformationssatz
- (english) https://en.wikipedia.org/wiki/Integration_by_substitution#Substitution_for_multiple_variables
where × is the cross product and · is the dot product
pardon my ignorance but what the fuck is a “cross product” or a “dot product”? I assumed at first this was multiplication, but then I saw the dot and realized this isn’t anything I’ve ever been taught
https://en.wikipedia.org/wiki/Cross_product
https://en.wikipedia.org/wiki/Dot_product
sorry i am too tired to explain in full detail rn
that’s fine, and thanks for the links! I just don’t quite agree that this is suprisingly simple mathematics :P
yeah it’s “surprisingly simple” in the sense that there’s a well-defined algorithm to do it. a computer can do it easily, with very little time/effort. anyways, you don’t need to think about every problem specifically. “now i got this parallelepiped, how do i calculate the volume?” you can just use the same formula every time.
it’s used to calculate volumes of curved objects. basically you chop down the object into a lot of small parallelepipeds (mentally), and then calculate the volume of each of them small things and sum over them. Done.
For anyone not quite getting this (like me), to calculate the area under a curve in 2D we were taught to cut it into tiny thin rectangles and sum those up; intergration when those rectangles have a width tending to 0.
For 3D (e.g. a pond ripple) or higher curves, a simple rectangle wont cut it as the length of the rectangle might only get the top of a wave, but not capture the crest of it tangential to it. So you create slopey rectangles to approximate that space, and bring the limit to zero to get the area (I think).
My only confusion now is, if I’m deforming a rectangle from one side to approximate the curve just above it, am I not also deforming the bottom of that rectangle in the same way (for the parralel strcture to hold true), and creating a forgotten space just above the axis plane?
well, almost. it’s a bit different than that.
what you mean is this:
you approximate an integral with a lot of small thin rectangles, but if the curve’s not entirely rectangular, there’s gonna be some error, which becomes smaller as the rectangles become smaller. this can be ignored if the rectangles are thin enough. and it’s not what i meant.
what i meant is something like this:
you take a piece of elastic fabric, and paint some squares on it. now, if you stretch the fabric, the squares change shape, they become approximately parallelepipeds. this is a good approximation. now, if we want to calculate the total area of that fabric (after stretching), we can calculate the area of each of the small parallelepipeds (which is easy to do with the formula in above comment) and then sum them.
Ohh! Calculating area, not volume under the curve – I see, thank you
Don’t forget the poor trapezoid.
RIP
Wasn’t she that actress from that 2008 movie Doomsday?
No, you’re thinking of Danny DeVito’s wife. They have separated but are still married.
Haven’t you guys used a kite before?
ILLUMINATI CONFIRMED!!1!one!!
There is an evil invasive plant called Sida rhombifolia that invades local habitats and is obnoxious to pull over here. Guess what shape the leaves are
edit: Acalypha rhomboidea is on the side of justice here though
OOP needs to eat baklava
isn’t that one of those mask things fascists use?
No, and the word is too Eastern European to keep a meme chain going in English if that’s what you’re going for
You know what they say, all busses go to Rhome
Rhombus deez nuts
A rhombus, otherwise known as a diamond. Its one of the more popular shapes past star, triangle, circle, and square.
Hexagon is bestagon
Hendecagon is the only one that is also a good source of protein though
Happy cake day!
I changed my excel cells to be rhombuses
Does it still work on vertical rows and columns, or are they both diagonal now?
Fun word for you, the up-down-left-right set of directions can be referred to as “orthogonal” - think “orthodox”.
