I went up to Calculus I in college (along with some physics and engineering), and I’ve done a bit of amateur math outside of school … but the concept of a rhombus was still used exactly zero times.
The engineers prefer triangles, the physicists prefer circles, and the pure mathematicians are usually dealing with weird shapes that don’t even have names. Who uses the rhombus?
A rhombus is just an equilateral parallelogram, and parallelograms are the easiest way to approach vector cross products. And you’re 3/4 of the way there when you do tip-to-tail vector addition. They should have come up at least a couple times in geometry. Rhombi pop up as faces in lots of polyhedra, and tiles in lattices. I don’t know much about crystallography, but I’m sure they make an appearance.
Now that I think of it, when doing lattice structures in Chemistry, many sections would have given rhombuses.
But still, never used the name even over there.
Right there with you, I didn’t use it in analytical geometry, calcs 1-3, diff eqs, or numerical methods. I don’t think it would have shown up in my other math classes either. I’m pretty sure the rhombus is nothing more than a happy shape.
As the other poster succinctly said, it’s about iterative solving problems. What might not be obvious is where you want to use this.
Let’s say you are holding onto a continuous, all-metal cooking pan’s handle and you turn on the burner; at what second will it be too hot to hold? If we assume the pan is starting at 20 C and it’ll be too hot for your leathery, lemmy-browsing hand at 100 C, then we at least have some basic bounds on the problem, but how do we think about the middle parts?
We could say that the burner is putting out about 5 kW of energy (or more simply, that the pan’s bottom has a fixed input of 5 kW (a so-called boundary condition). With a basic equation translating energy input into temperature (factoring in specific heat), we at least know the temperature of the bottom of the pan at a given moment, but what about the top of the pan and, most importantly, the handle? The temperature in these places isn’t climbing because they’re getting licked by flames, but due to thermal conductivity away from the bottom.
Thermal conductivity is actually one of the simplest questions in partial differential equations (PDEs), and is addressed with the aptly named heat equation. In extremely casual terms, the heat equation (du/dt = L u) is saying that the heat at a given position is determined by its past heat plus the past heats of positions around it. Our job as mathematicians is to express our problem using a so-called weak form of the heat equation that makes it possible for an iterative solution.
We can solve weak form equations with a framework like the Finite Differences method (which is a little dated but easier to understand than other methods, and also still quite effective). This very basically means we’ll create a fixed grid of points, then express the relationships between them with algebra.
Once we have our algebraic mesh and we are happy with all of our boundary conditions, we can start the simulation. We watch as the temperature immediately shoots up on the bottom of the pan (followed soon after by the top and edges), then as heat slowly transfers up the handle until it reaches 100 C at the end.
In doing these problems, we make as many simplifications as possible in the boundary conditions, because the equation’s solution space is unfathomably large. Everything we can do to constrain it (easy heat source, exact physical parameters of our materials, etc) shrink the solution space closer to one that matches our actual problem.
As a final thought, lots of very bright people have spent entire careers designing heatsinks using numerical methods, so don’t take my description of the heat equation as an easy problem to mean it’s not a great line of inquiry, but rather that the field is really fucking hard. Anyone who does this stuff can, in fact, hold a 90 C pan handle because their tolerance for pain is astronomical.
This is true for most parallelograms and later polygons.
Daughter just went through parts of middle school that seemed to focus on things like rhombuses and trapezoids. The only thing I"m thinking, as an engineer, is when was the last time I gave af about any of those shapes? I would just reduce them to triangles and rectangles.
Trig skills are WAY more useful than knowing how to calculate specific polygon shapes… I don’t mean to say that middle schoolers should be learning trig… just that the time spent learning how to do specific shapes is a literal waste of time that they could be using to prepare for later things (like trig).
As someone in education i would bet money that since it is a vocabulary word with a simple definition in a subject that is hard to teach, it is included to make things easier on the teachers and students. Mind you easier isnt often a good thing, and in this case I’d say its lazy and irresponsible to focus on things that don’t scaffold well or have much practical use
I went up to Calculus I in college (along with some physics and engineering), and I’ve done a bit of amateur math outside of school … but the concept of a rhombus was still used exactly zero times.
The engineers prefer triangles, the physicists prefer circles, and the pure mathematicians are usually dealing with weird shapes that don’t even have names. Who uses the rhombus?
A rhombus is just an equilateral parallelogram, and parallelograms are the easiest way to approach vector cross products. And you’re 3/4 of the way there when you do tip-to-tail vector addition. They should have come up at least a couple times in geometry. Rhombi pop up as faces in lots of polyhedra, and tiles in lattices. I don’t know much about crystallography, but I’m sure they make an appearance.
Yeah and we engineers love parallelograms, but not as much as triangles.
Now that I think of it, when doing lattice structures in Chemistry, many sections would have given rhombuses.
But still, never used the name even over there.
Right there with you, I didn’t use it in analytical geometry, calcs 1-3, diff eqs, or numerical methods. I don’t think it would have shown up in my other math classes either. I’m pretty sure the rhombus is nothing more than a happy shape.
What’s numerical methods all about?
I mean I think I can intuit a little and could Google a lot, but hearing from someone who seemingly somewhat recently studied is a unique chance.
As the other poster succinctly said, it’s about iterative solving problems. What might not be obvious is where you want to use this.
Let’s say you are holding onto a continuous, all-metal cooking pan’s handle and you turn on the burner; at what second will it be too hot to hold? If we assume the pan is starting at 20 C and it’ll be too hot for your leathery, lemmy-browsing hand at 100 C, then we at least have some basic bounds on the problem, but how do we think about the middle parts?
We could say that the burner is putting out about 5 kW of energy (or more simply, that the pan’s bottom has a fixed input of 5 kW (a so-called boundary condition). With a basic equation translating energy input into temperature (factoring in specific heat), we at least know the temperature of the bottom of the pan at a given moment, but what about the top of the pan and, most importantly, the handle? The temperature in these places isn’t climbing because they’re getting licked by flames, but due to thermal conductivity away from the bottom.
Thermal conductivity is actually one of the simplest questions in partial differential equations (PDEs), and is addressed with the aptly named heat equation. In extremely casual terms, the heat equation (du/dt = L u) is saying that the heat at a given position is determined by its past heat plus the past heats of positions around it. Our job as mathematicians is to express our problem using a so-called weak form of the heat equation that makes it possible for an iterative solution.
We can solve weak form equations with a framework like the Finite Differences method (which is a little dated but easier to understand than other methods, and also still quite effective). This very basically means we’ll create a fixed grid of points, then express the relationships between them with algebra.
Once we have our algebraic mesh and we are happy with all of our boundary conditions, we can start the simulation. We watch as the temperature immediately shoots up on the bottom of the pan (followed soon after by the top and edges), then as heat slowly transfers up the handle until it reaches 100 C at the end.
In doing these problems, we make as many simplifications as possible in the boundary conditions, because the equation’s solution space is unfathomably large. Everything we can do to constrain it (easy heat source, exact physical parameters of our materials, etc) shrink the solution space closer to one that matches our actual problem.
As a final thought, lots of very bright people have spent entire careers designing heatsinks using numerical methods, so don’t take my description of the heat equation as an easy problem to mean it’s not a great line of inquiry, but rather that the field is really fucking hard. Anyone who does this stuff can, in fact, hold a 90 C pan handle because their tolerance for pain is astronomical.
It’s all about solving things iteratively. Usually applied to how computers solve for things.
Thank you for the summary
This is true for most parallelograms and later polygons.
Daughter just went through parts of middle school that seemed to focus on things like rhombuses and trapezoids. The only thing I"m thinking, as an engineer, is when was the last time I gave af about any of those shapes? I would just reduce them to triangles and rectangles.
Trig skills are WAY more useful than knowing how to calculate specific polygon shapes… I don’t mean to say that middle schoolers should be learning trig… just that the time spent learning how to do specific shapes is a literal waste of time that they could be using to prepare for later things (like trig).
As someone in education i would bet money that since it is a vocabulary word with a simple definition in a subject that is hard to teach, it is included to make things easier on the teachers and students. Mind you easier isnt often a good thing, and in this case I’d say its lazy and irresponsible to focus on things that don’t scaffold well or have much practical use
Maybe it shows up in some of the dreaded standardized tests, and that’s why they’re sure to teach it?
What is a rhombus but two triangles, my dear