Pastelland Forum

Taalit wrote:Where does the factor of 3 come in?

Not too sure which method you are using here.
But essentially, using the area of a triangle formula, you can use radian polar coordinates to easily find the length of one side, the base of the triangle. (d=sqrt(r1^2+r2^2-2*r1*r2*cos(θ1-θ2), where both radii are 1 unit, angles are 0 and -2π/3 for simplicity's sake. It should equal sqrt(3).)
Then, you can use this length and the angle in a simple sine function to find the height. (Should yield 3/2...sin(pi/3)*sqrt(3)=sqrt(3)*sqrt(3)/2=3/2).
Plugging these into the triangle area formula, you get 3*sqrt(3)/2, or if you rather, 3^(3/2)/2.

Yeah, I saw a video on khanacademy about this exact problem and it helped. I need to work on my geometry, I've been skipping out on that stuff. All I know is integrals and whatnot
Speaking of integrals, I successfully solved this in school today:



It has quite an interesting value!

Taalit wrote:Yeah, I saw a video on khanacademy about this exact problem and it helped. I need to work on my geometry, I've been skipping out on that stuff. All I know is integrals and whatnot

Well, actually, all geometric formulas can be derived from integrals. (Although perimeter and circumference ones tend to be a bit tricky, especially since the circumference of a circle is usually derived in polar coordinates with radians, which is based on the circumference of a unit circle...>_>)

For example, in the triangle example, the area is merely the double integral of 1, with either the upper y limit or the upper x limit containing the equation for the hypotenuse.

This applies to all triangles, although any non-right triangle will have a continuous piecewise hypotenuse, and thus will require
you to split it into the sum of two separate triangles, which will give you the same formula in the end regardless.

So yeah, once you know integrals...(and polar, cylindrical, and spherical coordinate systems)...you don't need to know geometry anymore, as you have the tools to rigorously prove geometry.

Taalit wrote:Speaking of integrals, I successfully solved this in school today:



It has quite an interesting value!

Wait, are you using log to express the base 10 logarithm, or the natural logarithm?
Either one, that still is quite impressive. That is one I would have resorted to numerical approximations to derive...congratulations!

dVanDaHorns wrote:

Taalit wrote:Yeah, I saw a video on khanacademy about this exact problem and it helped. I need to work on my geometry, I've been skipping out on that stuff. All I know is integrals and whatnot

Well, actually, all geometric formulas can be derived from integrals. (Although perimeter and circumference ones tend to be a bit tricky, especially since the circumference of a circle is usually derived in polar coordinates with radians, which is based on the circumference of a unit circle...>_>)

For example, in the triangle example, the area is merely the double integral of 1, with either the upper y limit or the upper x limit containing the equation for the hypotenuse.

This applies to all triangles, although any non-right triangle will have a continuous piecewise hypotenuse, and thus will require
you to split it into the sum of two separate triangles, which will give you the same formula in the end regardless.

So yeah, once you know integrals...(and polar, cylindrical, and spherical coordinate systems)...you don't need to know geometry anymore, as you have the tools to rigorously prove geometry.

Yep. The reverse is true too, it helps me very much that I can visualize any differential equation of any order as a particle moving in a vector field

dVanDaHorns wrote:

Taalit wrote:Speaking of integrals, I successfully solved this in school today:



It has quite an interesting value!

Wait, are you using log to express the base 10 logarithm, or the natural logarithm?
Either one, that still is quite impressive. That is one I would have resorted to numerical approximations to derive...congratulations!

I think it's natural log. I (and from what I have heard, most mathematicians) always use "log" as the natural logarithm, and log_10 as the base 10 logarithm. That has caused quite a headache to some of my teachers XD
EDIT: I have a main reason why I use log instead of ln, but it's kinda silly so I better don't tell you
And yeah, that one is hard. It looks like it involves dilogarithms...

Yeah, my profs here tend to use ln for the natural logarithm and log_10 as the base 10 logarithm, just to avoid the confusion. Not that base 10 is used that much anymore, with the exception of for p values (pH, pKa, etc.)

Also, indeed it does involve a dilogarithm.

I use either log or ln to mean 'natural logarithm' depending on which one is prettier in the formula, although I try to keep consistent when I'm writing things on a single sheet of paper. There's not really ever confusion for me, because 99% of the time, you're using the natural log in any sort of calculus-related operation.

And the dilogarithm that shows up is exactly what makes that particular integral cool!

http://xaos.sourceforge.net/ there you go, some artsy math stuff

Erendis42 wrote:http://xaos.sourceforge.net/ there you go, some artsy math stuff

...in my field, more like artsy chemistry stuff.

Fractals =/= chemistry.

Isobel The Sorceress wrote:Fractals =/= chemistry.

Fractals are often used to describe the aggregation of nanoparticles, and in general, any system of atoms or molecules where no order can be found. We break them into systems of fractals.
Which is using fractals for chemistry.