#finite volume

[factsandalcohol]

This is Gabriel's Horn.

It has an infinite surface area but encloses a finite volume.

Wait, what?

[relatetonothing]

it's 1am, but i'm up reading maths. the area of fascination is: can there be an object of finite volume but infinite surface area?

thinking un-mathematically, the answer would be NO. imagine any object, like a book. it has a fixed volume and fixed surface area.

but maths explores more funky shapes that break the rule~ and of course the correct answer is YES, there is an object that fits the requirements.

imagine a trumpet like shape, more mathematically defined as a solid of revolution of the graph y=1/x domain x≥1 (obviously, to avoid the asymtote at x=0).

let the upper limit of this shape be a (you can choose any number, but you can see from the diagram that as a becomes larger, the cone becomes 'narrower'). the lower limit is of course 1.

so the VOLUME is defined as:

you can see here that the volume of the shape NEVER reaches Pi. because of the 1 - 1/a bit, which is always smaller than 1, making the product with Pi smaller than Pi.

However as a tends to infinity, 1/a tends to 0 and so V tends to Pi, a finite value.

In other words,

Let's now consider the surface area.

this is defined as:

There is no upper bound for lna as a tends towards infinity (as shown below)

therefore 2Pilna tends to infinity as a tends to infinity. In other words,

Here we have an object whose VOLUME is finite and SURFACE AREA is infinite.

Beautifullllllll <3

MATHS JOKE OF THE DAY (with respects to Mandelbrot):

His full name is Benoit B. Mandelbrot. EXPAND HIS NAME! XD

it would be Benoit Benoit B. Mandelbrot Mandelbrot, Benoit Benoit Benoit B. Mandelbrot Mandelbrot Mandelbrot (etc). an INFINITE series. JUST LIKE FRACTALS!! (the branch of mathematics he discovered) xDDDDD *dying*

NOTE:

maths is the foundation of life, and if you don't appreciate it, how can you appreciate life itself? ^^