Cool proof i found. seen it b4?

[Green Zach]

ok the proof is kinda complicated and i honestly don't want to wright it out lol but try it out because it works. So my equation states that an inscribed circle in a square has the surface area of the square x pi/4. So say you have a square with side lengths of 5 so the SA of that square is 25. if you had a circle inscribed into that square so thus the circle has a radius of 2.5, the surface area would be equal to 25 x pi/4. This also works with a cylinder inscribed into a 3D box. so say the box had a volume of base 5 x depth 5 x height 10 = 250, the inscribed cylinder would have a volume of 250 x pi/4. take out your calculators and compare it to the regular volume and SA equations for cylinders and circles, its true. Has anyone heard of or seen this before? its nothing amazing i am just a 12th grade student who got bored in class one day and did a math proof lol just thought it would be cool if i was the first to come up with it.

 

[HallsofIvy]

Yeah, it is kind of cool. How about showing us your proof? Also, have you considered the case with a sphere, rather than a cylinder, inscribed in a cube?

 

[ice109]

i don't believe it works for all cases. show me the proof.

 

[BlackWyvern]

I have also done a bit of fiddling around with this.

What else you will find very interesting, is that if you inscribe a square, into that circle that is inside the first square, the area of the smaller square is half the area of the larger square. This pattern keeps repeating whichever way you inscribe the squares.

It has been discovered before as well, all it is is a consequence of the definition of pi.

 

[Mentallic]

Using the fact that for a circle, $$A=\pi r^2$$ the proof is quite simple actually.

Let the side length of a square be s, therefore $$A=s^2$$ for the square.
A circle inscribed inside the square, touching all sides of the square's perimeter, will have a radius of $$s/2$$ since its diameter = s.

Hence, $$A=\pi (\frac{s}{2})^2 = s^2 \frac{\pi}{4}$$

but the square's area is $$s^2$$ , therefore, the ratio of the areas of the circle:square is $$\frac{\pi}{4}:1 \approx 0.78:1$$


The cylinder inside the cube (or rectangular prism with square base) isn't that interesting from here, since it won't change a thing.

For the sphere in the cube on the other hand, we have:

cube with sides s, therefore $$V=s^3$$

Using volume of sphere $$V=\frac{4}{3}\pi r^3$$

diameter of sphere is s, so radius is $$s/2$$

Hence, $$V=\frac{4}{3}\pi (\frac{s}{2})^3 = s^3 \frac{\pi}{6}$$

So the ratio of the volume of sphere inscribed in cube is $$\frac{\pi}{6}:1 \approx 0.52:1$$

Sorry if I spoilt any fun

 

[Green Zach]

so we know that depth x height x width = volume of a 3D cube or box. we also know that r²h$$\pi$$ = volume of a cylinder. so we can say that becuase its a cylinder being inscribed into a box, that the width and depth of the box are going to be equal. so instead of saying that its w x h x d, because depth and width are the same let's just call them $$\alpha$$₂ and multiply that by the height to get the volume. I use alpha mostly because i honestly just think that alpha is a totally sweet letter :P. So because the radius of the cylinder is going to be 1/2 that of the depth of the 3D box, we can call it (1/2$$\alpha$$)² so now the volume of the cylinder is (1/2$$\alpha$$)²$$\pi$$h. to get the ratio between the cylinder and the box we just divide the equation of the box over the equation of the cylinder. so as you can probably figure out, height cancels and we are left with $$\alpha$$²/(1/2$$\alpha$$)²$$\pi$$. but $$\alpha$$²/(1/2$$\alpha$$)² = 4 so we are left with 4/$$\pi$$ but this is the ratio of the box to the cylinder not the cylinder to the box so simply take the inverse of 4/$$\pi$$ and that's the thing you multiply a box by to get the volume of an inscribed cylinder

 

[Green Zach]

so we know that depth x height x width = volume of a 3D cube or box. we also know that r²h$$\pi$$ = volume of a cylinder. so we can't say that becuase its a cylinder being inscribed into a box, that the width and depth of the box are going to be equal. so instead of saying that its w x h x d, because depth and width are the same let's just call them $$\alpha$$₂ and multiply that by the height to get the volume. I use alpha mostly because i honestly just think that alpha is a totally sweet letter :P. So because the radius of the cylinder is going to be 1/2 that of the depth of the 3D box, we can call it (1/2$$\alpha$$)² so now the volume of the cylinder is (1/2$$\alpha$$)²$$\pi$$h. to get the ratio between the cylinder and the box we just divide the equation of the box over the equation of the cylinder. so as you can probably figure out, height cancels and we are left with $$\alpha$$²/(1/2$$\alpha$$)²$$\pi$$. but $$\alpha$$²/(1/2$$\alpha$$)² = 4 so we are left with 4/$$\pi$$ but this is the ratio of the box to the cylinder not the cylinder to the box so simply take the inverse of 4/$$\pi$$ and that's the thing you multiply a box by to get the volume of an inscribed cylinder

wow i clearly don't have this super script thing down. just assume that the only thing supposed to be in superscript is the 2... oops

 

[Bob3141592]

How about a 4D sphere in a 4D cube? A 5D sphere in a 5D cube? It's amazing that as the number of dimensions go up, the ratio of the space taken up in the volume of the sphere goes to zero. In higher dimensional spaces, all of the space is in the corners, leaving nothing in the middle. That sounds weird until you think about just how many corners there are.

This was talked about in the book Nonplussed! by Julian Havil. I think you might enjoy it.

 

[Mentallic]

How about a 4D sphere in a 4D cube? A 5D sphere in a 5D cube? It's amazing that as the number of dimensions go up, the ratio of the space taken up in the volume of the sphere goes to zero. In higher dimensional spaces, all of the space is in the corners, leaving nothing in the middle. That sounds weird until you think about just how many corners there are.

This was talked about in the book Nonplussed! by Julian Havil. I think you might enjoy it.

In other words you're amazed at the fact that these imaginary shapes in imaginary dimensions (warning: quantum theorists, keep out) behave in such extraordinary ways? Actually I would like to take a look at the theories that claim as the dimensions increase beyond #3, the sphere:cube ratio decreases further.

 

[Green Zach]

How about a 4D sphere in a 4D cube? A 5D sphere in a 5D cube? It's amazing that as the number of dimensions go up, the ratio of the space taken up in the volume of the sphere goes to zero. In higher dimensional spaces, all of the space is in the corners, leaving nothing in the middle. That sounds weird until you think about just how many corners there are.

This was talked about in the book Nonplussed! by Julian Havil. I think you might enjoy it.

I just did some thinking... could the fact that the ratio between a circle and a square is the same as the ratio between a cylinder and a cube indicate that a 3D circle is actually a cylinder not a sphere? I mean, I am no mathematician, I am just a 12th grade high school student who just barely started to learn calculus (which I find totally awesome by the way ) but as far as i know or think, a 3D object was once a point which got translated to the side to make a line which got translated up/down to make a square which translated forward/backward to make a cube so following the same translation principles doesn't this mean that a square is to a circle as a cube is to a cylinder? I mean, a sphere is a circle flipped around forward then back on itself in the z direction where a cube is a square pulled forward in the z direction... So a sphere should be compared to a square flipped forward and around in the z direction which would probably look something like a wheel on a car... and the ratio of these two things can probably be calculated easily... I wonder what I will find and what these ratio's might mean. I'll get to finding a relationship right now :tongue2: I'v always thought of pi as just "the way it is" but hmmmm

 

[Green Zach]

haha, i just realized that the square flipped forward would be a cylinder, well that makes coming up with an equation a lot easier! also kinda interesting how that worked out :D

 

[Green Zach]

well the relationship is that the cylinder is 1.5 (exactly) x bigger then the sphere which is a nice and clean relationship. I am starting to think that the value of pi might have something to do with the nature different kinds of transformations in dimensions so pi didnt originate from the relationship between a line and a sphere, it came from the relationship between different kinds of transformations so with this, i don't see it hard to think that maybe we can make an equation that can give us the relationship between any shape and any other shape quite easily based on the characteristics of space not the actual shapes!

 

[Bob3141592]

In other words you're amazed at the fact that these imaginary shapes in imaginary dimensions (warning: quantum theorists, keep out) behave in such extraordinary ways? Actually I would like to take a look at the theories that claim as the dimensions increase beyond #3, the sphere:cube ratio decreases further.

What can I say, I'm easily amused. You could say I'm also fascinated how the conventional 3-D projection from the higher dimensional reality does such a decent job at modeling physical processes where energy is near zero but is such an increasingly poor representation as we move away from that. But that's a topic for another thread in another forum.

In any case, the Havil book is accessible - you might even find it in your public library. The calculus in the chapter on the hyperspheres isn't at all difficult. I found some chapters in the book much more interesting than others, but where Havil is good, he is excellent.

 

[Green Zach]

Bob3141592, i have just fallen in love with your quote :P A ham sandwich is better than nothing. And, nothing is better than eternal happiness. Therefore, a ham sandwich is better than eternal happiness? hahaha :D