Online since 2002. Over 3300 puzzles, 2600 worldwide members, and 270,000 messages.

TwistyPuzzles.com Forum

It is currently Fri Apr 18, 2014 1:38 pm

All times are UTC - 5 hours



Post new topic Reply to topic  [ 14 posts ] 
Author Message
 Post subject: A New Conjecture of Volume?
PostPosted: Wed Apr 06, 2011 9:27 pm 
Offline
User avatar

Joined: Tue Jan 01, 2008 7:30 pm
Location: Texas, USA
Hello everybody,

I have recently created a new conjecture, however it is unlikely that my conjecture is original and unpublished. As I am still in high school, I don't know all of the theorems, conjectures, principles, laws, etc. I have attached my word document, but it is awkwardly worded. What I am asking help for is as followed:

1. Is their an equation or multiple equations that state everything my equation states? (novel part of my equation is as followed: calculates volume of irregular solids based on knowledge of Face Area and Face Location, and can calculate volume based on any point (x,y,z) [ie the point doesn't have to be within the solid or on the surface of the solid. The point can be outside of the solid]. )

2. Any recomendations on how to word this more properly/formaly.

3. How would one go about proving this?

4. If this is original and unpublished, where/which journal should this be published to?

All coments welcome.

While I would be suprised if such an issue of plagarism and copying would occur,

I have had this equation (concept, I am still allowing revisions to make the equation more formal) and supporting documents copyrighted and have had a statement signed and notorized (today april 6, 2011) saying that this is my original work and that I take full credit for everything published in my attached document such that this equation has not been previously published.

Any rewording and/or proof work that you submit will be credited as such in a form of your choice.

Attachment:
's Conjecture of Volume.doc [153 KiB]
Downloaded 93 times


Thanks for helping me out,
Tanner Frisby

_________________
"I discovered the triangle one day while shaving. I trimmed my beard like the intersection of three circles and noticed how I could unfog a square in the bathroom mirror by rubbing my beard circularly against the glass."-Franz Reuleaux


Top
 Profile  
 
 Post subject: Re: A New Conjecture of Volume?
PostPosted: Wed Apr 06, 2011 9:34 pm 
Offline
User avatar

Joined: Wed Jan 28, 2009 7:55 pm
Location: Montana
I'm having trouble understanding why the 3 is a constant. How does the denominator remain constant for any 3-d euclidean solid?

_________________
Andreas Nortmann wrote:
Things like this are illegal.
If not I will pass an appropriate law.


Top
 Profile  
 
 Post subject: Re: A New Conjecture of Volume?
PostPosted: Wed Apr 06, 2011 9:43 pm 
Offline
User avatar

Joined: Tue Jan 01, 2008 7:30 pm
Location: Texas, USA
Almost all of the teachers that I have shown this to have said the same thing. My equation turns a solid into a bunch of pyramids. The 1/3 comes from the formula for the volume of a pyramid.

Thanks for a prompt responce,
Tanner Frisby

_________________
"I discovered the triangle one day while shaving. I trimmed my beard like the intersection of three circles and noticed how I could unfog a square in the bathroom mirror by rubbing my beard circularly against the glass."-Franz Reuleaux


Top
 Profile  
 
 Post subject: Re: A New Conjecture of Volume?
PostPosted: Wed Apr 06, 2011 9:53 pm 
Offline
User avatar

Joined: Wed Jan 28, 2009 7:55 pm
Location: Montana
Well then, using the summation would neaten it:

V=

n
(Σ (FAnXd(x,y,z)))/3
1

(the n is subscript, as is (x,y,z))

_________________
Andreas Nortmann wrote:
Things like this are illegal.
If not I will pass an appropriate law.


Top
 Profile  
 
 Post subject: Re: A New Conjecture of Volume?
PostPosted: Wed Apr 06, 2011 10:06 pm 
Offline
User avatar

Joined: Mon Oct 18, 2010 10:48 am
I want to start out by saying I'm not criticizing, just trying to understand a bit better.

Quote:
A single predetermined Cartesian coordinate

What determines this coordinate?

Also, how do you prove 3 holds true for all solids?

How would the formula work for the Euclidean solids? Prove that your formula is reflexive for the Euclidean solids.

_________________
--Noah

I don't know half of you half as well as I should like and I like less than half of you half as well as you deserve.


Top
 Profile  
 
 Post subject: Re: A New Conjecture of Volume?
PostPosted: Wed Apr 06, 2011 10:16 pm 
Offline
User avatar

Joined: Fri Feb 18, 2011 5:49 pm
Location: New Jersey
A quick google search found this nifty javascript implementation of your conjecture (they describe how it works).

http://www.codeproject.com/KB/scripting ... eCalc.aspx

I don't know about whether a specific equation like yours has been published in any math papers, but I'm going to guess that this general method is known and/or used in certain applications already.

The way you defined negative and positive values for the the heights of the pyramids is interesting. I think it doesn't apply for concave polyhedra though (ex. a torus).


Top
 Profile  
 
 Post subject: Re: A New Conjecture of Volume?
PostPosted: Wed Apr 06, 2011 10:22 pm 
Offline
User avatar

Joined: Wed Mar 15, 2000 9:11 pm
Location: Delft, the Netherlands
In many cases the derivative of the volume of a solid w.r.t. the radius gives the formula for the surface area.
For example:

Cube volume: 8r^3
Cube surface: 24r^2

Sphere volume: 4/3 pi r^3
Sphere surface: 4 pi r^2

This only works if the shape has a centre point, and all the surfaces lie a distance r from that centre. The reason is that if you increase r by a small amount, s say, then it is like covering the shape with a layer of paint erverywhere of thickness s. The volume of that paint is approximately s times the surface area.

Since the volume of such shapes is proportional to r^3, the volume is always r/3 times the surface area.

You can extend this to other shapes where the surfaces are not all equidistant from a centre point. See for example here:
http://mathforum.org/library/drmath/view/51794.html
By differentiating w.r.t. each different radius you get the area of the planes specified by that radius. Adding all of them together pretty much leads to your formula: V = Sum( A_i r_i / 3 ) where A_i is a the area of a face and r_i the distance of that face to a centre point (which is arbitrary).

_________________
Jaap

Jaap's Puzzle Page:
http://www.jaapsch.net/puzzles/


Top
 Profile  
 
 Post subject: Re: A New Conjecture of Volume?
PostPosted: Wed Apr 06, 2011 10:25 pm 
Offline
User avatar

Joined: Sat Mar 24, 2007 6:58 pm
Location: Louisiana, US
Let me get this strait: You are trying to provide a formula to find the volume of a given solid object, correct?

I'm no math genius (though I do know some calculus), but it seems to me like you are calculating the volumes of tetrahedrons from origin (0,0,0) to each triangle in a solid mesh object. Assuming that all of the normals are facing the correct direction, the sum of all normals will equal the volume provided that the object is a solid unibody mesh.

It seems to make logical sense. Depending on the location of the origin, or the complexity of the object manifold, it seems logical that planes formed by the polygons which the origin falls to the outside rather than the inside, would achieve negative volumes and thus subtract from the total.

This concept can also be visually applied to 2 dimensions as well. While I don't exactly know how one should go about writing a rigorous proof, it seems that if it holds true for one object (non-self-intersecting manifold), then it should apply to all. The simplest polyhedron is a tetrahedron, with four polygons. If the origin is placed at the centroid, then the region can be divided up into four equal-volume tetrahedrons. They should each calculate to one fourth the volume of the full tetrahedron. That is the simplest example I can imagine.

This would be a very easy calculation to produce a computer script to find the volumes of large complex polyhedral objects. The equation would have to be calculated for every polygon, then summed. It also gives me insight as to why Shapeways is so picky about the submitted object manifolds: Even if an improperly formatted object could still be printed, the script would probably yield an incorrect value for volume. This is important for Shapeways since their pricing scheme is based on volume of material.

remember there have been in the past some notoriously simple conjectures that have deemed nearly impossible to prove. Take Collatz Conjecture, for instance:

N= positive integer

If N = odd, then N=N*3+1
If N = even, then N=N/2

Repeat indefinitely until N=1. The sequence always will eventually lead back to 1.

It has given me fascination due to the fact that the arithmatic is so decievingly simple, yet has yet be proven. Even some numbers reach much higher values before returning to 1. Example:

27, 82, 41, 124, 62, 31, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182, 91, 274, 137, 412, 206, 103, 310, 155, 466, 233, 700, 350, 175, 526, 263, 790, 395, 1186, 593, 1780, 890, 445, 1336, 668, 334, 167, 502, 251, 754, 377, 1132, 566, 283, 850, 425, 1276, 638, 319, 958, 479, 1438, 719, 2158, 1079, 3238, 1619, 4858, 2429, 7288, 3644, 1822, 911, 2734, 1367, 4102, 2051, 6154, 3077, 9232, 4616, 2308, 1154, 577, 1732, 866, 433, 1300, 650, 325, 976, 488, 244, 122, 61, 184, 92, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1

Source: Wikipedia
http://en.wikipedia.org/wiki/Collatz_conjecture

Pretty cool stuff. Keep playing with numbers. Good luck!

_________________
My Creepy 3D Rubik's Cube Video
cisco wrote:
Yeah, Uwe is Dalai Lama and Paganotis is mother Teresa of Calcutta.


Top
 Profile  
 
 Post subject: Re: A New Conjecture of Volume?
PostPosted: Wed Apr 06, 2011 10:28 pm 
Offline
User avatar

Joined: Tue Jan 01, 2008 7:30 pm
Location: Texas, USA
Thanks for such fast responses. I am texting from my phone right now, so I don't have copy paste or multi-tab browsing. So in an attempt to answer as many questions before going to bed tonight, I have created a list below.
1. Rentlix, while I don't understand summation, I do notice that your equation is strikingly similar to the Harmonic Parameters. HP does the exact thing my equation does, except is limited to points within (and maybe on the surface of) the solid. This should answer most of the questions listed above. That leaves the only novel aspect of my equation being the idea that the chosen point doesn't have to be inside the solid.
DKwan, I did see that, but can't read Java. Can you tell if they use The HP?

Good night/morning/afternoon/evening to all of you,
Tanner Frisby

_________________
"I discovered the triangle one day while shaving. I trimmed my beard like the intersection of three circles and noticed how I could unfog a square in the bathroom mirror by rubbing my beard circularly against the glass."-Franz Reuleaux


Top
 Profile  
 
 Post subject: Re: A New Conjecture of Volume?
PostPosted: Wed Apr 06, 2011 10:41 pm 
Offline
User avatar

Joined: Sat Mar 24, 2007 6:58 pm
Location: Louisiana, US
jaap wrote:
In many cases the derivative of the volume of a solid w.r.t. the radius gives the formula for the surface area.
For example:

Cube volume: 8r^3
Cube surface: 24r^2

Sphere volume: 4/3 pi r^3
Sphere surface: 4 pi r^2

This only works if the shape has a centre point, and all the surfaces lie a distance r from that centre. The reason is that if you increase r by a small amount, s say, then it is like covering the shape with a layer of paint erverywhere of thickness s. The volume of that paint is approximately s times the surface area.

Since the volume of such shapes is proportional to r^3, the volume is always r/3 times the surface area.

You can extend this to other shapes where the surfaces are not all equidistant from a centre point. See for example here:
http://mathforum.org/library/drmath/view/51794.html
By differentiating w.r.t. each different radius you get the area of the planes specified by that radius. Adding all of them together pretty much leads to your formula: V = Sum( A_i r_i / 3 ) where A_i is a the area of a face and r_i the distance of that face to a centre point (which is arbitrary).

Interesting point you bring up here. Through use of limits and definite integrals in Calculus, there are some forms which contain infinite surface area but finite volume. It creates an interesting paradox if you tried to paint such a surface.

For instance, take the equation 1/x^2. Now rotate it about the X-Axis. Now take the solid produced by this curve and use Integration to find the volume of this solid between +1 and +infinity. I forget the exact value, but let us suffice to say that the volume is finite, and a relatively small number at that. However, the surface area of this object is infinite. Now suppose you wanted to paint it. To cover an infinite surface area, would obviously require an infinite amount of paint. However, paint is measured by volume, not surface area, so you'll only need a relatively small amount of paint to completely fill it's volume! :P

_________________
My Creepy 3D Rubik's Cube Video
cisco wrote:
Yeah, Uwe is Dalai Lama and Paganotis is mother Teresa of Calcutta.


Top
 Profile  
 
 Post subject: Re: A New Conjecture of Volume?
PostPosted: Thu Apr 07, 2011 12:33 am 
Offline
User avatar

Joined: Wed May 13, 2009 4:58 pm
Location: Vancouver, Washington
stardust4ever wrote:
Through use of limits and definite integrals in Calculus, there are some forms which contain infinite surface area but finite volume. It creates an interesting paradox if you tried to paint such a surface.
My favorite example of this is the Mandelbrot fractal. It has finite area, but infinite circumference.

_________________
Real name: Landon Kryger


Top
 Profile  
 
 Post subject: Re: A New Conjecture of Volume?
PostPosted: Thu Apr 07, 2011 12:53 am 
Offline
User avatar

Joined: Wed Mar 15, 2000 9:11 pm
Location: Delft, the Netherlands
GuiltyBystander wrote:
stardust4ever wrote:
Through use of limits and definite integrals in Calculus, there are some forms which contain infinite surface area but finite volume. It creates an interesting paradox if you tried to paint such a surface.
My favorite example of this is the Mandelbrot fractal. It has finite area, but infinite circumference.

That's a fairly common feature of fractals in general, not just the Mandelbrot set. For example it is easy to prove for the Koch Snowflake.

_________________
Jaap

Jaap's Puzzle Page:
http://www.jaapsch.net/puzzles/


Top
 Profile  
 
 Post subject: Re: A New Conjecture of Volume?
PostPosted: Thu Apr 07, 2011 10:16 am 
Offline
User avatar

Joined: Tue Jan 01, 2008 7:30 pm
Location: Texas, USA
Rentlix wrote:
Well then, using the summation would neaten it:

V=

n
(Σ (FAnXd(x,y,z)))/3
1

(the n is subscript, as is (x,y,z))


This is similar to these first lines. However, I use V instead of h, no sumation =1 and I have a denomonator of 3 not a summation. Also, and the biggest point (lol, math pun)
Harmonic Paramater wrote:
distances from a fixed interior point to the faces
.

NType3 wrote:
I want to start out by saying I'm not criticizing, just trying to understand a bit better.

Quote:
A single predetermined Cartesian coordinate

What determines this coordinate?

Also, how do you prove 3 holds true for all solids?

How would the formula work for the Euclidean solids? Prove that your formula is reflexive for the Euclidean solids.


The coordinant is any (x,y,z) point you choose. You start by picking a single (only one) point, and fill in the x, y and z values of your point everywhere they appear in the equation. After you have done that, then you start to solve the equation.

I am not sure what a Euclidean solid is. I have looked on wiki, google and mathworld. Is it a set of polyhedra? Is it a definition of solids?

What I can tell you is that this formula works for all plactonic, Johnson, Archemedian solids, planar frustra, parallelepipeds, convex prisms and antiprisms, wedges, pyramids and dipyramids and zonohedron, etc.

Doesn't work for hyprebolic (non planar), Keplar-Poinsot or other stellations (concave), polyhedron compounds (concave), pentagramic antiprism (concave), etc.

Thanks,
Tanner Frisby

_________________
"I discovered the triangle one day while shaving. I trimmed my beard like the intersection of three circles and noticed how I could unfog a square in the bathroom mirror by rubbing my beard circularly against the glass."-Franz Reuleaux


Top
 Profile  
 
 Post subject: Re: A New Conjecture of Volume?
PostPosted: Wed Apr 13, 2011 12:08 am 
Offline
User avatar

Joined: Sat Mar 24, 2007 6:58 pm
Location: Louisiana, US
GuiltyBystander wrote:
stardust4ever wrote:
Through use of limits and definite integrals in Calculus, there are some forms which contain infinite surface area but finite volume. It creates an interesting paradox if you tried to paint such a surface.
My favorite example of this is the Mandelbrot fractal. It has finite area, but infinite circumference.
The Mandelbrot is my favorite fractal object. You can find some amazing patterns using deep zoom techniques. Not to derail this thread, but here is a link to my fractal gallery (mostly deep zooms of the classic Mandelbrot fomula) on Deviantart:
http://stardust4ever.deviantart.com/gallery/9952329

_________________
My Creepy 3D Rubik's Cube Video
cisco wrote:
Yeah, Uwe is Dalai Lama and Paganotis is mother Teresa of Calcutta.


Top
 Profile  
 
Display posts from previous:  Sort by  
Post new topic Reply to topic  [ 14 posts ] 

All times are UTC - 5 hours


Who is online

Users browsing this forum: No registered users and 3 guests


You cannot post new topics in this forum
You cannot reply to topics in this forum
You cannot edit your posts in this forum
You cannot delete your posts in this forum
You cannot post attachments in this forum

Search for:
Jump to:  

Forum powered by phpBB © 2000, 2002, 2005, 2007 phpBB Group