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Author Topic: What shape is formed at the intersection of 3 cylinders?  (Read 8462 times)

Offline chris

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I was asked a question on the radio today that got me thinking; I wasn't totally sure so had to check a maths textbook during the news break!

The question - from Julian, thanks - was:

"If you took a solid metal cube and drilled circular holes into the faces so that the holes united at the centre, what shape would be formed at the point in the cube's centre where all the drill holes met?"

I suggested that this could be thought of, analogously, as the shape at the intersection of 3 cylinders in the x, y and z axes. Initially I thought the product would be a sphere, but a book I found says the result is a curved rhombic dodecahedron, volume given by 8(2-√2)r³, which was the final answer I gave.

Have I got this right?

Chris


 

Offline syhprum

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What shape is formed at the intersection of 3 cylinders?
« Reply #1 on: 26/11/2009 11:11:17 »
I find this very hard to visualise the best I could come up with was to use a drill that drills square holes which would result in a cube, now substitute one for a round drill that would produce a cylinder OK so good but where do we go from here ?.
I think we are going to end up with a sphere with twelve pyramid like bumps on it, to get a perfect sphere we would have to drill an infinite number of holes !.
« Last Edit: 26/11/2009 14:10:43 by syhprum »
 

Offline LeeE

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What shape is formed at the intersection of 3 cylinders?
« Reply #2 on: 26/11/2009 12:10:54 »
It looks like this...

 

Offline syhprum

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What shape is formed at the intersection of 3 cylinders?
« Reply #3 on: 26/11/2009 14:06:13 »
I resisted the temptation to hunt google for a picture and tried to work it myself, if you drilled the block and filled the holes with some setting fluid and then dissolved away the block you would end up with a six spiked object.
You would have quite a difficult job cutting away the unwanted materiel to leave the curved rhombic dodecahedron object behind.

PS I guess a hollow cylindrical cutter would do the job.
« Last Edit: 26/11/2009 14:09:29 by syhprum »
 

ScientificBoyZClub

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What shape is formed at the intersection of 3 cylinders?
« Reply #4 on: 26/11/2009 15:30:41 »
wow I am really impressed. Chris loves geometry too WOW.

http://pentachoron.com/polyhedron/Nets/cylinders.pdf

The intersection of three cylinders is constructed from 12 curved rhombic shapes. The
construction has the structure of the rhombic dodecahedron, which you may want to
familiarize yourself with. Note that the rhombic dodecahedron is not to be confused with
the regular dodecahedron, they simply share the same last name because they have the
same number of faces.
http://www.thenakedscientists.com/forum/index.php?action=dlattach;topic=27020.0;attach=10655;image
« Last Edit: 26/11/2009 15:34:40 by ScientificBoyZClub »
 

Offline Geezer

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What shape is formed at the intersection of 3 cylinders?
« Reply #5 on: 26/11/2009 17:54:02 »
Here's another.

 

Offline Geezer

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What shape is formed at the intersection of 3 cylinders?
« Reply #6 on: 26/11/2009 18:20:34 »
and another.



This isometric view is interesting. It looks like a heaxagon. Perfectly straight edges too. No doubt there is a good mathematical reason for this.
« Last Edit: 26/11/2009 22:32:42 by Geezer »
 

ScientificBoyZClub

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What shape is formed at the intersection of 3 cylinders?
« Reply #7 on: 27/11/2009 09:26:41 »
Geezer you have used 3d max !

haven't you ? I know how to use it.
 

Offline Geezer

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What shape is formed at the intersection of 3 cylinders?
« Reply #8 on: 27/11/2009 17:53:51 »
Geezer you have used 3d max !

haven't you ? I know how to use it.

I used TurboCAD
 

Offline LeeE

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What shape is formed at the intersection of 3 cylinders?
« Reply #9 on: 28/11/2009 23:19:02 »
Heh - I used analytic solids in Realsoft3D.
 

Offline Geezer

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What shape is formed at the intersection of 3 cylinders?
« Reply #10 on: 28/11/2009 23:33:18 »
If you need to run motion simulations, here is a link to a free program

http://www.askoh.com/freecad/index.html

I've used it quite a bit and found it most helpful. The animations are not exactly photorealistic, but the simulations seem to be spot on.
 

Offline Mr. Scientist

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What shape is formed at the intersection of 3 cylinders?
« Reply #11 on: 29/11/2009 01:40:56 »
I was asked a question on the radio today that got me thinking; I wasn't totally sure so had to check a maths textbook during the news break!

The question - from Julian, thanks - was:

"If you took a solid metal cube and drilled circular holes into the faces so that the holes united at the centre, what shape would be formed at the point in the cube's centre where all the drill holes met?"

I suggested that this could be thought of, analogously, as the shape at the intersection of 3 cylinders in the x, y and z axes. Initially I thought the product would be a sphere, but a book I found says the result is a curved rhombic dodecahedron, volume given by 8(2-√2)r³, which was the final answer I gave.

Have I got this right?

Chris

I'm sorry i can't help. I hadn't even heard of one before this thread.So i looked it up; cannot tell you much more though.

http://pentachoron.com/polyhedron/Nets/cylinders.pdf
 

ScientificBoyZClub

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What shape is formed at the intersection of 3 cylinders?
« Reply #12 on: 29/11/2009 02:42:36 »
I was asked a question on the radio today that got me thinking; I wasn't totally sure so had to check a maths textbook during the news break!

The question - from Julian, thanks - was:

"If you took a solid metal cube and drilled circular holes into the faces so that the holes united at the centre, what shape would be formed at the point in the cube's centre where all the drill holes met?"

I suggested that this could be thought of, analogously, as the shape at the intersection of 3 cylinders in the x, y and z axes. Initially I thought the product would be a sphere, but a book I found says the result is a curved rhombic dodecahedron, volume given by 8(2-√2)r³, which was the final answer I gave.

Have I got this right?

Chris

I'm sorry i can't help. I hadn't even heard of one before this thread.So i looked it up; cannot tell you much more though.

http://pentachoron.com/polyhedron/Nets/cylinders.pdf
I gave the same link.
 

Offline LeeE

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What shape is formed at the intersection of 3 cylinders?
« Reply #13 on: 29/11/2009 11:58:11 »
If you need to run motion simulations, here is a link to a free program

http://www.askoh.com/freecad/index.html

I've used it quite a bit and found it most helpful. The animations are not exactly photorealistic, but the simulations seem to be spot on.

Heh - RS3D has its own built-in animation system, incorporating skeletons, IK, FK, footsteps, forces, friction and collision detection etc.  There are also a couple of plugins available to handle cloth and more advanced physics simulations.
 

Offline Geezer

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What shape is formed at the intersection of 3 cylinders?
« Reply #14 on: 29/11/2009 18:02:15 »
Heh - RS3D has its own built-in animation system, incorporating skeletons, IK, FK, footsteps, forces, friction and collision detection etc.  There are also a couple of plugins available to handle cloth and more advanced physics simulations.

OOOooo! It's probably a bit too spendy for me! I'll take a peek though.

Also, it would take me ages to learn a new tool. I've used turbocad for fifteen years, but only intermittently, so I keep having to re-learn how to use it, particularly the 3-D stuff. I was initially forced into using CAD because the patent office rejected my "kwality draughting"! But, having got the hang of it, I never draw anything by hand.
 

Offline LeeE

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What shape is formed at the intersection of 3 cylinders?
« Reply #15 on: 29/11/2009 23:31:01 »
Heh - RS3D has its own built-in animation system, incorporating skeletons, IK, FK, footsteps, forces, friction and collision detection etc.  There are also a couple of plugins available to handle cloth and more advanced physics simulations.

OOOooo! It's probably a bit too spendy for me! I'll take a peek though.hing by hand.

The linux version is quite a bit cheaper than the windows version and exceptionally good value.  The downsides are that you might have to do a bit of fiddling about to get it running, and while most of it works, some axis seem to have been transposed i.e. I was recently working with a complex skeleton and the banking, heading and pitch axis were a bit bum-about-face.  Still worked ok, so just a silly cross-platform coding error, most likely.

I won't spam a link, as you'll be able to google it.  I believe that the demo download (windows/mac) is either limited to 640x480 renders, or projects can't be saved, but are otherwise fully functional.  Yes, the learning curve is pretty steep at first, but once you've got a handle on the way it works it's pretty consistent (Lol - mind you, a lot of people still like Blender ???).
 

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What shape is formed at the intersection of 3 cylinders?
« Reply #15 on: 29/11/2009 23:31:01 »

 

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