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#1 2014-12-15 20:12:27

Agnishom
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From: Riemann Sphere
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Sphere packing

We have a cube of side 2, inside which a sphere or radius 1 is inscribed. What would be the radius of the sphere that would have been inside the cube touching the central sphere and the three walls of the cube?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
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#2 2014-12-15 20:34:19

Bob
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Registered: 2010-06-20
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Re: Sphere packing

hi Agnishom,

Is this what you wanted?

evRxg7d.gif

This is the view looking down from above.

This will not show the true lengths of AD and CD as these lines are 'slanting' in this view.

A is the centre of the sphere/cube and D is a corner of the cube.

The length CD appears to be  √2.r but the true length will be longer in the ratio AD:AB.

Then you can form an equation for r using AD = AB + BC + CD

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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#3 2014-12-16 00:11:34

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
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Re: Sphere packing

bob bundy wrote:

The length CD appears to be  √2.r but the true length will be longer in the ratio AD:AB.

I think it should be sqrt(3)*r

Then you can form an equation for r using AD = AB + BC + CD

Bob

How do we get information on CD?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#4 2014-12-16 00:44:17

Bob
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Registered: 2010-06-20
Posts: 10,052

Re: Sphere packing

hi Agnishom,

Sorry, I haven't quite made myself clear.

When you do a 2-D representation of a 3_D object, the lengths will only look correct if you are looking square on to them.  So both AD and CD are seen foreshortened due to the angle of view.

They both look √2 times something, but are in fact √3 times something.

So you need to scale them up in that ratio.  CD appears to be √2.r so you can take it that the true length is √3.r

Now can you finish it ?

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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#5 2014-12-16 03:02:17

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
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Re: Sphere packing

In the same way, AD should be just sqrt(3). Is it?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#6 2014-12-16 04:05:39

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
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Re: Sphere packing

Oh, yes. That works.

Now I need to do the same thing for a 2014-dimensional hypersphere.


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#7 2014-12-16 04:07:24

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
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Re: Sphere packing

Thanks, the same trick worked again.


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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