Help Needed: Show that r/R=.225

[Saladsamurai]

***First part solved skip to post #11

^^^^^^^^^^^^^^^^^^^^^^^^^^^It's a silly title, but I really need help. The four pink balls surround the purple one and all make contact.

Calling the radius of the purple ball r and the radii of the pink R:

Show that r/R=.225.

The Attempt at a Solution

I think that this can be done with simple plane trig, but I am having troubles with the visualization.Anybody have any ideas? I am dying here.

edit: I hijacked this image to simplify the problem. The original problem statement had no image and came from a Materials Science text: "Show that the minimum cation-to-anion ratio for a coordination number 4 is .225"

 

[LowlyPion]

You already get credit for explaining the problem so that even your grandmother can understand it.

As to the geometry I think you have to figure in several dimensions. But as I visualize it you can go a plane at a time.

For instance you know that the radii constrain the tetrahedral sides to be 2R.

From that I would examine the plane that intersects 2 spheres and the centroid of the tetrahedron, which is where the purple sphere is centered.

The centroid distance to a vertex - R is the radius of purple guy isn't it?

 

[Saladsamurai]

Okay LP. So I think I get you. r+R=height correct ?

And length = 2R

I am just not sure how to relate this to an angle?

 

[Saladsamurai]

I tried (r+R)/2R =sin 45

but this does not work. The angles should be 45 right?

 

[LowlyPion]

The line from a vertex to the centroid of the opposite face passes through the centroid of the tetrahedron.

It has the delicious property that it is in a ratio of 3 from the vertex to 1 from the opposite face. This 3:1 ratio is quite useful.

Because we can apply that to the height of the tetrahedron which we know (from looking it up) is (2/3)^½*(a) = (2/3)^½*(2R)

I think that means that r = (3/4)*(2/3)^½*(2R) - R

 

[Saladsamurai]

LP. Thank you for your help. This gets the correct answer, but I am still having some trouble understanding it because we only have words here. Did you get this info from a book or a particular website I could check out? (I will do some googling).

So the ratio 3:1 is:

(the distance from a Vertex to the centroid of opposite face)/
(distance from vertex to centroid of pyramid)

correct?

and you are saying that Height=(2/3)^1/2*2R

 

[LowlyPion]

http://en.wikipedia.org/wiki/Centroid#Centroid_of_triangle_and_tetrahedron

 

[Saladsamurai]

I guess I am just confused as to where the "3/4" is coming from ?

If h= r + R and h = (2/3)^1/2*2R

then r + R = (2/3)^1/2*2R

what detail am I missing?

 

[LowlyPion]

That comes from the 3:1 ratio of the distance from the vertex to the centroid of the opposite face.

1+1+1 : 1 is 3/4 of the way doesn't it?

That locates the position of the tetrahedral centroid.

 

[Saladsamurai]

Oh crapass. Thanks LP! I am with you now!

Casey

 

[Saladsamurai]

Now, if you were now asked "What are the angle between the the covalent bonds" in this structure (i.e., the lines connecting the center of the purple ball to the centers of the pink ones)?

Wouldn't those just be 90 degrees?

 

[LowlyPion]

Now, if you were now asked "What are the angle between the the covalent bonds" in this structure (i.e., the lines connecting the center of the purple ball to the centers of the pink ones)?

Wouldn't those just be 90 degrees?

No. I think you should draw it out.

 

[Saladsamurai]

I think you are right. I keep thinking that the centroid of the purple coincides with the centroids of the pinks, but it does not.

 

[Saladsamurai]

Also, now I have values for r and R... so r/R = .225 no longer applies.

I have that r= .04 and R=.14

Is this going to be a Law of Cosines problem by chance?

 

[Saladsamurai]

Where c²=a²+b⁶-2ab*cos(theta)

a = b = r+R

and c = 2R

so theta = 102 degrees.

Anyone confirm?

 

[gabbagabbahey]

Also, now I have values for r and R... so r/R = .225 no longer applies.

I have that r= .04 and R=.14

Is this going to be a Law of Cosines problem by chance?

That doesn't look right! You should not be able to solve for R and r, only for the ratio r/R. Using a somewhat different method than you, I get $$\frac{r}{R}=\frac{\sqrt{6}-2}{2}\approx0.224745$$

What I did was assign a coordinate system such that the center of the purple sphere is along the positive z-axis, 3 of the pink spheres lie in the x-y plane (with one having its center along the positive y-axis) and the 4th pink sphere's center lies along the positive z-axis. I then analyzed the 3 coplanar spheres, finding their equations, then found the equation of the 4th pink sphere, and finally the purple sphere.

Here is a mathematica plot of my solution, with R=3 in this plot:

http://img264.imageshack.us/img264/3426/5balls2yb8.gif

 

[LowlyPion]

That doesn't look right! You should not be able to solve for R and r, only for the ratio r/R. Using a somewhat different method than you,
I get $$\frac{r}{R}=\frac{\sqrt{6}-2}{2}\approx0.224745$$

That is apparently the same as the earlier answer.

r = (3/4)*(2/3)^½*(2R) - R ;; algebraically simplifies into your result.

 

[LowlyPion]

As for the angle, here is a pleasant little derivation I found:
http://mathcentral.uregina.ca/QQ/database/QQ.09.00/nishi1.html