Four Spheres Inscribed inside a Larger Sphere

[Holydog23]

Suppose there exists a sphere of a fixed volume V, and there also exists four equal spheres arranged as a tetrahedron that fit just inside this larger sphere. Now, suppose the four spheres are models for the 2 protons and 2 neutrons of a helium atom.

How would I find the distance between the centers of the two positively charged protons in this structure? I need help getting started on finding this distance. My 3D spatial thinking is not too advanced and I am honestly have trouble getting started on this problem.

My professor has talked to us about we think of a tetrahedron typically as 3 spheres on one lower plane with one in the center in a plane above the other three. He explained this way of going about it is quite hopeless and would result in a million triangles and trigonometric relationships. Instead, he suggested we take a cube, which has 8 vertices; he said to take four which do not share any edge of the cube and take these vertices as the vertices of the tetrahedron. The 6 diagonals that connect this vertices can reveal a 3D form in the shape of a tetrahedron. Now instead of having three spheres on one plane and the other above, the spheres now look like they are criss-crossed on two different planes. I can see what he is talking about but I have seen no progress in thinking about the problem in this manner either

 

[Dr.Peterson]

Suppose there exists a sphere of a fixed volume V, and there also exists four equal spheres arranged as a tetrahedron that fit just inside this larger sphere. Now, suppose the four spheres are models for the 2 protons and 2 neutrons of a helium atom.

How would I find the distance between the centers of the two positively charged protons in this structure? I need help getting started on finding this distance. My 3D spatial thinking is not too advanced and I am honestly have trouble getting started on this problem.

My professor has talked to us about we think of a tetrahedron typically as 3 spheres on one lower plane with one in the center in a plane above the other three. He explained this way of going about it is quite hopeless and would result in a million triangles and trigonometric relationships. Instead, he suggested we take a cube, which has 8 vertices; he said to take four which do not share any edge of the cube and take these vertices as the vertices of the tetrahedron. The 6 diagonals that connect this vertices can reveal a 3D form in the shape of a tetrahedron. Now instead of having three spheres on one plane and the other above, the spheres now look like they are criss-crossed on two different planes. I can see what he is talking about but I have seen no progress in thinking about the problem in this manner either

The basic idea, as I see it, is to work backward -- rather than start with a given sphere they are inscribed in, start with the four little spheres arranged as stated, and find how big a sphere they could be inscribed in. Then you can solve for the distance between two centers in terms of the radius of the big one (which I assume is what you are given).

Start by finding the radius of each little sphere, in terms of a side of the cube. Then find how far the center of each sphere is from the center of the cube (and therefore from the center of the big sphere). Then you should see what to do from there.

This general idea is useful in many real problems.

 

[Holydog23]

The basic idea, as I see it, is to work backward -- rather than start with a given sphere they are inscribed in, start with the four little spheres arranged as stated, and find how big a sphere they could be inscribed in. Then you can solve for the distance between two centers in terms of the radius of the big one (which I assume is what you are given).

Start by finding the radius of each little sphere, in terms of a side of the cube. Then find how far the center of each sphere is from the center of the cube (and therefore from the center of the big sphere). Then you should see what to do from there.

This general idea is useful in many real problems.

I've found a geometric model to showcase the tetrahedron with cubic symmetry

I've been trying to understand where to start in finding the radius of each little sphere in terms of the side of a cube. You are correct in that we are given the radius of the big sphere which is 2.4 fm. This is part of a larger problem and the part before had us calculate the radius of the nucleus if we were to treat the nucleus as a sphere. In this part then we fit the four spheres inside that larger sphere we calculated in the previous part. Analyzing the figure, I noticed the geometry doesn't appear to have any sense of symmetry. For example, the upper left sphere appears to be centered at the upper right corner of the cube but the lower left sphere does not. I do not know how start finding the radii of the four spheres from this point now.

 

[Dr.Peterson]

I've found a geometric model to showcase the tetrahedron with cubic symmetry
View attachment 10270
I've been trying to understand where to start in finding the radius of each little sphere in terms of the side of a cube. You are correct in that we are given the radius of the big sphere which is 2.4 fm. This is part of a larger problem and the part before had us calculate the radius of the nucleus if we were to treat the nucleus as a sphere. In this part then we fit the four spheres inside that larger sphere we calculated in the previous part. Analyzing the figure, I noticed the geometry doesn't appear to have any sense of symmetry. For example, the upper left sphere appears to be centered at the upper right corner of the cube but the lower left sphere does not. I do not know how start finding the radii of the four spheres from this point now.

That picture seems all wrong; I don't see a vertex of the cube at the centers of the lower two spheres.

I couldn't find any proper pictures of this arrangement, so instead just look at a tetrahedron in a cube:

Imagine putting a sphere at each of the four vertices, making them just large enough that each touches the other three. Taking the side of the cube as x, how far is it from the center of one sphere to the center of any other? From that, you can find the radius of each sphere.

 

[Holydog23]

That picture seems all wrong; I don't see a vertex of the cube at the centers of the lower two spheres.

I couldn't find any proper pictures of this arrangement, so instead just look at a tetrahedron in a cube:

Imagine putting a sphere at each of the four vertices, making them just large enough that each touches the other three. Taking the side of the cube as x, how far is it from the center of one sphere to the center of any other? From that, you can find the radius of each sphere.

By "putting a sphere at each of the four vertices," do you mean we must place the center of each sphere at the vertices? If so, I come up with this

I can't really see anywhere to go from here. I feel my ability to draw them just large enough to touch the other three is limited because I can't draw a perfect cube. I used a compass for the circles but, as you can see, they don't form a neat tetrahedral shape that we would like. I don't posses the skill to draw a more precise cube and much less be able to draw the four spheres with enough preciseness such that they just barely touch the other three without making some of them look like they are larger or smaller.

 

[Dr.Peterson]

By "putting a sphere at each of the four vertices," do you mean we must place the center of each sphere at the vertices? If so, I come up with this

I can't really see anywhere to go from here. I feel my ability to draw them just large enough to touch the other three is limited because I can't draw a perfect cube. I used a compass for the circles but, as you can see, they don't form a neat tetrahedral shape that we would like. I don't posses the skill to draw a more precise cube and much less be able to draw the four spheres with enough preciseness such that they just barely touch the other three without making some of them look like they are larger or smaller.

You don't really need a drawing that shows the spheres, just a little imagination. Just ask yourself where any two spheres will touch.

If two spheres whose centers are at opposite ends of a face diagonal touch, they must touch at the midpoint of that diagonal -- that is, the radius of each is half of that diagonal.