# Three touching spheres in d dimensions

There are two $$d$$-dimensional unit spheres centred at $$A$$ and $$B$$ in $$\mathbb{R}^d$$. They are touching at location $$I$$. How can I find the centre of a third $$d$$-dimensional unit sphere $$C$$ that touches the first two spheres? I know that:

• |A-B|=|A-C|=|B-C|=2,
• |A-I|=|B-I|=1.

I don't know how to proceed from here. I know there can be multiple locations for $$C$$. In fact the set of possible locations forms a $$(d-1)$$-sphere. But how to find its radius and centre?

• Hint: Reduce it to two circles touching in $\mathbb{R}^2$. Apr 19, 2021 at 12:04
• In 2 dimensions there are two locations for C and I know how to find them. But I don't know what to do in higher dimensions. Apr 19, 2021 at 12:05
• Is $d$ the dimension of ambient space $\mathbb R^d$ ? Apr 19, 2021 at 12:08
• @JeanMarie yes it is. Apr 19, 2021 at 12:09
• Why the minus? :( Apr 19, 2021 at 12:25

## 1 Answer

From the fact that

$$\lvert A - B\rvert = \lvert A - C\rvert = \lvert B - C\rvert = 2$$

you know that $$A, B, C$$ are vertices of an equilateral triangle of side $$2$$. It doesn't matter how many dimensions you are working in, three non-collinear points still make the vertices of a triangle and if all three sides are equal then the triangle is equilateral.

Given that $$\triangle ABC$$ is an equilateral triangle of side $$2$$, what is the relationship of $$C$$ to the midpoint of side $$AB$$?

The point $$C$$ must be on the perpendicular bisector of $$AB$$ at a distance $$\sqrt3$$ from $$AB.$$

Noting that $$I$$ is the midpoint of $$AB,$$ we have $$CI = \sqrt3$$ and $$C$$ lies on the $$(d-1)$$-plane through $$I$$ orthogonal to the line $$AB,$$ since that $$(d-1)$$-plane contains all lines through $$I$$ perpendicular to $$AB$$ and all lines through $$I$$ perpendicular to $$AB$$ are in that plane.

In summary, the sphere you are looking for has center $$I$$ and radius $$\sqrt3$$ and lies in the $$(d-1)$$-plane through $$I$$ orthogonal to $$AB.$$

• I believe the midpoint of $AB$ is $I$ and $|C-I|=\sqrt 3$. Apr 19, 2021 at 13:44
• That's most of the information you need to construct the $(d-1)$-sphere. Apr 20, 2021 at 0:23
• I think I found the answer, but not sure if it is correct. I need to construct a $(d-1)$-plane between the spheres going through point $I$. The location of $C$ can be all points on that plane that have a distance of $\sqrt 3$ from $I$. In other words the intersection of the plane and a sphere. By the way, this is not a homework assignment as it may seem. In fact I am working on the kissing number problem. Apr 20, 2021 at 6:42
• OK, I see this is not a question that needs to be drawn out in hints, so I've finished describing the locus of $C.$ Anyway it just confirms what you already figured out. Apr 20, 2021 at 12:19
• Thank you David. I really appreciate your help. Apr 20, 2021 at 12:30