The Apollonius Problem

Question 1.1

Let us consider three circles in the plane, as in the following picture:

$\includegraphics[scale=0.2]{Images/all8circles/all8circles-0.png}$

How many circles can we find that are simultaneously tangent to three fixed red circles?

Answer 1.2

The answer is 8, all shown in the following animation:

$\includegraphics[scale=0.2]{Images/all8circles/all8circles-0}$

Question 1.3

We can now ask what happens when we degenerate the three red circles, for instance, by letting their radius go to zero:

$\includegraphics[scale=0.2]{Images/all8circles/all8circles-0.png}$

In this situation, what happened to the 8 circles from before?

In order to understand this, we first need to revisit the definition of a circle:

Observation 1.4

Recall that a circle is determined by three parameters $(\text{center} (a,b),r)$, where $(a,b)$ represent the $xy$ coordinates of its center, and $r$ is the radius. But for the purpose of the computation, we need to enlarge the definition of a circle: we also accept lines as circles, seen as circles of infinite radius.

Définition 1.5

A (possibility degenerated circle) is a point in the complex projective space $\mathbb {P}^3$.

Construction 1.6

Every circle $C$, determines a subspace $Z_C\subseteq \mathbb {P}^3$: the subspace of all circles tangent to $C$.

$\includegraphics[scale=0.2]{Images/all8circles/all8circles-0.png}$

We can now formulate the Apollonius problem:

Problem 1.7 Apollonius

Fix three circles $C_1, C_2, C_3$. How many points lie in the intersection

\[ Z_{C_1}\cap Z_{C_2}\cap Z_{C_3} \, \, \, \, \, \, \, \, \text{ inside } \mathbb {P}^3 \]

Solution 1.8 to question 1.3

Out of the 8 circles, 7 dissipated. In fact, they became algebra and can only be traced back by contemplating the intersection within Grothendieck’s theory of schemes,

\[ Z_{C_1}\cap ^{Sch} Z_{C_2}\cap ^{Sch} Z_{C_3}=\mathsf{Spec}(\mathbb {C}[\epsilon _x, \epsilon _y, \epsilon _z]) \]

where the ring $\mathbb {C}[\epsilon _x, \epsilon _y, \epsilon _z]$ has three nilpotent generators $\epsilon _x$, $\epsilon _y$ and $\epsilon _z$, each encoding the collapse of one of the three red circles:

$\includegraphics[scale=0.2]{Images/all8circles/all8circles-0.png}$

Overall,the ring $\mathbb {C}[\epsilon _x, \epsilon _y, \epsilon _z]$ has 8 dimensions as a $\mathbb {C}$-vector space:

\[ \mathbb {C}[\epsilon _x, \epsilon _y, \epsilon _z]= \mathbb {C}\oplus \underbrace{\mathbb {C}.\epsilon _x \oplus \mathbb {C}.\epsilon _y \oplus \mathbb {C}.\epsilon _z \oplus \mathbb {C}.\epsilon _x\epsilon _y \oplus \mathbb {C}.\epsilon _x\epsilon _z \oplus \mathbb {C}.\epsilon _y\epsilon _z \oplus \mathbb {C}\epsilon _x.\epsilon _y.\epsilon _z}_{\text{algebraic infinitesimals}} \]

each counting for one of the tangent circles, 7 of which are purely algebraic.