Spinors and Descartes’ Theorem

Descartes’ theorem is a classical result in 2-dimensional Euclidean geometry, relating the curvatures of four mutually tangent circles (Figure 1) which form a 3-flower in the following sense.

Throughout, we denote the curvature of a circle $C_{\bullet}$ by $\kappa_{\bullet}$. Descartes’ theorem gives an equation satisfied by the curvatures in a 3-flower:

In this paper we present an explicit equation satisfied by the curvatures of an $n$-flower.

In other words, from the curvatures $\kappa_{\bullet}$ we define auxiliary variables $m_{\bullet}$ via (1.3), and the $m_{\bullet}$ satisfy polynomial equations. Since all $\kappa_{\bullet}>0$, each $m_{\bullet}$ is the square root of a manifestly positive number and we take $m_{\bullet}$ to be the positive square root.

In (1.4) and (1.5), $i$ is the usual square root of $-1$. Although complex numbers are crucial to the proof, in writing these equations they are merely a convenience. After expanding and cancelling terms, the resulting polynomials have integer coefficients. Indeed, writing $[n]$ for $\{1,2,\ldots,n\}$, then for any subset $K\subseteq[n-1]$, the term in the product $\prod_{j=1}^{n-1}(m_{j}\pm i)$ involving precisely the $m_{k}$ with $k\in K$ is given by $(\pm i)^{n-1-|K|}\prod_{k\in K}m_{k}$. Such terms come in pairs, one from $\prod_{j=1}^{n-1}(m_{j}+i)$ and one from $\prod_{j=1}^{n-1}(m_{j}-i)$. When $n-1-|K|$ is even, the terms in each pair are real, equal and cancel; when $n-1-|K|$ is odd, say equal to $2l+1$, then the terms in each pair are imaginary and conjugate, so since $\frac{i}{2}\left((-i)^{n-1-|K|}-i^{n-1-|K|}\right)=(-1)^{l}$, (1.4) and (1.5) can be written as

and

respectively.

In any case, making the substitutions of (1.3) in (1.4) or (1.5) (or (1.6) or (1.7)) yields an equation satisfied by the $\kappa_{\bullet}$, providing a generalisation of Descartes’ equation (1.2). This equation involves square roots, but upon multiplying by various conjugates (replacing various $m_{\bullet}$ with $-m_{\bullet}$) one may obtain a polynomial relation among the $m_{\bullet}^{2}$; after substituting and clearing denominators one obtains a polynomial relation among the $\kappa_{\bullet}$.

In this way, from equation (1.4) with $n=3$ one can recover Descartes’ theorem. Similarly, from (1.5) with $n=4$ we obtain the following equation relating the curvatures in a 4-flower:

For larger $n$ the polynomials grow rapidly in size and degree.

Flowers are a building block of circle packing theory [25]. It is known from the general theory of circle packing that once the curvatures of the petals of an $n$-flower are known, the curvature of the central circle is determined. Theorem Theorem gives an explicit equation for that central curvature.

Second, we use the spinor-horosphere correspondence recently proved by the first author in [15]. Building on work of Penrose–Rindler [20] and Penner [19], this correspondence provides an explicit, smooth, bijective, $SL(2,\mathbb{C})$-equivariant correspondence between nonzero 2-component spinors, and horospheres in hyperbolic 3-space $\mathbb{H}^{3}$, with a certain spinorial decoration. For us, spinors can be regarded simply as pairs of complex numbers $(\xi,\eta)$. When $\xi,\eta$ are real, the correspondence essentially reduces to 2 dimensions, yielding a correspondence between nonzero $(\xi,\eta)\in\mathbb{R}^{2}$, and horocycles in the hyperbolic plane $\mathbb{H}^{2}$, which is particularly simple in the upper half plane model. Under the correspondence, there is a simple way to calculate the distance between horospheres, and hence to know when they are tangent, using the bilinear form on spinors given by the $2\times 2$ determinant. Moreover, the curvatures $\mathring{\kappa}_{j}$ and $\kappa_{j}$ are both straightforward quadratic functions of $\xi,\eta$: they are each given, up to a constant, by the Euclidean square length $\xi^{2}+\eta^{2}$ of $(\xi,\eta)$.

Thus, the problem reduces to finding a relation among the Euclidean lengths of real 2-dimensional vectors $(\xi,\eta)$, given that they satisfy certain bilinear conditions equivalent to the tangency relations of a flower. The bilinear conditions are essentially that the vectors successively span parallelograms of area $1$. The desired relation is found by introducing complex variables $z_{j}=\xi_{j}+i\eta_{j}$.

The expressions for the $m_{j}$ in (1.3) arise naturally in calculations with the spinors $(\xi_{j},\eta_{j})$, as do the various products arising in the equations (1.4) and (1.5). We thus include some further calculations in Section 5 which help to motivate and explain these expressions, and explain how (1.4) and (1.5) were found.

Recent historical accounts of the the correspondence between Descartes and Elisabeth of the Palatinate, in which equation (1.2) first appeared, include [4], [18, pp. 31–33], and [21, pp. 37–38, 73–81]. Equation (1.2) was stated by Yamaji Nushizumi in 1751 [17], and rediscovered several times, including by Steiner (1826) [24], Beecroft (1842) [3], and Soddy (1936) [22]; the latter restating (1.2) in poetry.

Numerous generalisations of Descartes’ theorem are known; we mention a few. For instance, in 1936 Gosset generalised the result to $n+2$ mutually tangent spheres in $n$ dimensions, appending a verse to Soddy’s poem [1]. In 1962 Mauldon generalised these results to spherical and hyperbolic space [16]. In 2002 Lagarias–Mallows–Wilks [14] further extended these results, relating not only the curvatures but also the centres of the spheres involved. In 2007, Kocik [9] extended the result for $n+2$ Euclidean spheres in $n$ dimensions to the case where circles need not be tangent.

Apollonian circle packings, consisting of nested 3-flowers, are a research field in their own right; see e.g. Graham–Lagarias–Mallows–Wilks–Yan [6] or Aharonov–Stephenson [2] for general background. These packings have important number-theoretic properties and have seen recent breakthroughs, e.g. [7]. A special case closely related to our work, and mentioned by the first author in [15], is that of Ford circles [5]: these arise from integer spinors.

So far as we know, however, none of these works provide a generalisation of Descartes’ theorem to $n$-flowers for $n>3$. There are certainly results involving flower-like configurations, such as Soddy’s hexlet [23], but they do not provide an equation relating curvatures.

Flowers being simple examples of circle packings, the general theory of packing applies to them: see generally Stephenson [25]. In general, from a simplicial 2-complex $K$ triangulating an oriented surface, circle packing theory studies the existence and uniqueness of circle packings realising $K$, in the sense that vertices of $K$ correspond to circles, edges of $K$ correspond to tangencies, and triangles of $K$ correspond to oriented tangent circle triples. Flowers arise in the simple case when $K$ is a disc built from $n$ triangles around a central vertex. Perhaps most relevantly for our purposes, the “boundary value theorem” of [25, thm. 11.6] provides that when $K$ is topologically a disc, the curvatures of circles at boundary vertices, and branching structure, may be specified arbitrarily, and then a unique packing (in Eudlicean or hyperbolic geometry) exists. Our result gives the Euclidean curvature of the interior circle in the flower case.

To the best of our knowledge, perhaps the existing works most closely related to our approach are those of Kocik, who in several papers uses spinors to describe Descartes circle configurations and Apollonian packings [13, 12, 11, 10, 9, 8]. However in those works, spinors are complex numbers (defined up to sign) describing tangencies between pairs of circles. This is significantly different from our approach. We also mention that the linear algebra and bilinear forms of Lagarias–Mallows–Wilks [14] and Aharonov–Stephenson [2] are of a somewhat similar flavour.

Consider an $n$-flower ($n\geq 3$) in the Euclidean plane, as in Definition 1.1. Let each circle $C_{\bullet}$ have radius $r_{\bullet}$ and curvature $\kappa_{\bullet}=1/r_{\bullet}$.

Dilating the entire configuration by a factor of $\kappa_{\infty}$ sends the curvature $\kappa_{\infty}\mapsto 1$ and, for each petal circle, $\kappa_{j}\mapsto\kappa_{j}/\kappa_{\infty}$. We may thus assume the central circle has unit radius, so that $\kappa_{\infty}=1$, and for each petal circle $C_{j}$ write $\kappa_{j}$ for the resulting curvature.

Invert the configuration in the unit circle $C_{\infty}$. Each $C_{j}$ is mapped to a circle $\mathring{C}_{j}$ internally tangent to $C_{\infty}$. For each $j$ mod $n$ then $\mathring{C}_{j}$ is externally tangent to $\mathring{C}_{j-1}$ and $\mathring{C}_{j+1}$. See Figure 2. We denote by $\mathring{r}_{j}$ and $\mathring{\kappa}_{j}$ the (Euclidean) radius and curvature of $\mathring{C}_{j}$ respectively. (The superscript ring notation is intended to indicate “inside the unit disc”.)

We now consider the unit circle $C_{\infty}$ as the boundary of the disc model $\mathbb{D}^{2}$ of the hyperbolic plane. Each $\mathring{C}_{j}$ then appears as a horocycle.

Next, we convert from the disc model $\mathbb{D}^{2}$ to the upper half plane model $\mathbb{U}^{2}$ of the hyperbolic plane. Regarding $\mathbb{D}^{2}$ and $\mathbb{U}^{2}$ as subsets of the complex plane in the standard way, the two models are related by the Cayley transform $\mathfrak{S}\colon\mathbb{U}^{2}\longrightarrow\mathbb{D}^{2}$ and its inverse:

Observe that $\mathfrak{S}^{-1}$ sends $\partial\mathbb{D}^{2}$ to $\partial\mathbb{U}^{2}=\mathbb{R}\cup\{\infty\}$, and sends horocycles in $\mathbb{D}^{2}$ to horocycles in $\mathbb{U}^{2}$. Moreover, the point $1$ on $\partial\mathbb{D}^{2}$ corresponds to the point $\infty$ on $\partial\mathbb{U}^{2}=\mathbb{R}\cup\{\infty\}$. And as the unit circle $\partial\mathbb{D}^{2}$ is traversed anticlockwise, its image $\mathbb{R}\cup\{\infty\}$ under $\mathfrak{S}^{-1}$ is traversed in the increasing direction.

Before applying $\mathfrak{S}^{-1}$, if necessary we reflect the configuration in an arbitrary hyperbolic line through the origin, so that the centres of $\mathring{C}_{0},\mathring{C}_{1},\ldots,\mathring{C}_{n-1}$ are in anticlockwise order around $\partial\mathbb{D}^{2}$. This is a reflection in both hyperbolic and Euclidean geometry, so preserves all $\mathring{r}_{j}$ and $\mathring{\kappa}_{j}$. Then, if necessary, we rotate $\mathbb{D}^{2}$ about $0$ so that, proceeding anticlockwise from $1$ around $\partial\mathbb{D}^{2}$, we encounter the centre of $\mathring{C}_{0}$ first, then $\mathring{C}_{1},\ldots,\mathring{C}_{n-1}$, in order. Again, this is a rotation in both hyperbolic and Euclidean geometry, so preserves all $\mathring{r}_{j}$ and $\mathring{\kappa}_{j}$.

Now apply $\mathfrak{S}^{-1}$ to the entire configuration. Since no $\mathring{C}_{j}$ is tangent to $\partial\mathbb{D}^{2}$ at $1$, then no $\overline{C}_{j}$ is tangent to $\partial\mathbb{U}^{2}=\mathbb{R}\cup\{\infty\}$ at $\infty$. Thus each horocycle $\overline{C}_{j}$ has centre in $\mathbb{R}\subset\partial\mathbb{U}^{2}$, and appears as a circle in $\mathbb{U}^{2}$, with a finite radius $\overline{r}_{j}$ and curvature $\overline{\kappa}_{j}$. Moreover, since the centres of $\mathring{C}_{j}$ are in order around $\partial\mathbb{D}^{2}$ proceeding anticlockwise from $1$, the centres of the $\overline{C}_{j}$ are in increasing order along $\mathbb{R}\subset\partial\mathbb{U}^{2}$.

We thus obtain a “flat $n$-flower” where the original central circle has become $\mathbb{R}\cup\{\infty\}$, and each petal circle has become a horocycle in $\mathbb{U}^{2}$, with each $\overline{C}_{j}$ externally tangent to $\overline{C}_{j-1}$ and $\overline{C}_{j+1}$ (with indices taken mod $n$), and the horocycles increasing in index from left to right. See Figure 4. (The overline notation is intended to indicate “in a flat flower along the real line”.)

Finally, observe that the circles $\overline{C}_{j}$ can be translated horizontally without affecting the tangencies of circles or their or curvatures. We apply such a translation so that $\overline{C}_{0}$ is tangent to the real line at $0$. Since the horocycles $\overline{C}_{j}$ are tangent to the real line in increasing order, for $j\geq 1$ each $\overline{C}_{j}$ is tangent to the real line at a positive number.

We now introduce spinors for the horocycles $\overline{C}_{j}$.

As discussed in Section 2.1, each horocycle in the hyperbolic plane has two planar spin decorations, corresponding to two real spinors which are negatives of each other.

Thus each horocycle $\overline{C}_{j}$ corresponds to a real spinor $\alpha_{j}=(\xi_{j},\eta_{j})$, well defined up to sign. The centre of $\overline{C}_{j}$ is at $\xi_{j}/\eta_{j}$, which lies in $\mathbb{R}$, so each $\eta_{j}$ is nonzero. The Euclidean radius $\overline{r}_{j}$ and curvature $\overline{\kappa}_{j}$ of $\overline{C}_{j}$ are given by

(In general the Euclidean diameter of the horosphere corresponding to $(\xi,\eta)$ is $\frac{1}{|\eta|^{2}}$. Here $\eta$ is real.)

We now choose each $\alpha_{j}$ so that $\eta_{j}$ is positive.

Recall $\overline{C}_{0}$ is constructed to be tangent to the real line at $0$, and the centres of the $\overline{C}_{j}$ are in increasing order along the real line. Thus we have

Thus $\xi_{0}=0$ and, since all $\eta_{j}$ are chosen positive, for $j\geq 1$ we have $\xi_{j}>0$.

Calculating bilinear forms, for all $0\leq j<k\leq n-1$ we have

(This is the opposite of the totally positive notion of [15, sec. 6]; the $\alpha_{j}$ here are “totally negative”.)

Now for each $j$ mod $n$ we know that $\overline{C}_{j}$ and $\overline{C}_{j+1}$ are tangent. Thus the complex lambda length between planar spin decorations of $\overline{C}_{j}$ and $\overline{C}_{j+1}$ is $\pm 1$. Combining these observations we have the following.

The following proposition gives a useful relationship between each spinor $(\xi_{j},\eta_{j})$, and the Euclidean curvature $\mathring{\kappa}_{j}$ of the corresponding horocycle in the disc model.

Combining Proposition 4.3 with Lemma 3.1, we observe that $\kappa=\mathring{\kappa}-2=\xi^{2}+\eta^{2}-1$, so the curvatures of the original circles are also usefully expressed in terms of spinor coordinates.

In order to prove Proposition 4.3 we use some lemmas.

Let us briefly explain what is meant by rotating $H$ in the above proposition. As $H$ has a planar spin decoration, it corresponds to a spin-decorated horosphere $\widetilde{H}$ in $\mathbb{U}^{3}$. This $\widetilde{H}$ has a spin decoration $W$ consisting of associated inward and outward spin decorations, which are lifts of frame fields $f$ along $\widetilde{H}$. An orientation-preserving isometry $\phi\in\operatorname{Isom}^{+}(\mathbb{U}^{3})\cong PSL(2,\mathbb{C})$ may be applied to the horosphere $\widetilde{H}$ and (via its derivative) the frame fields $f$, yielding a decoration on a horosphere $\widetilde{H^{\prime}}$. If we specify a lift of $\phi$ to the spin double cover $SL(2,\mathbb{C})$, then there is a well-defined map of the spin decorations $W$ from $\widetilde{H}$ to $\widetilde{H^{\prime}}$. A lift of $\phi$ to $SL(2,\mathbb{C})$ can be specified by a path from the identity to $\phi$ in $PSL(2,\mathbb{C})$. Here, we have a rotation of $\pi$ about $i\in\mathbb{U}^{2}$, which naturally extends to a rotation in the 3-dimensional model $\mathbb{U}^{3}$ about the geodesic normal to $\mathbb{U}^{2}$ through $i$. This rotation is naturally lifted to the spin double cover of $\operatorname{Isom}^{+}(\mathbb{U}^{3})$ by taking the path of isometries $\phi_{t}$ over $t\in[0,1]$, where $\phi_{t}$ is a rotation of angle $\pi t$ about the same axis as $\phi$. The isometries $\phi_{t}$ then take the spin decoration on $\widetilde{H}$ to a spin decoration on $\widetilde{H^{\prime}}$, corresponding to the planar spin-decorated $H^{\prime}$ of the lemma.

Translating Lemma 4.5 into the disc model $\mathbb{D}^{2}$ via the Cayley transform (3.2), we immediately obtain the following.

We now present some calculations that are not strictly necessary for the main result, but may be of interest, and may motivate some expressions arising in the main result. The reader interested only in the proof of the main result may skip this section.