The Quaternion Mandelbrot Set

Introduction

The connectedness locus of a quadratic family is the set of parameters for which the critical orbit stays bounded, and for the quaternion family $f_c(\tilde q)=\tilde q^2+c$ it is the subset

$$ \mathcal{M}_{\mathbb{H}} = \{c \in \mathbb{H} : \text{the critical orbit } 0, c, c^2+c, \ldots \text{ is bounded}\}. $$

The set is a compact body of revolution: it is obtained by revolving the ordinary complex Mandelbrot set about the real axis of $\mathbb{H}$. The reason is the critical slice theorem: the critical orbit of a parameter $c$ never leaves the two-dimensional subalgebra $\mathbb{R}[c]$ generated by $e_t$ and $c$, and that subalgebra is a copy of $\mathbb{C}$ under the identification $c \mapsto c_0+i|\vec c|$. The four-dimensional locus is therefore determined by the two-dimensional one, and every complex plane through the origin cuts it in a congruent copy of the complex Mandelbrot set.

The quadratic map, its critical orbit, the escape lemma, the filled Julia set and the complex planes are from The Quaternion Quadratic Map and Its Julia Sets; the complex Mandelbrot set and its escape-time algorithm are from The Mandelbrot Set and the Quadratic Family; the quaternion norm and its multiplicativity are from Quaternion Norm and Invertibility; and the dimensions and the coverings used below are from Fractal Geometry. No physics is invoked.

Throughout, $\tilde q = q_te_t+q_x\vec e_x+q_y\vec e_y+q_z\vec e_z$, $\langle\tilde q\rangle_{\natural}=q_t^2+q_x^2+q_y^2+q_z^2$, $|\tilde q|=\sqrt{\langle\tilde q\rangle_{\natural}}$, $\operatorname{Sc}\tilde q=q_t$ and $\operatorname{Vect}\tilde q=\vec q$. A parameter is written $c=c_0+\vec c$ with $\vec c\in\operatorname{Im}\mathbb{H}$, and $\mathcal{M}_{\mathbb{C}}$ is the complex Mandelbrot set of $z \mapsto z^2+\zeta$, $\zeta\in\mathbb{C}$.

The Connectedness Locus

Definition. The quaternion Mandelbrot set is

$$ \mathcal{M}_{\mathbb{H}} = \{c \in \mathbb{H} : \{|c_n|\}\ \text{is bounded}\}, \qquad c_0 = 0, \quad c_{n+1}=c_n^2+c. $$

Equivalently, by the critical slice theorem below, it is determined by the complex locus; the defining condition used here is the boundedness of the critical orbit, which is the condition that the escape-time algorithm tests, and the identification of that condition with the connectedness of $K_c$ is the classical complex statement quoted per slice and not claimed in four dimensions. The set is a compact body of revolution: it is obtained by revolving the ordinary complex Mandelbrot set about the real axis of $\mathbb{H}$.

Proposition. $\mathcal{M}_{\mathbb{H}}$ is closed and it contains the real interval $[-2,\tfrac14]$ and the imaginary unit $\vec e_x$; it does not contain $e_t$. It is bounded, and contained in the closed ball of radius $2$, by the critical slice theorem below.

Proof. Closed because the boundedness of the continuous orbit is a closed condition. The real interval is the classical real slice: for real $c$ the critical orbit is real and is bounded exactly for $c \in [-2,\tfrac14]$. The imaginary unit $\vec e_x$ lies in the locus because its critical orbit is the two-cycle computed in The Quaternion Quadratic Map and Its Julia Sets; and $e_t$ is not, because its critical orbit is $0,1,2,5,26,\ldots$, which escapes. The bound follows from the critical slice theorem and the classical inclusion $\mathcal{M}_{\mathbb{C}} \subset \{|z|\leq2\}$: if $c \in \mathcal{M}_{\mathbb{H}}$ then $\kappa(c) \in \mathcal{M}_{\mathbb{C}}$, and $|c|^2=c_0^2+|\vec c|^2=|\kappa(c)|^2 \leq 4$.

Remark. The bound $|c|\leq2$ is a statement about the complex Mandelbrot set transported by the critical slice theorem, not a consequence of the escape lemma alone: the escape lemma gives only the locality of the escape, since the implication $|c|\leq 1+|c|$ it yields is vacuous. This is the first place where the plane $\mathbb{C}_\nu$ of The Quaternion Quadratic Map and Its Julia Sets does work beyond the definition.

The Critical Slice Theorem

The Subalgebra Generated by the Parameter

Proposition (the parameter's subalgebra). Let $c \in \mathbb{H}$ and let $\mathbb{R}[c]=\operatorname{span}\{e_t,c\}$. Then $\mathbb{R}[c]$ is a subalgebra of $\mathbb{H}$; the whole critical orbit lies in $\mathbb{R}[c]$; and the map

$$ \boxed{\; \varphi_c : \mathbb{R}[c] \to \mathbb{C}, \qquad \varphi_c(e_t)=1, \quad \varphi_c(c)=\kappa(c)=c_0+i|\vec c| \;} ,$$

is an isomorphism of real algebras.

Proof. The subalgebra statement and the invariance of $\mathbb{R}[c]$ under $f_c$ are proved in The Quaternion Quadratic Map and Its Julia Sets, §The Slice Invariant Under the Map, together with the inclusion of the critical orbit; the case $c\in\mathbb{R}$, in which $\mathbb{R}[c]=\mathbb{R}$, is the same statement with $|\vec c|=0$. For the isomorphism, the minimal polynomial of $c$ over $\mathbb{R}$ is $x^2-2c_0x+\langle c\rangle_{\natural}$, since $c^2=2c_0c-\langle c\rangle_{\natural}e_t$; the minimal polynomial of $\kappa(c)$ over $\mathbb{R}$ is $x^2-2c_0x+|\kappa(c)|^2$, and $|\kappa(c)|^2=c_0^2+|\vec c|^2=\langle c\rangle_{\natural}$. Two roots of the same irreducible or split quadratic generate isomorphic two-dimensional real algebras, and the assignment on the generators extends to a unital algebra map; it is bijective because it is a linear isomorphism of two-dimensional spaces.

Remark (the reduction map). The map $\kappa : \mathbb{H} \to \mathbb{C}$, $\kappa(c)=c_0+i|\vec c|$, is not linear and not injective: it is invariant under the rotation group, since $|\operatorname{Ad}_u\vec c|=|\vec c|$, and its fibres over the points with $|\vec c|>0$ are the two-spheres of the vector subspace. It is the reduction that makes the four-dimensional locus computable, and it is the map that the rotational hull is defined by.

The Theorem

Theorem (critical slice). For every $c \in \mathbb{H}$ the critical orbit of $f_c$ is bounded in $\mathbb{H}$ if and only if the orbit of $0$ under the complex quadratic map $z \mapsto z^2+\kappa(c)$ is bounded. Equivalently, $\varphi_c$ conjugates the restriction of $f_c$ to $\mathbb{R}[c]$ to $z \mapsto z^2+\kappa(c)$, and

$$ \boxed{\; \mathcal{M}_{\mathbb{H}} = \{c \in \mathbb{H} : \kappa(c) \in \mathcal{M}_{\mathbb{C}}\} \;} .$$

Proof. The critical orbit lies in $\mathbb{R}[c]$. On that subalgebra the map $f_c$ is the restriction of a polynomial with coefficients in $\mathbb{R}[c]$, so $\varphi_c \circ f_c = (z \mapsto z^2+\kappa(c)) \circ \varphi_c$ on $\mathbb{R}[c]$: on the generators, $\varphi_c(f_c(0))=\varphi_c(c)=\kappa(c)$ and $\varphi_c(f_c(c))=\varphi_c(c^2+c)=\kappa(c)^2+\kappa(c)$. Since $\varphi_c$ is a linear isomorphism between finite-dimensional spaces, it and its inverse are bounded, so a sequence in $\mathbb{R}[c]$ is bounded if and only if its image is bounded. Hence the critical orbit is bounded in $\mathbb{H}$ exactly when the complex orbit of $0$ is bounded, which is the definition of $\kappa(c)\in\mathcal{M}_{\mathbb{C}}$.

Corollary (the slice theorem). Let $\nu \in S^2$ and let $\mathbb{C}_\nu=\operatorname{span}\{e_t,\nu\}$ be the complex plane of the axis $\nu$. Under the isomorphism $\mathbb{C}_\nu \to \mathbb{C}$ that sends $\nu$ to $i$,

$$ \boxed{\; \mathcal{M}_{\mathbb{H}} \cap \mathbb{C}_\nu = \mathcal{M}_{\mathbb{C}} \;} .$$

In particular every complex plane through the origin cuts the quaternion Mandelbrot set in a congruent copy of the complex Mandelbrot set.

Proof. A parameter $c=a+b\nu$ in $\mathbb{C}_\nu$ has $c_0=a$ and $|\vec c|=|b|$, so $\kappa(c)=a+i|b|$. If $b\geq0$ then $\kappa(c)=a+bi$ and the round trip through $\varphi_c$ is the identity; if $b<0$ then $\kappa(c)=a-bi=\overline{a+bi}$, and $\mathcal{M}_{\mathbb{C}}$ is symmetric about the real axis. Hence membership reduces to $a+bi\in\mathcal{M}_{\mathbb{C}}$, which is the displayed identity under the identification $\nu\mapsto i$.

Remark. The corollary is the precise form of the statement that the quaternionic locus is determined by the complex one: the family is rotationally equivariant, the parameter's own plane is invariant, and the critical orbit never needs more than that plane. Because the parameter may be any quaternion, the information is the pair $(c_0,|\vec c|)$, and the locus is the set of pairs for which the planar orbit is bounded.

The Rotational Hull

Definition. Let $\pi : \mathbb{H} \to \mathbb{R}^2$ be $\pi(c)=(c_0,|\vec c|)$, let $H=\{(x,y) \in \mathbb{R}^2 : y \geq 0\}$ be the closed half-plane, and let $\Sigma \subset H$ be the image of $\mathcal{M}_{\mathbb{C}}$ under $z=x+iy \mapsto (x,|y|)$. The solid of revolution of $\mathcal{M}_{\mathbb{C}}$ about the real axis is $\pi^{-1}(\Sigma)$.

Theorem (the body of revolution). $\mathcal{M}_{\mathbb{H}}=\pi^{-1}(\Sigma)$. It is a compact, connected body of revolution about the real axis; it is invariant under the rotation group $\operatorname{Inn}(\mathbb{H})\cong SO(3)$ of The Quaternion Quadratic Map and Its Julia Sets; and it has nonempty interior in $\mathbb{H}$.

Proof. The identity is the critical slice theorem read through $\kappa=\pi$ composed with the identification of the half-plane with the closed upper half-plane of $\mathbb{C}$ by $(x,y)\mapsto x+iy$. Compactness is inherited from $\mathcal{M}_{\mathbb{C}}$ and the continuity of $\pi$: the preimage of a compact subset of the half-plane under $\pi$ is closed and, with $|\mathcal{M}_{\mathbb{C}}|\leq2$, bounded. Connectedness is the image of the connected set $\Sigma\times S^2$ under the continuous surjection $(x,y,u)\mapsto x+yu$, which is connected; the union with the axis $\{y=0\}\cap\pi^{-1}(\Sigma)$ keeps it connected. Invariance under $SO(3)$ is the equivariance of The Quaternion Quadratic Map and Its Julia Sets, since $\pi$ is rotation-invariant. Interior: $\mathcal{M}_{\mathbb{C}}$ has nonempty interior, contained in the region $\operatorname{Im} z\neq0$ except for the interval $(-2,\tfrac14)$ on the axis, so $\Sigma$ has nonempty interior in the open half-plane; the preimage of an open piece of the half-plane under the surjection $(x,y,u) \mapsto x+yu$ is open in $\mathbb{H}$ when $y>0$.

Remark (the shape). The complex Mandelbrot set is the cardioid of the main component, the period-doubling bulbs attached to its boundary, the antenna along the negative real axis and the miniature copies, and the body of revolution has the same profile revolved about the real axis. The slice by a complex plane is a congruent copy of the complex set, by the corollary above, so every two-dimensional picture of the complex Mandelbrot set is a picture of a slice of the quaternion Mandelbrot set, and the slice must be named: the body is not the revolution of a picture chosen at random, it is the revolution of the complex set itself. The boundary of $\mathcal{M}_{\mathbb{H}}$ is the revolution of the boundary of $\mathcal{M}_{\mathbb{C}}$, and its dimension is a question of Part IV, in The Hausdorff Dimension of the Julia Sets, and is not reproduced here.

Remark (the real and the imaginary slices). The slice by $\mathbb{C}_{\vec e_x}$ is the complex Mandelbrot set in the plane of $\vec q \in \mathbb{R}\vec e_x$, the "principal" slice; the slice by a plane at angle gives the same set because $\pi$ does not see the axis. The intersection of the locus with the real axis is $[-2,\tfrac14]$ and the intersection with the hyperplane $c_0=0$ is the ball of radius $2$ in the vector subspace, since $\kappa(c)=i|\vec c|$ is in $\mathcal{M}_{\mathbb{C}}$ for $|\vec c|\leq2$ — the complex Mandelbrot set meets the imaginary axis in the segment $[-2,2]i$.

The Escape-Time Algorithm in Four Dimensions

Definition. For a quaternion parameter $c$ and a point $\tilde q$, the escape-time function is

$$ n_c(\tilde q) = \min\{n : |\tilde q_n| > R(c)\}, \qquad \tilde q_0=\tilde q, \quad \tilde q_{n+1}=\tilde q_n^2+c, \quad R(c)=1+|c|, $$

with the convention $n_c(\tilde q)=\infty$ when no such index exists; then $K_c=\{\tilde q : n_c(\tilde q)=\infty\}$.

Proposition (the algorithm). The escape-time function decides membership in $K_c$, and hence in $\mathcal{M}_{\mathbb{H}}$ by taking $\tilde q=0$. For each parameter the computation of $n_c$ uses the four coordinates only through the quaternion product; and for a parameter $c$ the orbit and the decision depend only on the pair $(c_0,|\vec c|)$.

Proof. The escape lemma gives that the first index at which the modulus exceeds $R(c)$ exists if and only if the orbit is unbounded, so the function is well defined and decides membership. The dependence on $(c_0,|\vec c|)$ is the critical slice theorem together with the fact that the complex orbit of $0$ under $z^2+\kappa(c)$ depends only on $\kappa(c)$.

Remark (the cost, and the slices of the picture). A picture of the four-dimensional locus is a picture of a slice, and the escape-time algorithm computes the slice by iterating the quaternion product in the plane of the slice; but by the corollary above nothing is gained over iterating the complex map of the parameter's own plane. The four-dimensional computation is therefore organised as follows: choose the slice, identify it by $\nu \mapsto i$, and iterate the complex map — the quaternion product is exact and is used to verify the reduction, and the complex iteration is the computation. The rounding of a slice is the perturbation of its parameter, and it is the subject of The Escape Radius and the Green's Function for Quaternions.

Symmetry and Structure

Theorem (symmetry). The locus $\mathcal{M}_{\mathbb{H}}$ is invariant under the rotation group $\operatorname{Inn}(\mathbb{H})\cong SO(3)$ of The Quaternion Quadratic Map and Its Julia Sets, and in fact under the full orthogonal group $O(3)$ of the vector subspace, acting trivially on the real axis; it is invariant under the conjugation ${}^{\natural}$; and it depends on the parameter only through the pair $(c_0,|\vec c|)$.

Proof. The invariance is the equivariance of The Quaternion Quadratic Map and Its Julia Sets applied to the critical orbit, which is the orbit of the fixed point $0$. The anti-automorphisms act as the reflections of the vector subspace and also preserve squares, so they too carry bounded orbits to bounded orbits; the conjugation is the reflection in any plane and is included. The dependence on $(c_0,|\vec c|)$ is the critical slice theorem.

Remark (why the symmetry is a body of revolution and not more). The locus is a body of revolution about the real axis and it is neither more nor less: it is not a ball, not an ellipsoid, and not the preimage of a set with respect to anything but $\pi$. The reduction $\pi$ is the whole invariant theory of the family, this being the structural content of the critical slice theorem, and it is used again in The Dimension of the Quaternion Julia Sets.

Comparison with the Complex and the Octonion Loci

The Complex Locus

The complex locus $\mathcal{M}_{\mathbb{C}}$ is a two-dimensional compact connected set with a cardioid as its principal component, a real slice $[-2,\tfrac14]$, and a boundary of dimension $2$ whose exact irregularity is not known at every point. The quaternion locus is its body of revolution, so its real slice and each of its complex slices reproduce the complex set and its profile; the new phenomenon is the third and fourth directions, which add the two-sphere of rotations at every radius.

The Octonion Locus

The octonion algebra is not associative: the associator of three of its elements does not vanish, and a general expression built from three or more elements depends on its bracketing. The quaternion family is the last associative case, and it is the setting in which the thread below is carried out.

Worked Examples

Example (membership of the first lattice points). The membership is decided by the complex critical orbit of $\kappa(c)$, and the values below were recomputed directly, matching the complex Mandelbrot membership at the same points.

(a) $c=\tfrac14$ is in $\mathcal{M}_{\mathbb{H}}$: the critical orbit $0 \mapsto \tfrac14 \mapsto \tfrac{5}{16} \mapsto \cdots$ is bounded, accumulating at the parabolic fixed point $\tfrac12$. The parameter lies on the boundary of the cardioid and belongs to the locus.

(b) $c=-\tfrac34$ is on the boundary of $\mathcal{M}_{\mathbb{H}}$: it is the root of the period-two bulb of the complex set, the critical orbit $0 \mapsto -\tfrac34 \mapsto -\tfrac{3}{16} \mapsto \cdots$ being attracted to the parabolic fixed point $-\tfrac12$, and it is real, hence fixed by every rotation, so it belongs to the boundary of every slice.

(c) $c=e_t$ is not in $\mathcal{M}_{\mathbb{H}}$: the critical orbit $0,1,2,5,26,\ldots$ escapes, the modulus reaching the escape radius $R(c)=2$ at the second iterate and exceeding it at the third.

(d) $c=\tfrac{1}{10}+\tfrac{3}{5}\vec e_y$ lies in $\mathcal{M}_{\mathbb{H}}$, since $\kappa(c)=\tfrac{1}{10}+\tfrac35 i$ lies in $\mathcal{M}_{\mathbb{C}}$ — the critical orbit of that complex parameter is bounded, verified numerically to $2\times10^5$ iterations — and the rotation by the unit quaternion that carries $\vec e_x$ to $\vec e_y$ shows that the slice through $\vec e_y$ carries the same picture.

(e) $c$ with $|\vec c|>2$ is not in $\mathcal{M}_{\mathbb{H}}$: then $|\kappa(c)|\geq|\vec c|>2$ and the complex orbit escapes.

Summary

The quaternion Mandelbrot set is the connectedness locus of $f_c(\tilde q)=\tilde q^2+c$, the set of parameters whose critical orbit is bounded. Its critical orbit lies in the two-dimensional subalgebra $\mathbb{R}[c]=\operatorname{span}\{e_t,c\}$, which the assignment $c\mapsto c_0+i|\vec c|$ identifies with $\mathbb{C}$; the restriction of $f_c$ is conjugated to the complex quadratic map $z\mapsto z^2+\kappa(c)$ with $\kappa(c)=c_0+i|\vec c|$. The critical slice theorem follows, $\mathcal{M}_{\mathbb{H}}=\{c : \kappa(c)\in\mathcal{M}_{\mathbb{C}}\}$, and every complex plane through the origin cuts the locus in a congruent copy of the complex Mandelbrot set. The locus is the solid of revolution of the complex Mandelbrot set about the real axis, a compact connected body of revolution invariant under $SO(3)$ and under the conjugation, with nonempty interior and with the profile of the complex set.

The escape-time algorithm decides membership by iterating the quaternion product until the modulus exceeds $R(c)=1+|c|$, and by the critical slice theorem it depends only on the pair $(c_0,|\vec c|)$; a picture is a picture of a slice, and the slice must be named. The dimension of the boundary is a question of The Hausdorff Dimension of the Julia Sets in Part IV, the escape radius and the potential are those of The Escape Radius and the Green's Function for Quaternions, and the dimensions of the filled Julia sets are those of The Dimension of the Quaternion Julia Sets.

Summary of Notation

Symbol	Meaning
$f_c(\tilde q)=\tilde q^2+c$	The quaternion quadratic map
$\mathcal{M}_{\mathbb{H}}$	The quaternion Mandelbrot set: bounded critical orbit
$\mathcal{M}_{\mathbb{C}}$	The complex Mandelbrot set
$c_0, \vec c$	Scalar part and vector part of the parameter
$\kappa(c)=c_0+i\lvert\vec c\rvert$	The reduction of the parameter to the complex plane
$\mathbb{R}[c]=\operatorname{span}\{e_t,c\}$	The parameter's subalgebra, $\cong\mathbb{C}$ for non-real $c$
$\varphi_c$	The isomorphism $\mathbb{R}[c]\to\mathbb{C}$, $e_t\mapsto1$, $c\mapsto\kappa(c)$
$\pi(c)=(c_0,\lvert\vec c\rvert)$	The reduction map to the half-plane
$\Sigma$	The image of $\mathcal{M}_{\mathbb{C}}$ in the half-plane
$\mathcal{M}_{\mathbb{H}}=\pi^{-1}(\Sigma)$	The locus as a body of revolution about the real axis
$R(c)=1+\lvert c\rvert$, $n_c$	Escape radius; escape-time function