This section contains a brief introduction to quaternions and some explanations on how to use them to build Julia sets. The aim of this page is to provide a concise yet easy to understand explanation about the generation of quaternion Julia sets.
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What are quaternions?

The quaternions were first introduced by Sir William R. Hamilton, a scientist who lived in the first half of the past century. He worked on a wide range of subjects related to applied mathematics.

A quaternion can be written in the form

Q = x + yi + zj + wk

where x, y, z and w are real coefficients (called constituents) and i, j and k are three different imaginary units such that

Example:

Q = 0.845 -0.5i -0.113j -0.05k

Quaternions are added or subtracted by adding or subtracting their constituents:

The product of Q₁ and Q₂ is more complicated, but it's possible to obtain the formula by multiplying componentwise the expressions of Q₁ and Q₂ and then applying the formulas for i, j and k to simplify the resulting expression. Note that multiplication is not commutative.

Quaternions are a generalization of complex numbers: if we set the z and w constituents to zero, we get the ordinary complex numbers.
As complex numbers can be interpreted as rectangular coordinates in a plane, quaternions can be interpreted as rectangular coordinates in a 4-dimensional space.
For more information about quaternions, see Proceedings of the Royal Irish Academy, Nov. 13, 1843, vol. 2, 424-434.

What are quaternion Julia sets?

The process of building quaternion Julia sets is the same used for ordinary Julia sets, but quaternions are used instead of complex numbers. The only difference is the coloring method: if a point belongs to the set, it is assigned a colour using an arbitrary function, whilst points outside the set are left transparent.

A generic quaternion Q₀ belongs to the Julia set associated with the quaternion C if the series defined by the recursive formula Q_n+1 = Q_n² + C doesn't diverge when n tends to infinity.

As ordinary Julia sets are contained in a 2-dimensional space, quaternion Julia sets are contained in a 4-dimensional space, therefore we can only visualize their intersections with 3-dimensional spaces (hyperplanes of the 4D space).

Example:

The following series of images shows several intersections of the Julia set corresponding to C = 0.845 -0.5i -0.113j -0.05k with the hyperplane defined by the equation w=constant.

w = 1.10	w = 1.05	w = 1.00
w = 0.80	w = 0.50	w = 0.00

All the images in this page were made with Quat, a freeware 3D fractal generator by Dirk Meyer.

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