# Splits and Splits3D

Name:splits
Type: 2D
Author:Michael Faber

Name:splits3D
Type: 3D
Author:Chris Johns (TyrantWave)

## Overview

Splits splits the plane along the x and y axes, shifting the quarters apart to leave vertical and horizontal space, which can then be filled with other variations like elliptic. Splits3D extends the variation to three dimensions, shifting along the x, y, and z axes.

## Parameters

 variation amount Scale factor for the output; also scales the other variables. Usually set to 1. x Amount to shift in the x direction y Amount to shift in the y direction z (splits3D only) Amount to shift in the z direction

## Details

Setting a parameter to 0 means no shift in that direction. Setting all parameters to 0 makes splits the same as linear, or splits3D the same as linear3D. Setting a parameter to a positive value shifts both positive and negative directions away from the axis, leaving a space twice the parameter value. Setting a parameter to a negative value shifts the opposite direction, making the two halves overlap. Here are examples of splits and splits3D on a disk and a sphere of radius 1.

When splits is iterated, each iteration expands the previous one, creating a form of tiling. When x and y are both positive, the tiles form an X shape, as shown in the examples below. In the first, a circle of radius 1 is split, so the tiles overlap. In the second, the circle is smaller so it just fits in the split area, so the tiles don’t overlap but are smaller. In the third, the circle is replaced by a tall ellipse. If the y value is set to 0, the tiling becomes purely horizontal, as shown in the fourth example. (Setting x to 0 would make a vertical tiling.) All of this is with no scaling or rotation of the transform. The fifth example shows the result of rotating the splits transform 7° counter-clockwise, and the last of also scaling it down by 105%.

Different effects can be obtained by using different variations in place of the oval. Here are three examples using cylinder, elliptic, and sinusoidal.

Splits3D extends this to three dimensions. If the z parameter is 0, it works the same as splits but preserves the z value from other transforms. Other values will tile the fractal along the z axis just as x and y do. If the other variations used in the fractal are 2D (as all of the above are), this will add some depth that will be visible when the pitch is changed. If true 3D variations are used, a wide range of effects is available.

## Special considerations

Don’t confuse splits with the similarly named but very different variation split.

## Uses

• The basis for many styles, including splits-elliptic and splits-crop

## Mathematics

In the following formula, $x$, $y$, and $z$ are the inputs to the variation, $V_x$, $V_y$, and $V_z$ are the results of the variation and $w$ is the variation amount. To avoid confusion, the parameters x, y, and z are represented by $P_x$, $P_y$, and $P_z$. The last equation only applies to splits3D.

$V_x = \begin{cases} x+P_x,& x\ge 0 \\ x-P_x,& x=0 \end{cases}$
$V_y = \begin{cases} y+P_y,& y\ge 0 \\ y-P_y,& y=0 \end{cases}$
$V_z = \begin{cases} z+P_z,& z\ge 0 \\ z-P_z,& z=0 \end{cases}$

separation