Almond box

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Almond box viewed from the corner.

Frontal view.

In mathematics , an almond box is a fractal that has a box-like shape and was discovered by Tom Lowe in 2010. This fractal is defined in a similar way to the Mandelbrot set (hence the name), however, as with the Mandelbrot set, parameters will not tend towards infinity by means of geometric transformations. The almond box is defined as a "map" of continuous Julia sets , but the almond box is independent of dimensions, that is, it can also be represented in four dimensions. However, in order to better visualize the structure and to make it accessible to static media such as paper, the three-dimensional cube shape is preferred.

Generation

The iteration appears as a function of the vector z as follows:

function iterate(z):
    for each component in z:
        if component > 1:
            component := 2 – component
        else if component < -1:
            component := -2 – component

    if magnitude of z < 0.5:
        z := z * 4
    else if magnitude of z < 1:
        z := z / (magnitude of z)^2

    z := scale * z + c

In this case, c is the constant tested and scale is a real number .

properties

As with the Mandelbrot set or other fractals , parameter values ​​can be changed or manipulated with the Mandelbox.

A remarkable property of the Mandelbox, especially with the scale value ("scale") −1.5, is the approximate phenotype of known fractals within the Mandelbox fractal.

Scale = -1.5: IFS structures: on the right similar to the Menger sponge , in the middle an IFS structure.

When the scaling value amount is set between 1 and 2, the almond box contains a solid core.

For scaling values ​​less than −1, the almond box edges are each 4 length units long. For scaling values ​​between 1 and 4, the lengths 4 (scaling value + 1) / (scaling value -1) are to be found. ${\displaystyle 4{\sqrt {n}}+1}$

Related Links

• Mandelbox images at Wikimedia commons
• Gallery and description
• Images of some almond box cubes
• Video: zoom in the almond box cube

Individual evidence

1. ↑ Tom Lowe: What Is A Almond Box? . Archived from the original on October 8, 2016. Retrieved November 15, 2016.
2. ↑ negative almond box
3. ↑ more-negatives
4. ↑ mandelbox_3d_fractal
5. ↑ ^{a b} Rudi Chen: The Almond Box Set .