A Novel Hexagon/Rhombus Fractal Generated thru Recursion (Part II)

Construction of Non-similar Cantor Type Set of a Mingled Hexagon/Rhombus Fractal

The Sierpinski triangle, Wikipedia, and the Sierpinski square also in Wikipedia, are fractals with the overall shapes of either an equilateral triangle or a square, subdivided recursively into smaller equilateral triangles or squares. Part I showed that a square could also behave similarly to afford a new type of fractal. Applying the method of recursive subdivision to hexagons affords a new type of mingled (intermixed) fractal composed of three hexagons and three rhombuses.

To get the first iteration we start by taking a hexagon and connect opposite vertices to form R the circumradius equaling the side length of the hexagon t:

Three identical rhombuses are then formed by intersecting three of the sides as shown in the second structure. Three lines of equal length each separated by angles of 120° and then joined at the center, a structure which I call trigonal planar (TP) based on the similar chemical structure connecting three atoms with a central one Wikipedia. This TP is then joined to the rhombus at the vertex that points towards the center of the hexagon.

Removing the Rs and labeling rhombuses/hexagons as A,B,C,D,E and F at the vertices as shown, where B,D and F are identical hexagons colored gold to distinguish them from the rhombuses A,C and E but also to discern the hexagons from the rhombuses as they shrink in size as the number of iterations increases. Note that the figure has gone thru three iterations.

Performing a fourth iteration affords the figure below with a total number of 81 hexagons. Note that the figure seems to have a 3D look to it.

Performing an iteration on the hexagon affords three subrhombuses and three hexagons, a set dissimilar from the regular Cantor set:

where the first iteration hexagon is shown as a blue/black line composed of the six line segments standing for rhombuses A,C,E and the hexagon B,D,F. Since A,C and E are no longer within the hexagon group, the first, third and fifth line segment are deleted from row two. In rows three and four the process is repeated using shorter line segments.

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