The Koch curve in three dimensions

Transcription

1 The Koch curve in three dimensions Eric Baird The paper describes an approach to extending the Koch curve into three dimensions. An intermediate stage is a two-dimensional Koch leaf, a shape with zero area bounded by three Koch curves. The final delta-shaped fractal solid has an infinite number of planar facets, each of which is a Koch leaf. 1. Introduction Some of the best-known two-dimensional fractals have higher-dimensional counterparts the Boolean logic underlying the Sierpinski carpet can be extended into three dimensions to produce the Menger sponge (and can itself be considered as a two-dimensional extension of the Cantor Set), and the Sierpinski triangle can be extended into three dimensions in a slightly less obvious way to produce the Sierpinski pyramid (as used by Alexander Graham Bell for a kite design in the late C19th). Although there are many ways of producing fractal solids that generate Koch-snowflake-shaped profiles and silhouettes, the literature does not appear to mention a three-dimensional shape that could be considered to be an extension of the Koch curve. The current paper attempts to remedy this. 2. The Classic Koch curve The Koch curve is typically drawn as in figure 1. Figure 1: Standard Koch curve, and snowflake An initial baseline is replaced with a chain of four smaller same-size line segments, with the two endsegments lying on the original line and sharing its termination points, and with two additional linesegments forming a spike that protrudes away from this baseline. For the common Koch curve, the spike apex angle is 60, and the shape fits into a bounding triangle with angles 30, 30 and 120. Three outward-pointing Koch curves can be chained to enclose an area whose hexagonallysymmetrical outline is known as the Koch snowflake. Koch snowflakes of different sizes can be fitted together in a jigsawlike manner to tile an area. page 1 of 4

2 3. Generalising the Koch Curve Other variations on the curve are possible, for instance if we change the apex angle to 36, and the corresponding bounding-triangle's angles to 36, 36 and 108, we obtain a different style of Koch curve: Figure 2: Pentagon-proportioned Koch curve, and briar This can be used to create a five-sided counterpart to the Koch snowflake, the Koch briar, which has the same property of being able to tile with rescaled copies of itself. This generalisation allows for a continuous range of Koch curves by changing the proportions of the initial bounding triangle 4.The Koch leaf We'll now create an area-based counterpart of the generalised Koch curve. The Koch plane or Koch leaf is a closed shape that can be imagined as an isosceles triangle whose three sides have been replaced with inward-pointing Koch curves. The protrusions on the three bounding Koch curves interlock, and as the number of iterations increases, the enclosed area usually reduces towards zero (an exception being the special case of an apex angle of zero). We can create the shape by repeatedly removing wedge-shaped areas from each bounding region to produce smaller copies, or by repeatedly replacing each triangular bounding area with two smaller linked copies (sharing a vertex) that just fit into the same space. Figure 3: Koch leaf construction, with bounding area angles of 30, 30, 120 page 2 of 4

3 Although the Koch leaf may seem uninteresting in that it eventually generates a shape that is visually indistinguishable from a single linear Koch curve, its property of nominally enclosing a twodimensional area makes it a useful step towards the creation of a three-dimensional Koch-based solid. 5. Creating a Koch-based solid We can now extend the method into three dimensions. We start with a three-dimensional bounding polyhedron that plays an analogous role to the bounding triangle in figure 3. This three-dimensional boundary is a six-sided delta shape, whose faces or facets are isosceles triangles with sides of ratio 2:2:3. The solid has three long edges forming an equilateral triangle equator, and six shorter edges that meet in two groups of three at the shape's poles above and below the equatorial plane. If the equatorial sides have a length of 3 then the shorter-edged sides will have length 2, and the distance between the polar vertices will also be 2. Figure 4: Bounding volume, and first and tenth iterations We can convert this solid into a fractal by repeatedly removing wedge-shaped pieces of material to convert each piece into three smaller identical rotated copies, or by deleting the parent bounding polyhedron and replacing it with three smaller maximally-sized linked copies that fit into the same space and include the original vertices of their parent. These three copies are rotated 90 degrees out of the equatorial plane of their parent, share a common edge that links the parent's pole vertices, and are distributed around this new edge at 120 degrees with respect to their siblings. Each replacement creates new facets that are each have the same 2:2:3 proportions, and at each iteration, existing facets undergo the sort of division shown in figure 3, with the end result being that, with an infinite number of iterations, each facet ultimately becomes a Koch leaf. Figure 5: Two more views of the resulting solid page 3 of 4

4 Figure 6: One Koch leaf facet of the shape, with proportions 2:2:3 6. Conclusions This delta shape appears to meet the criteria for a higher-dimensional analogue of the Koch curve: it is self-similar at all scales, each of its edges is a Koch leaf, each of its edges is a Koch curve, it can be infinitely subdivided or extended in a broadly similar way to the Sierpinski pyramid, and all of its linear, planar and volumetric components are identical save for scalings and rotations. It would also seem to have zero volume (since each subdivision removes a significant fixed proportion of the total remaining volume). That the bounding polyhedron not a regular polygon is a consequence of the fact that our generalisation of the Koch curve and its extension to two dimensions does not allow an equilateral triangle as a bounding triangle. The delta construction can only be created using a Koch curve of the selected proportions (bounding triangle 2:2:3), and is therefore a unique solution. These zero quantities and the absence of any elements other than Koch curves and bounding areas and volumes suggest that the shape may also be a fundamental solution. However, since there is a continuous range of other possible Koch curves, it is not immediately obvious whether this solution is unique in a more general sense, or whether other fundamental solutions may exist, based on different construction methods and differently-proportioned bounding polygons. References Baird, Eric Alt.Fractals: A visual guide to fractal geometry and design (2011) ISBN , pages 29 & 161 Eric Baird, May 2014 page 4 of 4

5