Kochina
|
This is a crop circle that was discovered in Milk Hill in England on 8th August 1997. It resembles the Koch Snowflake fractal, which is an Iterated Functions System, or "IFS" fractal for short. The most common way to generate the Koch Fractal is to start from a triangle, or Star of David pattern (which results from the first iteration) and replace each line according to the following rule (called the "generator"): |
| The first 4 iterations (also called generations) of the fractal look like this: |
| There are actually quite a few ways of generating the Koch snowflake fractal. For example, you can invert the rule and generate it starting from a hexagon: |
| In tihs case, the first 4 iterations of the fractal will look like the diagram below: |
| This second rule is actually the same as the first rule turned upside down, but with the arrows pointing in the opposite direction, so you could generate the same pattern by using the first rule but orienting the lines in the initial hexagon so that they go anti-clockwise instead of clockwise. The first method of generating the Koch snowflake is additive - the total area grows outwards with each iteration (by a smaller amount each time), converging on a definite area value as the number of iterations tends towards infinity. The second method is subtractive - the area starts off as the area of a hexagon and shrinks inwards with each iteration, likewise converging as the number of iterations tends towards infinity. So we have "outies", and "innies". Just like your omphalos ;-). Note that after an infinite number of iterations the "innie" and "outie" snowflake will converge on the same (transcendental?) form. If we return to the original crop circle picture, the outer boundary of the crop circle (ignoring the smaller circles around it) is the second iteration of the "outie" method, and the boundary of the snowflake in the middle is the second iteration of the "innie": |
| Let's look at some other properties of the Koch snowflake. For starters, it's worth noting that it is possible to tile the plane with two different sizes of the snowflake, with one oriented upright (i.e. pointing up and down like the Star of David) and the other oriented sideways (i.e. pointing left and right, like a Star of David rotated 90 degrees). Another property is that it is possible to subdivide the area of a Koch snowflake in order to produce 6 smaller snowflakes (oriented the same way as the original snowflake) arranged around another slightly larger snowflake oriented at 90 degrees to the original. You can then subdivide this larger snowflake using the same scheme, to produce 6 more even smaller snowflake around one in the middle that is the same size as the 6 snowflakes you produced with the original subdivision. Here's a diagram to illustrate what I'm talking about: |
| The composite snowflake in the middle roughly corresponds to the layout of the crop circle, except for a couple of things: for one, the crop circle stops at two iterations, and secondly, the inner snowflake is rotated 90 degrees compared to how it would be oriented if you subdivided the larger snowflake according to the 6-around-1 scheme, such that it is oriented the same way as the outer snowflake. I'm not sure of what (if any significance) this has, but it's worth pointing out that the area or the two snowflakes is the same regardless of the orientation. You can generate the subdivided snowflake using the line method by starting with a Star of David pattern (made up of 6 triangles and 1 hexagon) and then applying the rules mentioned above. If you just want the outline, you don't actually have to do the hexagon because it's fractalised boundary overlaps exactly with the boundaries of the 6 snow-flakes generated from the 6 triangles around it. Here's an image of a Koch snowflake that has been subdivided according to the 6-around-1 scheme, applying the rule recursively to each of the smaller snowflakes that are generated after each subdivision: |
(Click on the image for a larger version, or click here)
| By applying the subdivision/recursion rules in reverse (or simply using the tiling/tesselation scheme mentioned earlier), it is possible to fill the entire plane with this fractally-subdivided Koch snowflake pattern. Now, how about some interpretation of this pattern. Does it mean anything, or does it just look cool? There are a few religious/esoteric/sacred geometry glyphs that can possibly be related to the Koch snowflake shape. One is obviously the Star of David (also "Vishnu's seal" in Hinduism), which has the same outline as the Koch snowflake after 1 iteration. Another is the "Seed of Life" pattern (and by extension the "Flower of Life" as well), which has the same 6-around-1 arrangement of its constituent circles as the Koch snowflake has of its subdivisions. Yet another is Metatron's Cube, in particular the arrangement of 7 circles that sits in the middle of the glyph, which is identical to the Seed of Life except that the circles are of half the radius. |
| Visually, it's quite easy to notice that all of these glyphs are interrelated. For example, the Star of David is found inside Metatron's cube, the Seed of Life is the inner part of the Flower of Life, the inner part of Metatron's cube is obviously in Metatron's cube, and is also related to the Seed of Life by being the same pattern but with circles of half the size. Another connection between the Seed of Life and the Star of David is via Circle Inversion. This is a geometric inversion transformation that transforms circles and lines about a reference circle (see that link for more details). If you apply a circle inversion transformation to the Seed of Life, using the outer enclosing circle as the reference circle, you get a pattern containing the Star of David. Since the transformation is reversible, it works both ways, so if you invert a Star of David, you get something resembling the Seed of Life. In the following two diagrams, the pre-transform glyph is rendered in pink, the reference circle for the inversion transform is in white, and the post-transform glyph is in blue, overlaid on the same diagram. |
 Inverted Seed of Life glyph
|  Inverted Star of David |
| Okay, so we've established that there are some simple geometrical relationships connecting all these glyphs. But how do they relate to the Koch snowflake fractal beyond a superficial resemblance? Let's look at another way to generate the Koch fractal, based around circles instead of lines. Suppose we use a circle for our starting shape, and replace the line generation rules from earlier with a simple substitution rule where at each iteration, we replace each circle with 7 smaller circles, arranged like the 7 circles from the inner part of Metatron's cube. Then in the next iteration we replace each of those 7 circles with 7 more circles (creating 49 new circles) and so on. Here's an illustration of the generation rule: |
| Here's what the fractal looks like after the iterated function system has been applied for 4 iterations, starting from a single circle: |
| The outer boundary of the fractal is shown in red, and quite obviously resembles the Koch snowflake. I've rendered the circles generated with each iteration over the top of the image in light grey, so you can see how the circles are generated at each step. Alrighty then, that shows that there is a connection between the Koch snowflake (a.k.a. the "Koch Curve") and the inner portion of Metatron's Cube (and indirectly the Seed of Life and Star of David) via an iterated function system fractal. So, what to make of all this? What are some of the meanings surrounding these glyphs, that when "fractalised" allow us to produce a Koch snowflake, and what if any interpretation can we make of the crop circle? Well, one of the meanings of the "Seed of Life" is that it represents the process of creation - the 6 circles around 1 central circle correpond to the 6 "days" of creation, and 1 day of rest in a 7-day creation "week". Sometimes the "days of creation" are referred to as "rays of creation", with the 7th one "hidden" (corresponding to the "day of rest" in the Genesis creation story). The number 7 is also related to the "Law of Seven" or Heptaparaparshinokh, which posits that 7 is a special number that describes the creation process of the Universe and patterns that appear within it. Fair enough so. Let's look at the crop circle again: |
| Looking at the crop circle and comparing it the inner-Metatron circle Koch snowflake fractal above it, there is one obvious difference (besides the orientation of the inner snowflake), which is that the inner, seventh, section is missing. Perhaps this is the missing "7th ray of creation"? Let's generate another fractal using our Metatron circles, but this time we'll start from 6 circles, with the central one missing, to represent the 6 "rays of creation" with the 7th missing. We'll keep the same generation rule as before, so 7 circles will be generated from these initial 6, and then 7 from each of them, and so on. Here's the new rule: |
| Let's iterate the rule once, and see what it looks like: |
| This resembles the crop circle a lot more closely, although the orientation of the empty section in the middle is still 90 degrees off. Starting with the original 6 circles, and creating 7 smaller circles from each of them, we now have 42 circles. So what does 42 mean? The secret of Life, the Universe and Everything, according to Douglas Adams :-) Here's one thing we can do with the number 42: use it as an exponent of Phi (the Golden ratio). Why phi? Well why not? Phi is something that crops up (no pun intended) a lot when you read into this sacred geometry stuff. It's supposedly a "divine ratio". So, what is the 42nd exponent of phi? Phi42 = 599,074,577.9999999983... Phi42 / 2 = 299,537,288.9999999991 Speed of Light = 299,792,458 metres per second That's neat. However, it should be said that metres and seconds are arbitrary units, so it doesn't necessarily mean anything. The standard metric metre is 1/10,000,000th of the distance from the North Pole to the equator (10,000 kilometres), running through the meridian line that passes through Paris. A second is 1/86,400th of a day (24 hours times 60 minutes times 60 seconds = 86,400). What about if we do 2 iterations of our fractal rule? (That's the number of iterations of the line-based rule you need to do to generate the Koch fractal in the crop circle) |
| Now we have 294 circles (6 times 7 times 7). What is the 294th exponent of phi? Phi294 = 2.7692759465 times 1061 The Planck length is approximately 1.6 × 10-35 meters. The estimated radius of the observable universe (4.4 × 1026 metres, or 46 billion light-years) is 2.7 × 1061 Planck lengths. This is a pretty startling correspondence, because it isn't based on arbitrary units, but on the Planck Length itself, which is basically the shortest length in the universe at which "traditional notions of space and time" apply. The prime factors of 294 are 2, 3 and 7 (twice). So if you are going to somehow encode the size of the universe geometrically in terms of logarithms of phi, you would need to use some kind of figure that divides itself according to these numbers, i.e. into 2 and 3 at one level (or levels), and then into 7 and 7 again. A Koch snowflake iterated for 2 generations with the middle bit missing at the top level would seem to be a perfect match. Here are another couple of crop circles. The first is another Koch fractal crop circle, which appeared near Silbury Hill, on July 23rd 1997, and the second is a square fractal that appeared at West Kennett in 1999: |
 Silbury Hill Koch Fractal Crop Circle (1997)
|  West Kennett Square Fractal Crop Circle (1999) |
| It is possible to create a 3-dimensional fractal object that resembles both of these crop circles in silhouette when viewed from different angles. The fractal object is generated by starting with a cube and adding 8 smaller cubes at each of the vertices, then applying the rule recursively: |
 Cube Fractal (Isometric view at 45 degree angle)
|  Cube Fractal (Isometric view from side-on) |
| Note that the proportions do not match up exactly the same for the square fractal, however they have to be "just so" for the fractal object to form the Koch Curve in silhouette, so it is not possible to make it match up exactly for both of them. The resemblence is still fairly strong however. I actually made this fractal before seeing a picture of the West Kennet crop circle, it was quite startling to see it for the first time. Here is the rule for generating the fractal, and a perpective render of the object: |
| It is also possible to create 3D fractals from spheres, octahedrons, and the stella octangula (two interlocking tetrahedrons) that have a Koch Curve silhouette when rendered in an isometric projection. The fractal generated from spheres is called the Sphereflake fractal. Click on the thumbnail images below for larger views: |
 Sphere fractal (6 children) Isometric
|  Sphere fractal (6 children) Perspective
|  Sphere fractal (8 children) Isometric
|  Sphere fractal (8 children) Perspective |
 Octahedron fractal (6 children) Isometric
|  Octahedron fractal (6 children) Perspective
|  Octahedron fractal (8 children) Isometric
|  Octahedron fractal (8 children) Perspective |
 Octahedron crop circle. A template for an octahedronal fractal form? |
 Stella Octangula fractal Isometric
|  Stella Octangula fractal Perspective
|  Stella Octangula fractal Perspective (underside) |
| Link to a thread on Fractal Forums with more technical detail about generating some of these fractals. My MySpace page where I have a blog about this kind of stuff. Forum discussion thread in the Ziggurat subforum of The Phora on this and related topics (particularly esoteric/occult interpretations). Note: links to a "free speech" forum where politics is also discussed. Some of the opinions expressed there are politically incorrect (to say the least). Star Nation Gallery article about the Milk Hill Koch Snowflake crop circle (and others). |
Return to Index
- Chris Hayton 2007