Hackpen Hill Koch fractal, Wiltshire 18th August 1997 Geometrics   Outliers ('grapeshot')

Flattened Grain in Crop Fields Exalting the checksum and exterminating the message? A call for better record-keeping of crop circle features By now, some people feel that crop formations express mathematical relationships, some even novel. Are these formations from 'other', and do they contain a message for us?

Crop circles manifest in the dark night with sheer size, complexity, precision and speed. If this does not blow you away just buy a crop circle calendar and enjoy pretty pictures. The rest of us should become systematic in researching them, however.

Is flattened crop a low-bandwidth and noisy communications medium? Certainly. But the message might not be complex, should not violate our free will, and is best targeted to those ready to accept it, not those in authority. Until we ourselves extract the full bandwidth that IS there, we are not worthy of asking the question, and until we prove that, the bandwidth won't likely increase.

Croppies agree that to decode a message any authentic formations must be well differentiated from cargo cult artifacts. However, there's little agreement on authenticity, nor message.

We are slow learners. Eventually, an astrophysicist realized the Barbury Castle formation encoded pi to ten digits. Aliens are teaching us pi? Doubtful. But we can be sure that what Lucy Pringle hails as "a seminal event", Suzanne Taylor is "almost certain" is an artifact. But Suzanne, the skeptic, elegantly says this:

"... speculation about whether they are real or not is the wrong conversation -- just what our gossipy society likes to dwell on, instead of on the majesty of life. Let's just hang out in majesty to see what that's all about and what we should do."

Some try to authenticate circles by finding features difficult to lay out on the ground with string and poles:

1. complex curves
2. orthogonal lines
3. precisely repeated patterns
4. true rather than approximated spirals
5. trisected angles
6. curves the center of which lie far outside the figure (requires trampling some crop)
7. no construction lines, traces that are used as anchors to construct locations for lain crop, but which are not part of the final figure

More esoterically, one might douse or sense strange energy fields emanating from them.

All these determinations, however, are subjective.

Many formation features are a subset of the 'fully constrained' variety that drafters use, being geometric (used by the ancients who could not reproducably divide measuring tapes) rather than numeric (our feet & inches or meters). They have points precisely attached to other previously laid down feature or construction lines. You may find the two circles defining the inner and outer bounds of a crop ring can be perfectly filled with an unseen inscribed contruction-line regular polygon. Finding this is a delightful exercise, but what it really conveys is the ratio of the circles. The coarse manifestation of laid grain reveals, with analysis, an infinitely precise design intent.

1. East Field at Alton Barnes, Wiltshire, July 9, 1998
2. Avebury Circle, near Marlborough, Wiltshire August 2, 1998
3. Taw[n]smead Copse, Alton Priors, near West Stowell, Wiltshire August 9, 1998
4. Sugar Hill, nearr Aldbourne, near Swindon, Wiltshire, July 21, 1999

Zef Damen's beautiful renderings of these into rules that can be reconstructed with straightedge-and-compass are a vital resource.

With computers, we can numerically reproduce these geometric features to far greater precision than is found in the crop. Consider the Chilbolton 2000 formation, interesting for its location in both time and space.

How long would it take to flatten this much crop on the ground, given cookbook instructions such as Zef Damon's? I have no idea, but I do know that it took me over two hours to make a model that you can download and view with E-Drawings of only the geometric parts, using a fast computer and a sophisticated drafting program. I actually constructed only one quarter of this dually symmetric formation. Simply mirroring the rest saved even more time compared to flattening crop. Rendering Zef's complex deconstructions and the lack of construction lines in actual crop convinces me that while crop circles contain precise classical relationships, laying crop out on the ground with straightedge/compass technique would have been nearly impossible.

Further evidence for this conviction is lent by Bert Janssen's work on squaring the circle. His formation measured on the ground formed a squared circle, but if you construct the same figure with the implied tangencies (that is, the square intersects the point of tangency of the small guide circle and the derived circle), you arrive at only a 98.97% accuracy, pretty good but not infinitely perfect: further evidence that at least part of the formation was created on the ground numerically rather than geometrically.

Crop formations seem to have these general characteristics:

• Linear structures such as squares and complex regular polygons are done without construction lines.
• Circles and arcs often have a length and center derived from pre-existing geometry, so they can be formed without the use of construction lines.
• Construction lines and the centers of circular or arc elements almost always lie within the laid crop, not outside the geometry.
• Outside the 'infinitely perfect' geometry, the small outlying circles almost always seem to lose a connection to the formal geometric part.
• The formations are done in stages, with smaller more complex parts being done after, and dependent on, the larger more basic elements.

In radio communications, it is the stable reference of the carrier wave that enables us to decode (or de-modulate) the modulation on that wave that contains the message.

Did we send our Aceribo SETI message out into the far reaches of outer space with error detection and correction elements? No mention of that in the Wikipedia article -- without adding error control bits, each pixel failing to make it to it's destination makes the message exponentially more unrecoverable. Given the distance/power product of this signal, the correction bits should probably have outnumbered data bits! No matter; the Aceribo message was just a photo-op anyway: it was only sent once, over a period of three minutes, to a star system that will have moved out of the beam by the time the message gets to it!

In contrast to the message we sent, the much more elegant Circlemakers

• include an elaborate verification system with almost every formation
• send multiple messages
• include time and location relationships in their communications

The Noise Figure of a crop formation is simply a single number that expresses the degree to which the flattened crop deviates from the design intent, and each formation will contain its own characteristic number. Non-geometric formation elements are not incorporated in the determination. The Noise Figure lets us determine the granularity of the formation, that is, the level of confidence that a measurement we derive from a feature therein is information, rather than noise. In which asci = area of standing crop inside the design intent, in pixels afco = area of flattened crop outside the design intent, in pixels adi = area of the design intent, in pixels

1. Scale, rotate and keystone (normalize) a circle aerial photo so that it most closely fits a pixel representation of the design intent
2. The most correct normalization will yield the smallest noise figure
3. By photo analysis, count the pixels that are not within the design intent
4. Divide by the number of pixels that are within the design intent

1. These formations are done with quantifiable precision.
2. By carefully measuring photographs taken with as high resolution as possible, we could determine the granularity of features in a given formation. This is the boundary which defines data versus noise, and is the first thing a communications system has to resolve before decoding can occur.

   I am not aware that anyone has yet done this kind of analysis, let alone even the data acquisition that would make it possible.

3. It would be a judgement call, but simply chosing a noise figure level, and discarding all formations above that level as hoaxes would be very simple to do. I suspect they would sort into two distinct populations, however, which would make the call more reliable.

   They must be measured to the precision dictated by the noise figure, or we have no hope to decode them.

Image analysis software is readily available.

Here is a crude normalized derivation of the 2000 Chilbolton formation, of which Zef Damen has done a phenomenal deconstruction of the geometric parts.

Chilbolton, August 13, 2000

Zef rightly omitted the small detached circles from his deconstruction, because they are obviously asymmetric, especially evident top-to-bottom. Because they are not checksum, they are likely message. To determine whether the left-right features are asymmetric too, I copied, rotated 180° and overlaid on the other half, in blue. While these are apparently asymmetrical along both axes, without a noise figure, we can't tell if the asymmetry is intentional or accidental.   Unfortunately, we don't have photos sharp enough to determine this.

Referring to the Koch fractal at the top of this page, note the phenomenal number of outliers that could serve as message.

The number of different outlier parameters may represent a modulation mode. Since outliers are part of a rough geometric pattern with a definable center, defining the deviation from a mean (rather than the absolute macro measurement) would drop out the geometric contribution to these parameters. This is analogous to subtracting the carrier wave from a radio signal. Diameter D Cartesian displacement X or polar displacement R[adius] Cartesian displacement Y or polar angle (alpha) Major to minor axis length ratio There are often even small 'tufts' of crop in the centers, the shadows of which can be used as precise reference points. Often, the lay center and the outline center are not concentric. This could be quantized, with all of the parameters above. There may also be information encoded in the direction and percentage of the lain crop as well as the outline. Clockwise? Counterclockwise?

Referring to our Chilbolton formation above, we find 2 sets of 13 3-outlier strings, or 78 circles. Multiplying by the 9 parameters above yields 702 information entities, each of which may possibly have the resolution of an ASCII character.

The Big Question Of course, no one is voicing the question will they farm us for food? But for anyone who sees a hierarchy of being extending up past our presumed position as the top of the food chain, it is never completely off the table. Indeed, the formation that appeared at Lockeridge, Wiltshire on August 6, 1998, "The Queen" so resembled a contented chicken that at least one croppie denounced it as a hoax, despite its flawless execution and beautifully lain crop.

In conclusion, the human race has been mainly just gawking at these formations, and this has to change. The tourist photos pilots have been taking of these structures so far are inadequate to decode them, and therefore the information content of much of these formations may be already lost.

What we immediately must do is insist that pilots who provide the crop formation research community with photos give us (sell us) images taken from directly overhead, with as high an altitude (to minimize existing geographic features and optical distortion) and resolution as humanly possible, not the artful isometric calendar shots, or the grainy internet jpgs.

Recommended reading:

Anything by Freddy Silva. While his book Secrets in the Fields is not recommended for anyone made queasy by every possible numerological or mathematical significance slung about like so much grapeshot, it is an eloquent, information-packed labor of love.