EVOLUTION AND GEOMETRY

Early Evolution

The history of the crop circle phenomenon might well have started as long ago as August 1678, according to two sources: a book entitled "Witchcraft in Hertfordshire", and a leaflet dated 22nd August, 1678, headed "The Mowing-Devil: Or, Strange News out of Hartford-Shire". These tell how several witnesses saw a round flaming object descending into a field of oats, which was seemingly set on fire. On the next morning, instead of finding fire damage, the farmer saw an area of "cut down" plants in the shape of something which might have been a ringed circle.

Rare verbal accounts were handed down from the 1890s up to the 1960s about circles in cornfields in Britain, Ireland, the mainland of Europe and further afield. Even an aerial photograph is known to exist (but unpublished) which shows an equal-sized pair of circles in a field close to RAF Tangmere, near Chichester in West Sussex, and taken in 1943. However, the more recent history of crop circles, which is now regularly reported on and published, begins in 1978 when the first quintuplet was photographed in a wheatfield near Headbourne Worthy in Hampshire. From that year until 1986, single circles and complex groupings of circles were registered in that area. Although the number of crop circle events increased during these eleven summers, the variety of designs was limited to just five discernibly different types (as shown in fig. A from left to right):



fig. A


the single circle,

the quintuplet (a group of five circles arranged symmetrically like the five dots on a dice. Usually, the central circle was considerably larger than the four outlying, equal-sized, circles),

the pair of unequal-sized circles,

the triplet (Three circles placed in a line. Also here the circle in the middle used to be larger than the two outlying, equally sized, circles.) and

the pair of equal-sized circles (Normally these circles were not larger than the outlying circles of the triplets and the quintuplets.).

The first main sighting of a quintuplet was in 1978, although a single earlier example was seen being formed by whirlwinds in 1946. The first main pair of unequal-sized circles followed in 1979 except for one reporting in 1945. The triplet and pair of equal-sized circles came in 1981, although a possible triplet is said to have been found in the early 1940s, and a pair of equal-sized circles has been photographed in 1943, as mentioned above. For a complete account of all the crop circle events before 1978, which includes some exceptions from that quite repetitive rule of crop circle evolution, see "The Secret History Of Crop Circles" by Terry Wilson, published by the Centre for Crop Circle Studies in 1998.



Tangential Geometry

A straight line is called a tangent when it touches a curve and does not cut it. In connection with crop circle formations, a straight line that touches at least three geometrically significant points of the formation, such as centres and edges of circles or corners of rectangular features, is also called a tangent.



fig. B


In fig. B it is obvious how four straight lines running through the centres and edges of the two larger circles and the oval become a tangential frame for the placing of the four smaller circles. Two additional straight lines, tangents in the classical sense, and running parallel and vertical to the tractor's tracks and crossing between the oval and the large circle, signify the connection between the whole formation and the tracks.



fig. C


In fig. C a pencil of tangential rays forms a grid below the circular components of the formation in which the human-shaped figure is positioned. (This figure is a common design named "Indalo" and is adapted from a Neolithic rock carving found in a cave named "Cueva de los Letteros" near the village of Velez-Blanco in Almeria province in southern Spain.) The crop circle formation displayed in fig. B was found in a field of Barley on the Baltic Farm near Bishops Cannings, Wiltshire, Great Britain, on the 12th of June 1992. Fig. C shows a formation allegedly made by Douglas Bouwer and David Chorley on the 21st of July 1992 on Hyden Hill, Hampshire, Great Britain.



Pentagonal Geometry

Pentagonal geometry is a development of the tangential geometry mentioned before. Two tangents crossing at an angle of 36 degrees can easily be completed to describe a pentagram by adding three more tangents which cross the first two lines at angles of 72 and 108 degrees, respectively. Fig. D shows how two tangents touching the perimeters of the crop circles and the circular patch of bare earth around the pole of the powerline can be added up to a pentagram, which relates the formation to the tracks (the single black line close to the curved track stands for a drainage channel).



fig. D



fig. E



fig. F



fig. G



fig. H



fig. I



fig. J



fig. K



fig. L



fig. M



fig. N




Figs. D to N all display the role of pentagrams and their circumscribing circles in the geometrical construction of a crop circle formation. Figs. G, H, I, J and L also show the connection between the pentagonal grid of a crop circle formation and the adjacent tractor's tracks: Some of the circumscribing circles run into the tracks on opposite sides of the circle, so that the circle is spaced between the tracks, thereby defining position and size of the formation. Furthermore, figs. F, G, and K display the pentagonal frame's orientation towards north.

Addendum to fig. F: The lower circle circumscribed by two pentagrams is not a part of the original formation. It is included in this drawing to give the "master diameter" from which all pentagonal ratios of the formation can be derived.



Barbury Castle, Wiltshire, GB, 17. 7. 1991

Found in a gently sloping field north of an Iron-Age hill fort, this pattern was the most complex of all found up to that date. In fig. O it can be observed how the formation is also constructed and positioned with the help of an underlying pentagonal frame, which consists of pentagrams with their lines crossing on the perimeters of the crop circles. The whole frame is precisely fitted between two tracks approx. 133 metres apart. North is defined by one side of the largest pentagram, and the length and distance from the centre of one of the six short straight pathways of the segmented spiral (the pathway that lies just outside the large pentagram's diameter) defines a ring with the same radius of the formationŐs other two outlying rings.



fig. O


With the arrival of this formation appeared the obvious possibility of interpreting the designs and their geometry as projections of three-dimensional bodies. At first glance (and disregarding pentagonal geometry) the design's central circle, and rings in reference to the outlying circles and their centres, display the proportion of a circle inscribed within an equilateral triangle. However, they also describe the ratio of tetrahedrons and their inscribed spheres. Although the formulae for both proportions cannot be derived from each other directly, they give similar results when filled with the same numbers. (For a complete deduction of both formulae from the Barbury Castle crop circle formation see W. Schindler: Mathematische Harmonie, in "Spuren im Korn", edited by J. Krönig 1992.) Using the radius found in the wheatfield to calculate the difference between the results for the sidelength of the circumscribing triangle and the tetrahedron would be fifteen centimetres, which is only a third of the width of the formation's thinnest pathway. Therefore the difference is not even visible in the scale of the formation. Fig. P displays the derivable tetrahedrons and their inscribed spheres seen from the front, above and the side (f. l. t. r.). The given distance [1] is the unit used for both formulas and is in fact the radius of the two outlying circles of the formation.



fig. P




Milk Hill, Wiltshire, GB, 16. 7. 1992

This design was probably the only crop circle formation constructed with the help of an invisible "tracks-related" pentagonal frame discovered in the summer of 1992 (and seemingly also the last one until Roundway Hill, Wiltshire, GB, 23. 7. 1995 appeared, see fig. L). Fig. T shows the formation and its underlying grid.



fig. T


The shape does not fit into its framing grid precisely: The diameters of the corn circles are approx. three to seven percent larger than the circles inscribed into the pentagrams of the frame, which itself deviates slightly from its correct geometrical construction (fig. U). Whether this was intended, or was a result of aligning the design with certain features in the surrounding landscape, is difficult to say.



fig. U


These alignments are described in figs. V and W. A straight line running north through the centres of three circles is a tangent to the Avebury Stone Circle and Silbury Hill and runs through the Swallowhead Spring. A second line running through the centre of the formation's small middle circle and touching the thinner curved path, connects the Alton Barnes White Horse, which is located 800 metres north-east of the formation, with a bridge that crosses the Kennet & Avon Canal and lies 1600 metres away from the formation to the south-west. Seemingly that bridge, which is named "England's Bridge" in the Ordnance Survey Pathfinder Map 1185, played a role for the choice of the formation's position. A line drawn from the bridge through the centre of Silbury Hill, 6600 metres further north, also runs through the two central stones of the northern inner stone circle of Avebury. This line crosses the connection between the bridge, the formation and the chalk horse at an angle of approx. 36.5 degrees, which is very close to the angle of a pentagram's tip (36 degrees).



fig. V


A coincidence worth mentioning is the similarity of the thinner curved path (in which the two lines that connect the formation with Silbury Hill/Avebury, the Alton Barnes White Horse and England's Bridge cross at an angle of 45 degree) to the very part of the Kennet & Avon Canal which is crossed by EnglandŐs Bridge. If a transparent drawing is laid over a map that displays this part of the canal, in a way that the path and the waterway curve both have the same size, the cross-point of the two lines that run through the curved path lies on the position of EnglandŐs Bridge. So probably the thinner curved path was meant to stand for that part of the canal (if so, the scale in which it was shown in the wheatfield was approx. 1: 44), thereby giving a reference both to the position of the design and to its internal geometry.



fig. W




A Sequence

Almost all existing crop circle formations were separated from each other by the relatively large distances between them, or at least by the borders of the cornfields and other dividing marks in the landscape. Two or more crop circle formations within a single field were quite rare and therefore it might seem plausible that the arranging of the isolated shapes in logical sequences was not intended by the unknown artist(s) behind the phenomenon. Still, there are clues which make the idea of linking the formations more reasonable. All the designs in fig. X are shown in the same scale. They were found in Hampshire and Wiltshire between 1990 and 1992 and, although scattered over a large area during three successive summers, they have several features in common that other crop circle formations do not have. The circles' sizes of different formations relate to each other; also features like two or four parallel bars, thick and thin rings, one-, two- and three-fingered flags and smaller satellite circles are repeated throughout the different formations. Most of the shapes can be put into groups of three according to similarity. (The total number of formations put into this sequence is twenty-seven, which is the product of three to the third power.) Finally, most of the patterns are constructed by the use of pentagonal geometry or at least display a discernible reference to it.

All this allows the assembling of a sequence that, seen from left to right, seemingly describes the transformation of a relatively simple pattern into another, quite different one, through various stages of growing and diminishing complexity in which circles are added or merged into each other.



fig. X


In fig. X nearly all formations are standing "upright" on this page, which means their top was close to the nearest field border. The lines drawn between the figures make clear which circle conforms to which in the next formation. Small arrows between the figures indicate whether a formation came after (>) or before (<) its neighbour on the left side.

The whole sequence can be repeated endlessly by connecting the first formation on the left (Chilcomb Down, Hampshire, 23. 5. 1990) to the last formation on the right (Hazeley Down, Hampshire, 3. 8. 1990). The diameters of the large circle in the Chilcomb Down formation and the ring in the Hazeley Down formation are almost the same, also the width and the position of the four small curves included in the Hazeley Down formation corresponds with the four parallel bars in the Chilcomb Down formation.



fig. Y


The Hazeley Down formation of August 1990 appeared in the same field as the third formation seen from the left (Hazeley Down, Hampshire, 2. 6. 1990). They were just approx. 35 metres away from each other and connected by a track left earlier by a tractor. To express this in fig. Y the sequence is doubled, so there is another reversed endless row below the first and the two formations are linked by vertical lines.

As mentioned before, it is not clear whether sequencing of several formations was intended by their author(s) and, even if it were, nobody can imply that the sequence displayed here is the right answer to the puzzle. Since there is no obvious translation for a single crop circle formation, combining twenty-seven of them into a sequence and interpreting this sequence as meaningful is pure speculation.

The formations included in the sequence shown in fig. X are (f. l. t. r.):

Chilcomb Down, Hampshire, 23. 5. 1990,
Litchfield, Hampshire, 23. 6. 1990,
Hazeley Down, Hampshire, 2. 6. 1990,
Chilcomb Down, Hampshire, 6. 7. 1990,
Longwood Estate, Hampshire, 28. 6. 1990,
Crawley Down, Hampshire, 13. 7. 1990,
Lockeridge, Wiltshire, 21. 6. 1991,
Ogbourne Maizey, Wiltshire, 11. 7. 1991,
Alton Barnes, Wiltshire, 2. 7. 1991,
Avebury Trusloe, Wiltshire, 9. 6. 1991,
Froxfield, Wiltshire, 19. 8. 1991,
Hackpen Hill, Wiltshire, 12. 7. 1991,
Alton Priors, Wiltshire, 18. 7. 1991/Clatford, Wiltshire, 14. 8. 1991,
East Kennett, Wiltshire, 27. 7. 1991/Clench Common, Wiltshire, 13. 8. 1991,
Preshute Downs, Wiltshire, 17. 7. 1991/Silbury Hill, Wiltshire, 27. 7. 1991,
East Kennett, Wiltshire, 27. 7. 1990/East Kennett, Wiltshire, late July 1990,
Milk Hill, Wiltshire, 12. 7. 1990/Milk Hill, Wiltshire, mid-July 1990,
Alton Barnes, Wiltshire, 12. 7. 1990,
Milk Hill, Wiltshire, 16. 7. 1992/Morestead Down, Hampshire, 30. 6. 1990,
Westbury, Wiltshire, 7. 8. 1990,
Hazeley Down, Hampshire, 3. 8. 1990.


© 1999 wolfgang schindler

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