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Balls of Light at Hoeven?

An analysis of the Hoeven crop circle case: is there really any evidence that it has been created by a ball of light?
(October 2005)
[Updated June 2006]

by Paolo Russo

Those who think that not all crop circles are made with simple tools, such as ropes and wooden planks, cite some alleged anomalies that would be identifiable in "genuine" crop circles (whatever that word may mean) in support of their view. Stem node lengthening stands out among all anomalies, not only because it is one of the most frequently cited, but also because, according to the Dutch engineer Eltjo H. Haselhoff, a mathematical analysis of node lengthening would let us deduce its origin: a radiation emitted by some unspecified kind of ball of light (BOL).


Nodes and Lengthening

Nodes are small stem segments with particular features and look like rings whose color is different from the rest of the stem. They are usually a few (two-four) millimeters long. One of their functions consists in allowing the stem to bend, which is useful in some situations: the plant may bend towards the light and can rise again (partially at least) after it has been flattened to the ground. In these cases the nodes change their shape and bend the stem towards the desired direction. The shape change is actually a lenghtening of one side of the node. In the plant there is a hormonal control system that controls node lengthening and is sensitive to light (phototropism) and gravity (geotropism); both tropisms may get active in the case of a crushed plant.

Some crop circle researchers noticed long ago that nodes look very long in crop circles - in some of them, at least. Although this fact is often mentioned as an anomaly, it does not look very strange, since lengthening is a normal function of nodes and they look very long also in plants that have been flattened to the ground by wind, rain and bad weather in general. Somebody, such as Nancy Talbott of the BLT group (Burke, Levengood & Talbott, one of the main crop circle research teams), tries to cope with that observation by claiming that irregular areas of flattened plants, that peasants and agronomists ascribe to bad weather or excessive fertilization, are not due to these causes at all but to the same phenomenon that produces the geometrical formations, i.e. the real crop circles. This idea implies a few conceptual and methodological problems, the most important of them being the risk of circular reasoning: a circle is considered genuine because nodes are lengthened, lengthening is an anomaly because it occurs only in genuine circles... The basic thesis that node lengthening is an anomaly should be demonstrated some other way.

In 1994 Levengood published an article in the scientific journal Physiologia Plantarum [1]. The article was about various anomalies that would have been found in crop circles; one of them was node lengthening, of course.

The subsequent article published in 1999 by Levengood and Talbott [2] was entirely devoted to node lengthening. It reported the results of an experiment of corn bending with ropes and planks: after the bending, a progressive node lengthening was observed, about 2% a day. After a whole week the lengthening was still less than 20%, while in circles considered genuine the authors claimed to have measured lengthenings between 30% and 200% in just three days since the circle creation. Is this the so desired proof that the anomaly is anomalous? We'll see later on. The paper by Levengood and Talbott cited three crop circles (Devizes 1993, Chehalis 1994 and Sussex 1994) where node lengthening seemed to vary according to the distance from the center of the formation; longer nodes in the center, less long near the borders. The paper also suggested a plasma vortex mathematical model to address this dependence on the distance. The idea was that a hypothetical plasma vortex in the middle of the circle, besides bending plants with the air displacement, radiated electromagnetic waves and that these caused the node lengthening due to thermal dilation. Under this respect it may be the right time to discredit a rather widespread myth: Levengood did put some stems in a microwave oven, but never wrote in any of those articles that he could reproduce the node lengthening. On the contrary, he claimed he reproduced the enlargement of cell wall pits, microscopic structures that have nothing in common with nodes. If he ever reproduced node lengthening too, why would he have omitted so important a detail from his paper? Another misunderstanding too is rather widespread among crop circle enthusiasts: some snapshots of stems from crop circles show nodes that are not only lengthened, but bent as well, so there are people who think that some mysterious phenomenon acts on nodes, lengthening and bending them just in order to create the circle. Those snapshots actually show the longest nodes, which, according to Levengood's data, are the last ones (those highest along the stem) or the penultimate ones; they are not in the bottom part of the stem, where plants have been bent to create the circle. The tropisms activated by the plant after being crushed to the ground often cause the top nodes to bend: it is a well known phenomenon. In 2001 Haselhoff wrote a comment [3] to Levengood and Talbott's last article, published in the same journal, where he (correctly) criticized their analysis, pointing out a couple of serious errors, and put forward his own model: not a plasma vortex but a point-like or spherical source of radiation (fig. 1).

Figure 1.
BOL radiating the ground.


According to this model, node lengthening should be proportional to the inverse square of the distance r; thus, it should be maximum just under the BOL and decrease with distance, following a bell-shaped trend. Haselhoff used his model as a base for a new analysis of the data of the three formations published in the 1999 article. He then compared them to the data he collected in the Nieuwerkerk circle (aka "Dreischor circle"), whose human origin is commonly accepted, where he did not find the same mathematical trend.

All of these (questionable) papers and comments have been published in a peer-reviewed scientific journal, which is extremely rare in the field of crop circle studies. Node lengthening is thus the anomaly with the highest claim of scientificity. Therefore, it can be considered convenient to criticize those articles in the pages of a scientific journal. Is is just what Francesco Grassi, Claudio Cocheo and I did not long ago [4]. For this reason, the rest of this article will not deal with those three articles, which have been already analyzed, but with some other texts by Haselhoff, not published in any scientific journal.

The Hoeven Circle (aka Noord-Badant Circle)

Haselhoff wrote [5],[7] that a young Dutch man claimed he saw, at about half past one AM of June 7, 1999, a small pink light that moved in the air, then turned into an elliptic shape and slowly faded, leaving a crop circle of about nine meters of diameter, still hot (literally). The following morning the young man saw a second circle of about 60 cm of diameter near the first circle. Five nights later, the supposed eyewitness saw a flash of pale blue light coming from a single point; this time the witness found another circle of nine meters of diameter, still hot. The following day Haselhoff arrived and collected 25 plant samples from each of the major circles, two from the small circle and nine far from the circles as a control group. Every sample consisted of about twenty stems collected close to one another. The samples in the major circles were collected following a star-like scheme. The star of the oldest circle is composed by the "transects" (sets of samples collected in straight lines) A, B and C, while the second circle star consists of transects D, E and F (fig. 2). Transects A, B and C share the central sample (A4=B4=C4); transects D, E and F do too.

Figure 2.
Sampling scheme of the two major circles. The small circle (transect G) and control samples are not shown.


After allowing some months for the samples to dry, Haselhoff measured the nodes, computed their lengthening with respect to the control samples, compared the trend with his radiating sphere model and concluded that there was a significant match. In his book The Deepening Complexity of Crop Circles [5], Haselhoff supplies the data about the node lengths of transects A, B and C as colored bar charts (pp. 84-85). Those data are shown in fig. 3 in a similar form; for simplicity, I omit the standard deviations, not used in this analysis.

Figure 3.
Node lengths in transects A, B and C.


No graph about transects D, E and F is present in the book; their patterns are left to the reader's imagination. Haselhoff reports the result of a statistical analysis of the single transect B (p. 88): the maximum degree of matching between the BOL model and the data corresponds to a hypothetical BOL floating at 4.1 meters above the ground. The resulting Pearson coefficient (what it is will be clearer later on) is 0.988, a value which is very close to the unity that would be obtained if the match were perfect. Haselhoff wrote he also performed a similar analysis on the data of a surely man-made crop circle (Dreischor, 1997); he did not report the obtained coefficient, but he claimed it was "absolutely unsatisfactory" (my translation from the Italian edition, p. 89).

Haselhoff does not put forward any hypothesis about how a BOL can bend the plants; of course he does not claim that the BOL radiation is responsible for it, it would be quite physically implausible. Neither he specifies the nature of this radiation; he suggests that it might be a mix of infrared or microwaves and ionizing radiations (these last ones are called in to justify other alleged anomalies, p. 130).

The Enquiry Begins

I could find the following sources of information about the Hoeven circle and Haselhoff's analysis:

  1. Haselhoff's aforementioned book [5]. Published in several languages and editions, largely advertised in the site of the Dutch Center for Crop Circle Research (; ON-LINE UPDATE: in the last few months the DCCCS web site and other on-line sources of Haselhoff-related issues seem to have become unaccessible; the bibliography at the end of this article has been amended with alternative links where possible) and quite famous in the crop circle world, it can be certainly considered as the magnum opus of the Dutch researcher (i.e. his largest and most outstanding piece of work);
  2. some on-line articles [6] written by Haselhoff that just report his conclusions, but almost nothing about the data and calculations they should be based on;
  3. an on-line report [7] available in the DCCCS web site that describes in good detail Haselhoff's analysis of the Hoeven crop circle. There are at least two versions of this report; they differ just in the page layout and header. A version looks like a private communication to Levengood, the other version has a more generic header and looks like a real article. Apart from the version, this text is the most complete source I could find about the Hoeven crop circle and is the basis of my analysis.

Getting the Data

Unfortunately, data have not been supplied in numerical form, neither in the book nor in the report; it has been necessary to read them back from the bar charts. Of course, this procedure may have introduced some small errors of about one pixel, and the JPEG format certainly did not help (a GIF would have had sharper edges), but... this is the form chosen by Haselhoff to publish his data and this is all that anybody has at his/her disposal to verify Haselhoff's conclusions. I wrote to Haselhoff asking for the original data, but I never received a reply. The values I obtained are in tab. 1.

Table 1.
Node length in millimeters. "Co" = controls. The last digit is not significant but it has been included in the table for a scruple of perfect reproducibility of the analysis. A4, B4 and C4, which should coincide, give some clue about the error introduced while reading back the data from the charts. E4 and F4 are mere copies of D4.


It is evident at a first glance that transects D, E and F (fig. 4), present in the report but omitted from the book, look very different from A, B and C.

Figure 4.
In the original chart the samples of F are numbered 0, 7, 6, 5, [4], 3, 2, 1, 8, for unknown reasons. I assumed that the correct spatial order corresponded to the chart and numbered again the samples from 0 to 8.


The nodes in these transects are just a little longer than the control ones (which are about 2.054 mm on average) and there is no trace of the bell curve that should be typical of a BOL. That's probably why the Pearson coefficients of these transects are not even mentioned in the report. As opposed to the Hoeven circle, I did have the original data of the Dreischor circle, sent to me by Francesco Grassi, who received them from Haselhoff.

Method of Analysis

After obtaining the data, the next step consists in verifying the calculations.

In order to evaluate how a mathematical model fits the real data you have to compute the values predicted by the model and compare them to the real ones, calculating a unique number that summarizes all differences. This number is called "mean square error"; I will indicate it as S. The smaller is S, the better is the model. The "correlation coefficient" (or "Pearson coefficient") R is often preferred to S, because its value is always between -1 and 1 and it is easier to interpret: 1 means that there is a perfect correlation between the two data sets, 0 means that there is no linear correlation, -1 means that correlation is negative (the model seems wronger than a random one). The square of R, indicated as R2, is often adopted instead of R. Haselhoff uses R or R2 in his works. The Pearson coefficient lets you compare models using different data, but it has a disadvantage: a greater R (or R2) does not always imply a smaller S, i.e. a better correspondence between the model and the data. When different models are compared using the same data (in our case, the data of the same crop circle) it is better to look at S. This is the model proposed by Haselhoff:

y = k / r2 = k / (d2 + h2)

that is to say, the node lengthening (y) is assumed proportional to various factors (total emitted energy, degree of absorption by the vegetable tissue...) represented by a unique parameter k, and inversely proportional to the square of the distance r (this would be the "signature" of a spherical radiation source). The distance r between the BOL and the plant, according to Pythagoras' theorem, can be obtained from the distance d of the plant from the center of the circle and the height h of the BOL, which is supposed to be floating over the center of the circle (fig. 1). The problem is that we do not know h nor k, so what did Haselhoff do? He actually computed the values for h and k that a BOL should have had in order to produce the observed effects, or, more exactly, to produce the effects that are most similar to the observed ones (smallest S when comparing the model to the data). This procedure is named regression, it is widely employed in science and is completely proper; however, we must consider that, the more numerous free parameters a model has, the easier it is to adapt the model to any data set and - as a consequence - the less significant the result is from a statistical point of view. Therefore, some caution is mandatory when judging the results.

But how can these h and k be computed? If the model is simple enough you can do it analytically, i.e. working on the model with paper and pencil to deduce a formula for computing them. It is the best way, because it yields the exact solution. If the model is too complex you can go the number-crunching way, letting a computer try a (huge) lot of possible combinations of values of h and k and pick up the best combination. As a last resort, you can look for the best values by trial and error, by hand, with paper and pencil (and a calculator), which is the most uncomfortable and imprecise method. In the case of the BOL model I deduced a formula that computes the best k for any h and then looked for the best h with a computer.

Once the optimal values for h and k have been found, it is easy to compute S or R; but then? It is not so easy to understand how significant a value of S or R is in absolute terms; for example, if R=0.9, does this let us conclude that the data confirm the hypothesis? Or, better, that they demonstrate it? In order to have a touchstone I compared Haselhoff's model to the simplest possible two-parameter model:

y = ad + b

which is a trivial straight line. I think it is reasonable to demand that Haselhoff's model fits the data better than a straight line, otherwise I cannot really see how it can be claimed that the data yield an empirical evidence supporting that model. The comparison with the straight line is aimed at answering a question: is the BOL curve objectively deducible from the data, or is it just a subjective interpretation? Here I am not even requiring that the BOL model gives the best data fit among all two-parameter models; as a preliminary test, I just expect it to be better than the simplest one.

Data Processing

50 samples are available, 25 for each circle. Since the BOLs possibly responsible for the two circles might have had different parameters h and k, the two sets of 25 samples should be analyzed independently from each other. For each circle the optimal values for h and k and the resulting S and R should be computed. The method adopted by Haselhoff, on the contrary, seems affected by three serious errors:

1) The 25 samples of the second circles are ignored; they are not analyzed and not even shown in the book. It is not very difficult to understand the reason: their correspondence to the BOL model is very low. It is not allowable to ignore the data unfavourable to a theory.

2) The six most external samples of the 25 remaining samples are excluded from the analysis. Indeed, in every transect the samples from 1 to 7 were taken at regular intervals of one meter and a half. Samples 1 and 7 were taken near the circle border, while samples 0 and 8 were taken just out of the circle, as close to 1 and 7 as possible (that is what Haselhoff wrote in the report, without specifying the exact distance; my request for details remained unanswered). This sampling scheme is very good, since the direct comparison between 0-8 and 1-7 allows to understand whether the node lengthening abruptly ends at the circle border or goes on a little farther. The latter is what one would expect of an energy source with spherical symmetry: a slow decay following the inverse square of the distance. Data speak for themselves: the lengthening abruptly ceases at the circle border. This fact is not evident at a first glance in Haselhoff's charts because, strangely, they do not show the line of the control sample average length, which would allow for an easier comparison. Moreover, strangely, they are bar charts instead of Cartesian - as would be normal in these cases - and thus they do not show the real distances among samples. In the book, strangely, under any bar chart there is even a sampling diagram indicating wrong distances among samples: the distances between samples 0 and 1 and between 7 and 8 seem equal to all other distances. A few pages earlier (p. 81), thus harder to look up, there is a small diagram which is both three times smaller and less misleading than those under the charts, but probably still not scale-drawn. Curiously, the text in the book does not mention the difference in distances. There is just an enigmatic paragraph: "The sample a0 (20-25 stems) was collected at the edge of the circle; samples a1, a2 and a3 were taken at the same distance from one another, towards the center of the circle; a4 was collected at the center; a5, a6 and a7 in the opposite direction, off the center; a8 at the opposite edge of a0" (my translation from the Italian edition, p. 84). Haselhoff excludes all external samples from the analysis, in spite of their considerable relevance.

3) The remaining 19 samples are incorrectly processed. They are split up in three distinct sets of seven samples (transects A, B and C, with the central sample in common) and each set is analyzed independently, finding three different couples of parameters h and k. That would be fine if each set were supposed to have been caused by a distinct BOL, but in Haselhoff's model a circle is created by one BOL radiating one energy (k) at one height h, thus it is meaningless to perform three distinct analysis. Even if it were correct, the model fits only the data of B and C better than a straight line; much better for B, not so better for C. The seven internal samples of transect B are the only ones shown in Haselhoff's "popularizing" web pages [6]; however, the remaining 43 samples tell a different story.


Tab.2 reports the results of the analysis on both circles, with or without external samples (I assumed a distance from the internal samples of 30 cm), on single transects or putting them together.

Table 2.
The best values (smaller S or greater R2) are indicated in bold, the values corresponding to meaningless analyses are in italics.


The BOL model never works better than a straight line, except for the (meaningless) cases of transects B and C singled out and with the external samples removed. The minimum I can say is that the data do not support the BOL hypothesis. The data of A, B and C and the lines of the BOL and the straight line that most closely approximate the data are shown in fig. 5, so that it is possible to evaluate at a glance too how much the BOL "springs out" of the data.

Figure 5.
Graphs with external samples (top) and without them (bottom). Data of transects A, B and C are shown as dots, their averages as circles. The horizontal line represents the average level of control samples.


A few comments are mandatory. The values of R2 for A and C, as computed by Haselhoff, do not coincide with those in tab. 2, computed by me. The differences may be due to the data having been read back from the charts or to a more accurate optimization procedure for h and k; in fact, the values I obtained are more favourable to the BOL hypothesis than the original ones. The greatest difference is about transect C: R2=0.95 (h=3.3 meters) for me, R2=0.85 (h=6.6 meters) for Haselhoff. I too obtain a similar value (0.86) for h=6.6 meters, therefore I suppose that the researcher simply missed the best parameter combination, which I suppose he searched by hand, by trial and error.

It should be pointed out that the BOL analysis of the Dreischor data which have been published in the book yields a higher R2 (0.549 with external samples, 0.505 without them) than transects D, E and F. Therefore, if Haselhoff regards this value as "absolutely unsatisfactory", he should have a similar opinion about the DEF circle, but an explicit admission cannot be found in the book nor in the report. ON-LINE UPDATE: afterwards, Haselhoff claimed that the Dreischor data published in the book are incorrect because an incident mixed them up; he actually wanted to publish another data set, but made a mistake (more details here). Even accepting his point, the "correct" data set gives a lower R2 (0.410 with external samples, 0.312 without them) but still comparable to D, E and F.

Haselhoff remarks the symmetry present in any single transect, but that symmetry does not correspond to the BOL model. Haselhoff admits it in the report (and only there), suggesting that the model may be just an approximation of a more complex reality. Maybe, but a completely different reality may be responsible for those data as well. If and when Haselhoff will have a better model, we'll discuss it; currently, the claim that the data support the BOL model looks unjustified.

Conclusions and Hypotheses

At this point one may wonder what the collected data actually demostrate, assuming they are fully reliable. Not much, actually:

  1. nodes of bent plants are lengthened;
  2. the lengthening is correlated to the distance from the center of the circle;
  3. there is some symmetry, not perfect and not circular;
  4. the lengthening abruptly ends at the circle border.

Haselhoff introduces point 1 as a mysterious effect. He claims that tropisms can lengthen the nodes but not so much in so little time; at most 10-20% in a week (p. 84). The only source of this estimate is Levengood's plant bending experiment [2]. However, the occurrence of natural factors cannot be excluded on that basis; tropisms are known to be affected a lot by contingent factors such as light, wind, moisture, planting thickness [8] and more; Levengood's estimate was obtained in unknown conditions that might have been not optimal for the lengthening. Is there any evidence at all about this possibility? As of this writing, there is one: the Dreischor circle, made by people who got the field owner's permission, where Haselhoff found a lengthening of 30% in just three days. Strangely, the Dutch researcher cites Levengood but not his own data. Tab. 3 reports the maximum lengthening for Hoeven and Dreischor, the (claimed) time between formation and measurement and the ratio of those two numbers. It can be noticed that the computed lengthening paces are not so different as would be required for suggesting radically different mechanisms, especially considering that the Dreischor conditions are not known too.

Table 3.
Data about node lengthening in various formations.


Points 2 and 3 just demonstrate that the lengthening is affected by some factor that varies according to the position in the circle. Haselhoff writes that a plant cannot know where in the circle it is, but this claim is somewhat excessive. As an example, the wind intensity may change (bent plants closer to the border are less exposed); the wind might bend the plants that try to rise again and cause the nodes to lengthen again. Maybe whoever bent the stems stomped the central ones more than other ones, probably stimulating a greater reaction. The temperature of the plants or the soil might change; after all, horizontal stems intercept the light differently (in fact crop circles are clearly visible at a noticeable distance); since heat tends to flow from hot regions to cold ones, a gradient of temperature might form between the inner and outer areas of the circle. Even rain might be collected differently... there are lots of possibilities. As already noted, it is impossible to demonstrate the inexistence of unspecified factors.

Point 4 just proves that, if a BOL created the circle, it would not have radiated in all directions but just the circle, in a cone. That does not disprove the BOL hypothesis, but demolishes the idea of a BOL that bends plants with a mysterious power and additionally emits light and radiation in all directions just because of its very nature (because it is "of light"). One would be then forced to think that the emission is intentional, but for what purpose, since it cannot bend anything on its own? Just for lengthening the nodes and making people who measure them happy?

But there is another pretty serious problem. Haselhoff's thesis - which was also Levengood's - is that node lengthening is due to thermal dilation of the liquids inside the nodes (basically water). Here we have two possibilities. If the alleged heating just heated water a bit without turning it into vapor, there would be something wrong: the thermal dilation coefficient of water is far too low to justify the measured lengthening. On the other hand, if water boiled, the plants, that cannot bear more than about 70 degrees centigrade, would immediately die. It is useful to remember that Levengood, who put plants into a microwave oven, neither wrote that nodes lengthened, nor that the plant did survive, nor of course that both things ever occurred at the same time. In fact, somebody who tried reports a very different outcome [8], although an ultimate conclusion cannot be reached, since it is impossible to test all possible conditions: as already pointed out, a negative assertion cannot be demostrated on these bases.

Anybody can draw his/her own conclusions. Mine are that the Hoeven circle data do not support the BOL hypothesis and do not disprove more "traditional" possibilities such as the use of strings and planks.



  1. Levengood, W. C. 1994. "Anatomical anomalies in crop formation plants". Physiologia Plantarum 92: pp. 356-363.
  2. Levengood, W. C., Talbott, N.P. 1999. "Dispersion of energies in worldwide crop formations". Physiologia Plantarum 105: pp. 615-624.
  3. Haselhoff, E. H. 2001. "Opinions and comments on Levengood WC, Talbott NP (1999) Dispersion of energies in worldwide crop formations. Physiol Plant 105: 615-624". Physiologia Plantarum 111: pp. 123-125.
  4. Grassi, F., Cocheo, C., Russo, P. 2005. "Balls of Light: The Questionable Science of Crop Circles". Journal of Scientific Exploration (19) 2: pp. 159-170.
  5. Haselhoff, E. H. 2002. La natura complessa dei cerchi nel grano, Reggio Emilia: Natrix Edition.
    In English: The Deepening Complexity of Crop Circles,
  6. Typical "popularizing" web pages, in English and in Italian:
    Haselhoff, E. H. Scientific Studies Confirm: Crop Circles Made by "Balls of Light".
    Haselhoff, E. H. GLI STUDI SCIENTIFICI CONFERMANO: i Cerchi nel Grano sono creati da "Sfere di Luce".
  7. Haselhoff, E. H. 1999. Node Length Measurement.
    also as: Haselhoff, E. H. 1999. Hoeven Report.
  8. Campaniolo, M. Cum Grano Salis. (In Italian)

Paolo Russo
Programmer, develops system and real-time software.
He is member of CICAP Friuli Venezia Giulia and, as could be argued, of the CICAP Study Team on Crop Circles.

Article published onto Scienza & Paranormale (S&P N. 63 - Anno XIII - Set/Ott 2005).

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