Posted on 9 September 2015 by Anton Alipov

Last edited on 20 December 2015 by Anton Alipov

Categories: Art, History

Contextual Analysis of Voynich Objects - Part I

1. Introduction

This is the first of several posts dedicated to the contextual analysis of some of the Voynich Manuscript objects. As I write these lines, the scheduled research is far from being finished, so "how the subject theme may gang, let time and chance determine"; although a certain research plan, of course, lies before me.

Let me, in the first place, explain what I mean by the "contextual analysis of the Voynich objects". This is nothing groundbreaking, but just an expression of a certain research approach. By "Voynich objects" I mean various graphical (like images) or textual (like text patterns) objects that have been identified, or (which is, alas, much more common) are yet subject to identification, in an attempt to reverse engineer the VMS script. For example, there are plants, stars, animals, geographical places etc. There are also sequence-like character patterns (like in f49v) or right-margin-aligned text patterns (like in f1r) etc. 

Much of the Voynich research is aimed at identification of Voynich objects, especially the graphical ones. Traditionally, the approach to identification thereof has been to seek for a direct match against natural objects (such as plants or celestial bodies) or against patterns contained in other manuscripts. Unfortunately, very few objects have been more or less reliably identified by this method, and they still failed to help us in the reverse engineering process, because the VMS author was careful not to label obvious objects (like Sun, Moon or the "cloudband" pattern). It is sufficient to say that none of more than a hundred Voynich plants has been reliably identified up to date.

In contrast to this "direct match" approach, the "contextual analysis" approach tries to take advantage of the fact that if the VMS is meaningful text discussing objects therein depicted, then discussions of those objects are most probably not randomly scattered across the corpus, but are in some way interrelated, structured, and existing in a single context. So far that we cannot directly reveal that context because we do not understand Voynichese, we are left with indirect methods of revealing it: statistical analysis and comparative analysis.

Statistical analysis tries to discover trends of how Voynich objects are discussed and mentioned, and what affinity they exhibit to each other - hence, what for they are mentioned. Comparative analysis - by means of comparison with (supposedly) thematically similar literature - tries to suggest or predict how Voynich objects should be discussed and mentioned.

Revealing the context in which the given object is mentioned or discussed delivers additional information about that object and thus may greatly assist in the object's identification.

In this article we speak of object names' occurrences throughout the VMS as of "applications" of those objects. Among those applications are: botanical section, Zodiac section, balneological section etc. For example, if stars are systematically mentioned in the botanical section, we recognize that stars have some particular botanical application, albeit of nature yet unknown to us.

The planned series of posts deals with two groups of Voynich objects: plants and stars. The first two posts in the series utilize statistical analysis; the third one, as I expect, shall explore comparative analysis.


2. Hypotheses

As usual with the VMS, this investigation is impossible without some initial hypotheses. Hypotheses adopted for the sake of this article are listed below.

H1. The VMS is a meaningful text and not a random text hoax

This is the fundamental hypothesis, the invalidity of which would just make all the below research meaningless. There has been research showing valid ways to produce Voynichese-like hoax, such as works of Rugg [1] or Timm [2], but recent articles of Cham [3] and especially Montemurro & Zanette [4] strongly suggest that the VMS is not random text but a meaningful message. So hypothesis H1 is very reasonable.

H2. The Voynichese labels represent words (or combinations thereof), and not just parts of words

This is an essential hypothesis; its invalidity would render useless the statistical analysis performed herein. However, the hypothesis is at least natural to propose: an idea that the labels are not directly related to the labeled objects and constitute fragments of a mosaic of longer words connected under some complex rule - such idea is a bit complicated and suffers under the Ockham's razor.

H3. A given Voynichese "word" represents the same plain text on any page

This is a hypothesis as essential as H2 above, but less reliable, because we are not sure that each folio is not encoded in its own way. However, H3 is supported by the work [4] which detects systematic affinities between Voynichese "words" throughout the whole VMS.

H4. The Voynich "botanical section" folios discuss plants

This may seem obvious, but in fact it is not. Supposing the enciphered text, the writer who managed to develop such a cunning cipher (and evidently with some good reason to conceal his plain text) would certainly possess enough imagination to additionaly mask the real subject of the text with irrelevant images - presenting the VMS as an innocent herbal, for example. Again, ceteris paribus, the Ockham's razor suggests the discussion of plants; so H4 looks at least reasonable.

The validity of this hypothesis does not affect the results obtained in this article; rather, it may affect deductions built on top of those results. 

H5. The Voynich "stars" are celestial objects indeed

This hypothesis deals with the left and central pages of f68r, representing what looks like star charts or maps. The labeled objects contained therein and resembling stars are considered as "Voynich stars" for the sake of this series of posts. (Of course, not only literal "stars" are covered, but celestial bodies on the whole - including planets or even constellations, such as decans; we use the term "stars" for brevity). These two diagrams are subject of intensive discussion and attempts to identify the stars. The blog of Prof. Bax is a good example of such community discussion.

The look of the diagrams suggests the labeled objects to be "stars" as the first choice, indeed, - mainly because they are depicted in the company of the (obvious) Sun and Moon. Nevertheless, we must admit that there are aspects that do not support H5. For example, in contrast to f68v,  there is no Earth in this allegedly cosmological picture. Next, the supposed celestial sphere is not represented by the cloudband pattern (again, in contrast to f68v). This suggests that the diagrams may represent something different from the celestial sphere and the labeled objects - something different from celestial bodies. For example, they might represent certain "spirits" or "angels". David Jackson makes (below in the comments) an interesting suggestion of a lapidary.

One thing we can say for sure: these diagrams present a system of objects that is relevant to the subject of the corpus. Granted that H2 and H3 above are true, this follows from the fact that many of the labels occur only once in the VMS - that is, in f68r itself. It does not make sense to include objects that do not relate in any way to the subject of the VMS, hence they are included solely to present the complete picture of some system of objects that does relate to the subject.

In fact, what are those objects, - are they stars, spirits or otherwise, - does not affect the validity of the statistical results. As in the case with H4, this would affect only deductions made out of those results. If such deductions based on H5 come to a deadend, this would be a good reason to re-consider H5. There is a good test, however, for H5, which I would like to cover in the next section, before we move to the main subject of this article.

H6. The left and central pages of f68r represent a "full" set of stars

This means that the set of stars in the two diagrams is complete and no other diagrams in the VMS add to this set of stars. This is not obvious, and in fact some researchers (such as, e.g., S. Bax) consider the right page of f68r to also contribute to the set of stars. Until f68r is decoded, I am afraid, H6 will stay undecided. However, there are some observations against f68r3 contributing to the full set of stars. First, in contrast to f68r1 and f68r2, it does not have a text header - which in the two mentioned pages looks like serving the purpose of introduction into the nature of the depicted sets. Second, the general composition of f68r3 does not look like f68r1 and f68r2, casting it out of the row: the circle is divided into sectors, there is no Sun, Moon is positioned in the centre. All this is conceptually different from f68r1 and f68r2.

Invalidity of H6 would not affect absolute figures of our statistics, but would affect relative figures: for example, when we speak of percentages or discuss concentration.


3. The mysterious star of "otol"

The diagram in f68r1 contains the star labeled as otol. It is the rightmost star in the diagram, and it is the most frequently mentioned star in the whole VMS (see Section 6 below). What is more interesting, we meet otol in some remarkable contexts throughout the VMS.

First, we meet it as early as in f1r - and not in the regular flow of the text (which would be nothing more than just statistically common for an "important" star frequently referred to), but in the righ-margin-aligned text block in line 18 (the last line of paragraph 3), which reads otol daiiin. The word "daiiin" occurs 18 times in the VMS, but in no case except f1r it is preceded by a star name.

We do not know what is the designation of these right-aligned patterns. They have been proposed to represent section titles (the section description being subject of paragraphs of f1r, thus f1r serving as a "table of contents"). Another possibility is that they represent page numbers in the table of contents. Within the framework of the "section title" hypothesis, a star name used in the generalized decription (which section titles would usually be) deserves attention. What word could be used as a star name and, at the same time, as some generalization? The Arabic "Kochab" (which literally means "star") comes to mind; are there any other options?!

Moving to f55r, we find, in the last line of that folio, another remarkable phrase: ockhy daiin otol, where ockhy is another Voynich star. So we have two star names separated by daiin. Of course, we do not know whether some punctuation mark is not supposed here in between.

otol is not uncommon in astrological folios: beside f68r, it is twice encountered in f68v (right page), then in f68v (left page), f69v (central and right pages). It is introduced several times in the Zodiac pages, among those in the "grey Aries" page in "otol ypsharal" - the label for the star held by one of the persons.

So far we considered only exact matches, but there are also some "star labels" in the Zodiac pages beginning with "otol": like otolaiin in grey Aries or otolchd, otolchey and otoloaram in white Aries; also there are some "person labels" beginning with "otol": like otolam in Gemini or otoly in Scorpio. In addition, two star names in f68r also begin with otol - that is, otolchedy and otolchcthy.

Back to exact matches, we find even more striking appearances of otol - in the visual contexts seemingly unrelated to the star topics. See Fig. 1.

Fig. 1. Appearances of "otol" in unexpected visual contexts: f77r (left), f77v (centre), f102r (right)


So otol appears to be a widely used notion, suitable for the following applications:

  • serving as the name of some quite important celestial object;
  • possibly also serving as morphological part of other star names;
  • possibly serving for a certain generalization;
  • serving for description (direct or by means of association) of a number of unidentified objects: anatomic/distillation (?), leaf/root of some plant (?);
  • possibly serving as morphological part of description (direct or by means of association) of "personified" objects in the Zodiac pages.

What celestial body (Sun and Moon excluded) in what language could serve so many purposes at once? If a positive answer to this question is given, then that would be the confirmation of H5, let alone a huge advance in the Voynich research. If no valid candidates are discovered, it would be worthwile to re-consider H5 and look at some other system instead of "stars".


4. The two groups of Voynich stars

The Voynich stars are grouped in two diagrams: in f68r1 and f68r2, respectively. The stars are not duplicated between these two diagrams; the set of stars in each of the diagrams is unique. This suggests that all stars are grouped in two distinct sets, by the criterion unknown to us. In the first diagram, Sun is depicted at the top, and Moon at the bottom; while in the second diagram, vice versa, Moon is at the top and Sun - at the bottom.

Various suggestions have been proposed as to the meaning of the two diagrams; but we do not touch this question. Our research only needs to take into account that two sets of stars do exist - and to distinguish between them. Whether the true difference is related to day/night, north/south, equinox or solstice, we simply take the terminology of "day" and "night" which, to make it clear, we use only for the sake of formally referring to one of the two sets of stars. Since in the first diagram the Sun is "dominating", let us call these stars "dayside" stars. Respectively, we call the stars in the second diagram "nightside" stars, because Moon is "dominating" there.

The dayside diagram contains 29 stars, all of which are labeled. The nightside diagram contains 59 stars, of which 24 are labeled and 35 are unlabeled. The stars do not seem to be arranged in any particular order, except for the ordered outer circle of 23 unlabeled stars in the nightside diagram. The nightside diagram also suggests another (inner) concentric circle consisting of both labeled and unlabeled stars, but it is "spoilt" by some stars residing not exactly on the circumference and thus making the evidence of the second circle less apparent.

Both diagrams have a central star, which size is larger than the size of any other star in the diagram.

Different stars vary in size and in paint: some are left unpainted, some are ochre-painted (as all dayside stars are); some have a little circle in the centre which can be either ochre-painted or outlined and unpainted. Two of the dayside stars have a black circle in the centre - but that can be attributed to blots in unpainted outlined circles.

For sure, the described peculiarities of the two diagrams are not occasional and have some special meaning. I remember reading that celestial bodies had certain characteristics in medieval astrology; however, we do not touch that. Moreover, for the sake of this post we are interested only in labeled stars, because what we would actually do is finding those Voynichese words throughout the corpus.


5. Methodology

The Voynich Query Processor resource (hereinafter - "VQP") was used to obtain frequency counts and to track Voynich words occurrences. Unless explicitly specified otherwise, only exact matches were accounted for. Unfortunately, VQP suffers from some transcription errors. Those errors were checked and dealt with in the following way.

  • Only the occurrences reported by VQP were subject to check (if any). If VQP misses any occurrences, those fall completely out of the stastistics reported herein - unless, of course, those missed occurrences were not occasionally detected in the course of data collection.
  • Voynich star occurrences reported by VQP were generally manually checked across the botanical section (because this article has some special interest in the botanical folios - see Section 10 below and further); occurrences beyond the botanical section (except, of course, f68r itself) were generally not checked, and thus were simply trusted.
  • If the reading of a Voynich star name or the occurrence thereof, as reported by VQP, raised doubts, then the decision was made whether to accept the VQP reading or to correct it; all cases of such correction are listed in the provided dataset file in the ways that are described below in the relevant Sections.

It is important to note that various web resources use slightly different folio numbering. Unless explicitly stated otherwise, the VQP notation is used in this article. This would be convenient for readers who would like to follow with VQP the observations discussed herein.

Unfortunately, VQP does not provide the opportunity to search through the "Rosettes" folio. Hence this folio falls out of our statistics, notwithstanding that it does contain certain star names, such as otol or odaiin.


6. Star frequency

We begin with observing the general frequency count of Voynich stars.

The data retrieved from VQP and used throughout this article is summarized in the attached MS Excel file. The frequency count for dayside and nightside stars is represented in the "Dayside" and "Nightside" sheets of the file, respectively. (Other sheets refer to further Sections of the article and are explained therein). In each of these two sheets, the first column contains the name of the star, the second column - its total frequency count, the third column gives any side notes that I considered appropriate, the fourth one lists all appearances of the given star in the botanical section of the VMS (we will need that below).

As noted in Section 5 above, some errors in VQP were corrected while gathering data. First, if the VQP reading of the star name in f68r was deemed wrong, it was corrected to what I consider the correct reading. Such star names, in their correct reading, are marked with red font in the discussed sheets. The frequency count in column 2 was retrieved against this proposed correct reading, not against the VQP reading. There are three differences in reading both for the dayside and nightside sets. All differences in the dayside set affect the frequency count; while in the nightside set none of the differences affects the frequency count. 

Second, if some occurrence of a star name in the botanical section reported by VQP was deemed inexact or erroneous, this occurrence was deducted from the frequency count reported. Column 2 gives the count thus corrected, which circumstance is marked with red font in respective cases. Those occurrences that were thus excluded from consideration are specifically listed in column 5, so that the reader may ensure that they are not appropriate exact matches indeed.

The raw results of the frequency analysis are as follows. 53 Voynich stars are mentioned 414 times, of which 260 times (or 62,8%) are for 29 dayside stars and 154 times (37,2%) are for 24 nightside stars. Since dayside stars count for only 54,7% of the total number of stars, but are mentioned in 62,8% of the cases, we may conclude that they are more "important" than the nightside stars. For the sake of brevity, let us introduce the formal term of a star's "importance": the more frequently the star is mentioned, the more "important" it is. Let us, then, consider the relative importance of stars - both broken down by sets and in total.

The most important dayside stars are:

  • otol (32,3% of all dayside star occurrences)
  • otor (17,3%)
  • olor (11,5%)
  • okeor (8,5%)
  • ockhy (5,0%)

Other dayside stars exhibit less than 4% of occurrences each.

The most important nightside stars are:

  • odaiin (39,6% of all nightside star occurrences)
  • dchol (15,6%)
  • okchor (12,3%)
  • chodar (9,1%)
  • shdar (5,8%)

Other nightside stars exhibit less than 4% of occurrences each.

The most important stars overall are ranked as follows:

  • otol (20,3% of all occurrences)
  • odaiin (14,7%)
  • otor (10,9%)
  • olor (7,2%)
  • dchol (5,8%)
  • okeor (5,3%)
  • okchor (4,6%)
  • chodar (3,4%)
  • ockhy (3,1%)

Other stars exhibit less than 3% of occurrences each.

Thus 17% of dayside stars provide 75% of dayside occurrences, 21% of nightside stars provide 82% of nightside occurrences, and 17% of all stars provide 75% of total occurrences. This corresponds well with the Pareto principle.

Many stars are mentioned only once - that is, in f68r itself. Let us call such stars "useless", because nowhere else in the corpus the author did feel the necessity to refer to them. As I noted above, most probably they are included "to complete the picture" of some system. Respectively, we will call stars mentioned beyond f68r "useful". It turns out that 47,2% - almost a half! - of all stars are useless. Broken down by sets, 34,5% of dayside stars and 62,5% of nightside stars are useless. Hence not only are nightside stars less important - they are also more useless (or less useful, if you wish). It is also worth noting that both dayside and nightside central stars - ofcheor and oydchy, respectively, - are useless.


7. Morphology of star names

We are not going to delve deep into the morphology of the Voynich star names, just make two observations. The first one is the old and common observation that most star names begin with EVA "o". Indeed, 79,3% of dayside star and 66,7% of nightside star names begin with "o". In total, 72,2% of all star names begin with "o". Other starting EVA characters include "c", "d", "t" (both dayside and nightside), "y" (only dayside) and "s" (only nightside). As for "t", the nightside todaraiily has a common "t", while the dayside toeeodcthy begins with that strange form of the "gallows" glyph which has its right leg connected to another glyph towards the end of the word.

Interestingly, those stars that do not begin with "o" exhibit some weak clustering in the charts of f68r, especially in the nightside chart. Refer to Fig. 2, where those stars are emphasized with white circles. The clusters are meant to be where you can "reach" a star from another star in "one hop". For example, cholar is adjacent with dcheoldy. Perhaps we will return to the predominance of EVA "o" as the starting character of the star names in later articles of this series.

Fig. 2. Positioning of stars beginning with characters other than EVA "o"


The second observation considered in this Section is that stars that have shorter names seem to appear more frequently in the corpus. Let us check if it is so. Of course, the EVA-based transcription does distort the "real" name length, but since we have nothing else, then let us take it as the first approximation of the "scribal complexity" of the star names. The lengths of the names, measured in EVA characters, are listed in column 6 of the sheets. Mean length for dayside is 6,4, for nightside - 7,2, overall - 6,8. Both dayside, nightside, as well as the overall distribution, have their main modes at 6-character length. To investigate whether there is correlation between the star name length and the star frequency count, let us estimate the correlation coefficient. It results to be -0,48 for the total set of stars and -0,55 if we consider only useful stars, which can be characterized only as weak to moderate correlation, providing no solid grounds for positive assertions. (This even to omit that the distribution of the frequency count is far from normal, which renders it unreliable to speak of statistical dependency or independency based on the correlation).


8. Importance chart

It is interesting to cast a glance on how star importance does map onto the diagrams of f68r. For mapping importance onto diagrams we use the rainbow colour scheme with violet for least important and red for most important stars. We take the range of frequency counts which is 1 to 84, break it down into seven intervals of equal length, and then mark each star with the respective colour. This is just a most rough visual representation, but in principle it is analogous to a histogram. The result is presented in Fig. 3.

Fig. 3. Star importance chart


The diameter of the circles does not mean anything - it was chosen as large as to not obscure the star labels. The colour of the circles represents the stars' importance, as described above. Mostly we have the violet field, because most of the stars have low count. Interestingly, most stars with high count tend to cluster in the same region both in the dayside and nightside diagrams - roughly between four and eleven o'clock. There is one outsider of the cluster in each case, however, - otol in the dayside and chodar in the nightside chart. I make no conclusions but leave this chart to art and science historians with hope that they make something of this. Perhaps the clusters are just accidental.


9. Stars in f1r

Folio 1r is generally supposed to represent some "introduction", "summary" or "table of contents" for the VMS, which would be natural indeed. Let us note if any stars are mentioned in this "summary". As described in Section 3 above, the most important star of otol is mentioned in paragraph 3 in the right-aligned text block. It turns out that one more star is also mentioned in f1r, and it is... odaiin, the most important of the nightside stars and second most important overall! Interestingly, it occurs in the same paragraph 3, in the context of "chol odaiin chodain".

If paragraphs of f1r correspond to different thematical "sections" of the manuscript, then the third section is "balneological", is it not? What does it have to do specifically with odaiin and otol? Of course, those stars are mentioned in balneological folios, but not with overwhelming count. Also, they are not the only stars mentioned in balneological section. One more question left open.


10. Stars and plants: the general idea

We now move to the part of the article which tries to investigate the mutual context of stars and plants. The idea is that stars and plants may be associated with each other. For example, each plant may have its associated star. Or each plant should be gathered only on days "ruled" by a specific star. And so on. Art historians, for sure, will suggest how it should be, based on similar literature (and thus contribute to the comparative analysis). The fact that, at the moment, we know little of precedents only frees us from bias: we search for associations not being burdened with any specific to-be explanations. Generally we will look at whether the stars are mentioned in the botanical section and, if so, then how they are mentioned.

By the "botanical section", for the sake of this article, we mean all folios that seem to contain the picture of a single plant and thus are supposed to discuss those plants. We distinguish 129 botanical folios, which are: f1v, f2r - f11v, f13r - f56v, f57r, f65r, f65v, f66v, f87r, f87v, f90r1, f90r2, f90v1, f90v2, f93r - f94v, f95r1, f95r2, f95v1, f95v2, f96r, f96v.


11. Botanical affinity of stars

Voynich stars are mentioned in different portions of the manuscript. Let us have a look at what stars tend to be attracted to the botanical section, in contrast to other sections of the manuscript. This might designate that those stars are more important in regard to plants than other stars are. In other words, they are more "botanically important".

In this, we naturally exclude useless stars from consideration, because they occur nowhere except f68r. Also, we do not count the occurrency in f68r for the affinity count explored in this section. This is because we consider f68r as star "catalogue" serving just to list all the stars and not having any associative significance in regard to the subjects of the VMS sections. Thus, we deduct 1 from the frequency count of useful stars. For brevity, let us call such count "meaningful count" (MC) and respective occurrences - "meaningful occurrences". We then count the occurrences of a star in the botanical section (column 7) and divide it by the meaningful count of that star. This represents the percentage of the botanical count -  the "botanical affinity ratio" (BAR).

Do stars exhibit strong botanical affinity on the whole? At first glance, they do not. Of 231 meaningful occurrences of dayside stars only 83 (35,9%) fall into the botanical section. Of 130 meaningful occurrences of nightside stars 61 (46,9%) fall into the botanical section. The two sets considered together, only 39,9% of all meaningful occurrences are "botanical".

However, one would reasonably argue that botanical folios are more laconic, thus any star occurrence in the botanical section is more "valuable" in terms of affinity than, say, its occurrence in the balneological section which simply has far greater word density per folio. Let us account for this circumstance. In total there are 37886 words in the VMS, of which 11385, or 30,1%, accrue to the botanical section. (These figures were retrieved from VQP and trusted without further check; as noted above, they do not account for the "Rosettes" folio). Thus if BAR is much greater than 30%, then we can speak of conspicuous positive affinity. By analogy, if BAR is much lower than 30%, then we can speak of negative affinity - that is, the star tends to be mentioned mostly outside of the botanical section. Let us consider "two times" synonymous to "much". Even in this light, as follows from the figures above, stars on the whole do not exhibit strong botanical affinity.

Let us extend this method on individual stars and look if any of them exhibit strong positive or negative botanical affinity.

The following dayside stars exhibit positive botanical affinity:

  • chocphy (BAR = 100%, MC = 2)
  • otydy (BAR = 100%, MC = 2)
  • okshor (BAR = 100%, MC = 1)
  • or daiin (BAR = 100%, MC = 1)
  • ytchody (BAR = 75%, MC = 4)

The following nightside stars exhibit positive botanical affinity:

  • okchor (BAR = 72%, MC = 18)
  • dchol (BAR = 70%, MC = 23)

The following dayside stars exhibit negative botanical affinity:

  • dolchedy (BAR = 0, MC = 2)
  • otcheody (BAR = 0, MC = 1)
  • otchdo (BAR = 0, MC = 1)
  • otochedy (BAR = 0, MC = 1)
  • otshey (BAR = 14%, MC = 7)
  • okoldy (BAR = 14%, MC = 7)

The following nightside stars exhibit negative botanical affinity:

  • shdar (BAR = 0, MC = 8)
  • shchy (BAR = 0, MC = 4)
  • cheorol (BAR = 0, MC = 1)
  • cholar (BAR = 0, MC = 1)

It is worth noting that there is no correlation between BAR and MC.

What is important is that there are stars meaningfully occurring only in the botanical section, and there are stars meaningfully occurring only outside of the botanical section. Remembering that the VMS introduces a "full" set of stars (this was explained in Section 2, H5 above) and, most probably, the author aspired to introduce the "full" set of plants (which we derive from the fact that some botanical folios occur outside of the main botanical corpus, as if added afterwards as necessary addenda), the following conclusions can be made:

  • Association with plants is one distinct application of stars;
  • There are other applications of stars beside association with plants;
  • Not all useful stars are associated with plants.

"Association" here, as well as hereinafter, is considered in the broad sense of the word - actually, in terms of contextual affinity.

In the light of the suggested "full sets" concept, we should not hesitate at low MC values of some stars above while considering their BAR. If a star meaningfully occurs only once, doing that in the botanical section, we should recognize that it is mentioned in the botanical section for a reason, and at the same time there is also no reason to mention it elsewhere in the corpus - i.e. it is irrelevant to other "applications". (Of course, one application of all stars is the f68r diagrams, but we explicitly excluded it from consideration in this Section, because it is trivial.)

What is the share of stars having non-zero BAR in the total amount of stars having non-zero MC? In other words, how many stars that are meaningfully mentioned on the whole are, at the same time, mentioned in the botanical section? For dayside, this is 15/19 = 78,9%. Thus, if dayside stars are meaningfully mentioned at all, they tend to be mentioned at least in the botanical section. For nightside, this is 5/9 = 55,6%, which is not so decisive. For stars on the whole it is 20/28 = 71,4%.

For the end of this Section, let us look at which stars have the highest botanical count. The following dayside stars have high botanical counts:

  • otol (39,8% of all dayside botanical count)
  • otor (22,9%)
  • olor (7,2%)
  • okeor (6,0%)
  • ockhy (4,8%)

Other dayside stars provide for less than 4% of the count each.

The following nightside stars have high botanical counts:

  • odaiin (39,3% of all nightside botanical count)
  • dchol (26,2%)
  • okchor (21,3%)
  • chodar (11,5%)

Other nightside stars provide for less than 2% of the count each.

The total botanical count rating, without breakdown by day/nightside, looks like this:

  • otol (22,9%)
  • odaiin (16,7%)
  • otor (13,2%)
  • dchol (11,1%)
  • okchor (9,0%)
  • chodar (4,9%)
  • olor (4,2%)

Other stars provide for less than 4% count each.

We see that in the breakdown by sets, the botanical count top ratings fully correspond to the total count top ratings (see Section 6 above). Without breakdown, the top three stars are the same, but then nightside stars dchol, okchor and chodar crawl before the dayside olor. This goes side by side with the above observation that greater percentage of nightside star count falls onto the botanical section.

We see that 82% of all botanical count is provided by seven stars (four dayside and three nightside), which is 35% of the total of 20 stars having non-zero BAR. Seven is 13,2% of the whole amount of 53 VMS stars, while 20 is 37,7% of 53. Thus, we can make the following conclusions:

  • plants are associated with less than 40% of all available stars;
  • plants are mostly associated with the tiny minority of seven stars.


12. Mapping of stars across the botanical section

In the previous Section, we ensured that association with plants is one application of Voynich stars. Let us consider if there is any regularity in this association. Actually, in view of the suggestions of Section 10 above, I anticipated straightforward regularity in the following way: each botanical folio will mention a star, and only one star. Thus, each plant would have been nicely associated with one star. At most, I would have agreed to one dayside star and one nightside star per botanical folio. Of course, as usually with the VMS, this assumption failed. The actual picture is described below.

In the Excel file attached above, the special sheet "Mapping" is dedicated to star mapping across the botanical section. Column 1 lists all botanical folios. If a botanical folio mentions a star, the cell is presented in green fill. (For brevity, we call such botanical folios "star-aware" folios). The second column marks dayside star occurrences in respective folios, with one "+" sign for each occurrence. Respectively, column 3 marks nightside star occurrences.

Of 129 botanical folios only 89 (or 69,0%) are star-aware. Thus, almost a third of Voynich plants do not exhibit association with stars. (It is worth noting that even engagement of distinctly labeled "stars" from f68r3 - see discussion of H6, Section 2 above - does not yield full coverage of plants). In relation to the above assumption that each plant should have a star association, the following explanations can be proposed:

  • E1: There are such "hidden" stars not taken into account as Sun and Moon. Indeed, both Sun and Moon are present in the diagrams of f68r, but we do not know how to label them, so they fall out of our statistics. Perhaps it is Sun and Moon that provide the "lacking" star associations;
  • E2: The plants that do not have star associations are rare or exotic plants which the author was not very familiar with. Hence he simply did not know what stars are "connected" with those plants;
  • E3: The plants that do not have star associations are plants which are "useless" from the perspective of process or phenomenon which connects them with stars. For example, imagine that stars designate the time period at which to pick the plants. But some plants are not used in practice, so they do not need to be picked at all. They are included only for the purpose of completeness of the plant catalogue;
  • E4: The plants that do not have star associations are simply imaginary;
  • E5: Some folios contain grammatical forms of star names or/and some spaces went undetected. For example, the star-unaware f35v contains the word "dchold". This may be a word form of dchol, or this may me actually "dchol d"; but in our statistics it does not contribute to the frequency count of dchol.

Of these five, E1 is highly probable (given that H5 is true), because many plants in Middle Ages were associated with Sun and Moon. I also think that there is a way to cross-test E4, but this lies out of the scope of the present article. Perhaps I will return to it in later articles of the series.

Note that we should be careful with the very assumption that botanical folios should contain references to stars. I would not say that 69% is so large a figure for to assert this as the rule; hence, the proposed explanations E1-E5 may be a bit ahead of the game. Nevertheless, the fact is that some botanical folios do mention stars, and some do not; I expect to return to this in the second article of the series in relation to the future concept of "usefulness" of plants. Based on E2-E4, I would recommend to use primarily star-aware folios for plant identification attempts; perhaps this recommendation will be altered or withdrawn based on further research, though.

One thing is for sure: these "rare", "useless", "imaginary" (or whatever they are) plants - in other words, plants that are not associated with stars - are not clustered. In particular, they are not in any way clustered beyond f57r (at which point the contiguous portion of the botanical section terminates), neither they are clustered in any part of that contiguous portion (e.g. in the end thereof). Refer to Fig. 4 which shows the star count for each botanical folio. It is worth noting that in this and subsequent figures the horizontal scale is not perfectly linear and even is not monotonous, because some folios are missing and some are probably out of their initial order due to re-binding. Hence one needs to be careful in reporting any regularities based on this visual look.

Fig. 4. Star counts for botanical folios


33 of 110 botanical folios before f57r (that folio included), which is 30,0%, do not mention stars. 7 of 19 botanical folios beyond f57r, which is 36,8%, do not mention stars. So there is no evidence that the author intended to group (for whatever reason) plants by the criterion of whether they are associated with stars or not.

The density of mapping is as follows. 57,3% of star-aware folios mention stars only once, 28,1% - twice, 11,2% - thrice. Two folios (2,2%) - f28r and f55r - have four star occurrences and one folio 17v has five star occurrences. This, however, does not defeat our assumption yet, because not all star occurrences are unique stars. For example, f95v2 has two star occurrences, but they both are for olor.

Let us, therefore, consider the density of unique star mapping - i.e., how many unique stars are mentioned in a folio. 59,6% of star-aware folios mention only one star, 32,6% mention two different stars, and 7 folios (7,9%) mention as many as three stars. So the assumption that one plant is associated with one star finally fails.

Fig. 5 shows unique star count for each botanical folio. Again, no clustering is observed.

Fig. 5. Unique star counts for botanical folios


The amount of repetition is not great: of 144 total star occurrences in botanical folios 132 (or 91,7%) are folio-unique. No star is mentioned more than twice in a single folio. Only five stars can be mentioned twice in a single folio: three dayside (otol, otor and olor) and two nightside (odaiin and okchor). They all belong to most frequent in their sets. Almost half of all repetitions (5 of 12) accrue to otol. Other stars have no more than two repetitions.

Let us then move on to the breakdown of dayside and nightside stars. 41 star-aware folios (46,1%) mention only dayside stars, 28 star-aware folios (31,5%) mention only nightside stars, and 20 folios (22,5%) mention both dayside and nightside stars. Various combinations are in place. There are plants associated with two dayside stars, with two nightside stars, or with one dayside and one nightside star. More than that, there are plants associated with two dayside and one nightside star, and, vice versa, with one dayside and two nightside stars. This utterly ruins our vapid assumption that the association would be no more complex than one plant to 1+1 dayside and nightside star.

Fig. 6 shows unique dayside (top) and nightside (bottom) star count for each botanical folio. Here we can note one potential instance of clustering - that between f38v and f48v, where dayside stars occur exclusively, with one exception of f44v.

Fig. 6. Unique dayside (top) and nightside (bottom) star counts for botanical folios


Let us now explore if individual stars exhibit any mapping irregularity across the botanical section space. As the number of repetitions is very low, the unique botanical count rating looks much like the total botanical count rating in the previous Section 11:

  • otol (28 unique occurrences)
  • odaiin (22)
  • otor (17)
  • dchol (16)
  • okchor (11)
  • chodar (7)

Together these six stars provide for 76,5% of all unique star occurrences in the botanical section, which is 101 of 132. We take these six individual stars and plot for each of them the graph similar to Fig. 5. Refer to Fig. 7. No clustering is visible.

Fig. 7. Individual star unique counts for botanical folios. From top to bottom: otol, odaiin, otor, dchol, okchor, chodar


13. Mapping of stars within individual botanical folios

Those who have not fallen asleep yet would naturally anticipate the next level of detalization - namely, how the stars are mapped within a single folio? Do they exhibit any regularity in this, like being listed in the first paragraph exclusively or something like that? Or they are randomly scattered across the folio space?

Any regularity, if detected, would, of course, allow us to raise assumptions about the general structure of narration for the botanical folios. That is why search for this kind of regularity is so exciting.

To investigate this, we need some kind of coordinate measure to compare different botanical folios with each other. To obtain the "coordinate" of the star occurrence in a folio, the most granular way would have been to count the relative number of the respective word. However, the problem is that too many spaces are uncertain, which could introduce a significant error. And manual check of VQP for all spaces across the botanical section is quite wearysome. So I decided to rise a level upwards and measure the coordinate in lines. Of course, since folios have different numbers of lines, we need to express the coordinate in the relative fashion. E.g., if the folio contains 23 lines, and the star is mentioned in line 15, then the line coordinate ("L-coordinate", for brevity) is 15/23 = 0,65. As the sideway, I also recorded the coordinate measured in paragraphs ("P-coordinate"), applying the same principle. In contrast to lines, paragraph separations are in some places not as distinct; in each case of uncertainty a decision was made whether the paragraph separation is there or is not (such cases are marked in red font in the "Mapping" sheet).

The raw data are collected in the "Mapping" sheet. It gives the total number of lines (column 9) and paragraphs (column 15) in each folio, lists absolute coordinates of star occurrences measured in lines (columns 4-8) and in paragraphs (columns 10-14), and also lists what stars and in what order occur in the folio (columns 16-20). (As noted above, there are up to five star occurrences per folio, so five columns are asssigned in each case).

Fig. 8, chart A, shows the histogram of the star occurrences' L-coordinate. We can see that the coordinate is more or less spread across the possible range from 0 to 1, and there is no simple case like "stars are mostly discussed in the beginning of the folio, and then the author proceeds to discuss other things."

Fig. 8. Histograms of the star occurrences' L-coordinate: total occurrences (A), primary occurrences (B), secondary occurrences (C), singular primary occurrences (D)


It is interesting, nonetheless, to have a look at where (within a star-aware folio) the stars tend to be mentioned for the first time. Let us call the first occurrence of stars in a folio the "primary occurrence" and all subsequent occurrences (if any) - the "secondary occurrences". The chart B in Fig. 8 shows the L-coordinate histogram for primary occurrences. It is notably declining towards larger coordinates; 80% of primary occurrences take place within the first 60% of the folio space. The peak at the interval of (0,5; 0,6] is also notable - it is the major input into the main peak of the histogram A.

It is worth noting that, while larger coordinates yield less primary occurrencies, the latter are perfectly possible in the later portions of a folio. They can even appear as the very last word of a folio, as it is in the case of f1v which is ended with "chodar". On the opposite side of the coordinate range, the primary occurrence can also appear close to the edge of the folio. For example, in f90v1 it is as early as the second word of the folio; and, interestingly, it is the same "chodar". But in no case a star name appears as the first word of a folio.

The histogram of secondary occurrences (Fig. 8, chart C) reveals some interesting particulars. Its peak at (0,5; 0,7] is well supported by the main peak of the preceding histogram B. It is just natural that any subsequent mentions of stars would be somewhere in the vicinity of the primary mention. However, it has a secondary peak towards the end of the folio at (0,9; 1] - 8 secondary occurrences fall into there. Furthermore, it has also a tertiary peak at (0,2; 0,3] - which, of course, relates to those cases when the respective primary occurrence appears in this very interval or even earlier - and we can see from the primary occurrence L-coordinate histogram that such cases are also numerous indeed.

All this does not reveal any inner structure of the "narration" and suggests one (or several) of the following possibilities:

  • there is no intended structure; the narration flows in arbitrary way;
  • there is structure, but it is complex: like, within the folio, at first some stars are mentioned in one respect, then other stars are mentioned in another respect, and so on;
  • word shuffling is in place as part of the cryptographic algorithm.

Of these three, we can easily cross-test the second possibility if we build the coordinate histogram accounting for the singular primary occurrences exclusively - that is, for those folios which contain only one star reference. It is presented in Fig. 8, chart D. According to this histogram, even a single star occurrence can well be anywhere in the folio space. It is not that single stars are mentioned, for example, only in the beginning of the folio. Hence the second possibility expressed above does not find any positive substantiation; and we are left with the loose structure of narration or (and) with shuffling.

We can try to discover traces of structure on the paragraph level. It is natural to suppose that paragraphs serve to distinguish between different theses expressed in a folio. To this end, we build histograms similar to those in Fig. 8, but this time for coordinate measured in paragraphs, and not in lines.

Of course, we need to exclude all one-paragraph folios from consideration - because all P-coordinates in such folios will equal 1 simply by definition, be the star mentioned in the beginning, in the middle or in the end of the folio; there is no sense in seeking any paragraph-level structure in folios with less than two paragraphs. There is 21 star-aware 1-paragraph botanical folio out of 89, so we are left with 68 folios, of which 47 are 2-paragraph, 18 are 3-paragraph, and three folios are 4-paragraph.

Fig. 9. Histograms of the star occurrences' P-coordinate: total occurrences (A), primary occurrences (B), secondary occurrences (C), singular primary occurrences (D)


Refer to Fig. 9. Before discussing the results, it is important to say a couple of words about the proper interpretation. Unlike the above situation where the coordinate was referenced to lines, here the set of possible coordinates is quite scarce - simply due to the fact that the possible number of paragraphs in a folio is only 2, 3 or 4. The recorded  P-coordinate thus takes one of the following values: 0,25, 0,33, 0,5, 0,66, 0,75, and 1. The leftmost bar in the histograms of Fig. 9 is comprised of the occurrences of the value of 0,25 which corresponds to the first paragraph in a 4-paragraph folio; as noted above, there are only three such folios, so the leftmost bar is necessarily negligible. The second bar (counting from the left) corresponds to the first paragraph of a 2-paragraph folio, to the second paragraph of a 4-paragraph folio, and also to the first paragraph of a 3-paragraph folio (with its coordinate of 1/3 = 0,33). The third bar collects the second paragraph of a 3-paragraph folio and also the third paragraph of a 4-paragraph folio. Finally, the rightmost bar embraces last paragraphs of all multi-paragraph folios. From this explanation it should be clear that in our search for the paragraph-level structure we should look primarily at the difference between the second and the fourth bars. If the former is much greater, then we can say that the prevalence rests with the earlier paragraphs of the folio; in the opposite case it rests with the later paragraphs.

The picture is again not very decisive, nonetheless. The totals histogram (Fig. 9, A) does not exhibit clear dominance of the second bar - the difference is only 27%. Chart B, which shows the distribution of the primary occurrences' coordinates, is encouraging: it indeed suggests that primary occurrences of stars tend towards the early paragraphs. Likewise, chart C shows that secondary occurrences tend towards the very last paragraph. So far, so good. But what about singular occurrences? If we consider only folios mentioning stars only once (chart D), then we get the distribution much like chart A, with no decisive dominance of the second or the fourth bar; this means that in such folios the star well can be mentioned in any paragraph.

One factor could potentially distort the picture and mask the structure; and, unfortunately, this factor cannot be compensated for. This is the "noise" introduced by the possible general meaning of the star names. Imagine, for example, that there is a star called "dog". Sometimes the author would refer to this star (when he needs that), and the word "dog" would appear in the botanical folios - most probably, when the respective plant is associated with that star. But from time to time the author may need to refer to the actual dog - not the star, but the animal called "dog". The word would appear again, but in no way as a reference to the star. However, since our statistics do not (and cannot) distinguish between the two "dogs", the "dog" as an animal would distort the actual count of the "dog" as a star. I suppose this might be the case with otol, because, as shown in Section 3 above, otol is several times encountered in quite unexpected, non-astrological contexts.


14. Summary and further directions

In this article, we took the set of object labels from f68r1 and f68r2 (organized into two respective subsets) and investigated various patterns of their occurrence throughout the VMS. In respect to the thematic context in which those "stars" are mentioned, we introduced the notion of "application" of Voynich objects. We focused primarily on the botanical application of Voynich stars. The following main results were observed through the simple statistical analysis (which was performed with the help of the "VQP" online tool and therefore suffers from a number of limitations):

  1. The most "important" (i.e. frequently mentioned) stars are otol and odaiin. Also, they are the only stars mentioned in f1r.
  2. "Dayside" (f68r1) stars tend to be more important than "nightside" (f68r2) stars.
  3. The stars' importance exhibits significant concentration: the minority of stars gives the majority of the count - in line with the Pareto principle. This is true for dayside stars, for nightside stars, and for all stars in total.
  4. Almost a half of all stars are "useless" (i.e., not mentioned beyond f68r). In this, nightside stars are more useless than dayside stars. Interestingly, the central stars in f68r1 and f68r2 are both useless.
  5. The correlation between the star name length and the star's importance is only weak to moderate.
  6. Stars' importance plotted onto the diagrams of f68r exhibits some clustering, albeit not without outsiders.
  7. "Botanical application" (i.e. their being mentioned in botanical folios) is one distinct application of stars.
  8. Not all useful stars are associated with plants; however, the majority of them are.
  9. There are stars exhibiting positive "botanical affinity", stars exhibiting negative botanical affinity, and stars exhibiting no decisive affinity to (or against) the botanical section. The botanical affinity is not correlated with the stars' importance.
  10. Plants are associated with less than 40% of all available stars; and, more than that, they are mostly associated with the tiny minority of seven stars.
  11. Almost a third of Voynich plants do not exhibit association with stars. No clustering of such plants was detected.
  12. No distinct regularity was detected in the distribution of stars across the corpus of botanical folios - be that the total of star occurrences, only dayside, only nightside, or only unique occurrences. Individual stars also do not exhibit any regularity.
  13. The overwhelming majority of star ocurrences in the botanical section are folio-unique; in other words, a star is rarely mentioned again in the same folio.
  14. Investigation of the stars' mapping across the folio space did not detect strong evidence towards any structure of narration. The only positive result is that, in multi-paragraph folios, the first star occurrence tends to be in earlier paragraphs while the subsequent star occurrences (if they do exist) tend to be in the last paragraph; however, folios with only one star occurrence do not support this picture - the star is very often mentioned in the last paragraph.
  15. The effect of the "general meaning" of some stars' names may introduce noise masking the narration structure.
  16. The Sun and Moon potentially missing in the discussed statistics may introduce significant corrections into the results, since they generally are the most important celestial bodies in astrology.

In the vein of the proposed methodology, the following directions of further research are envisioned:

  1. Analysis of other applications of Voynich stars: Zodiac section, balneological section, Rosettes folio, recipe section.
  2. Identification of other appropriate "full sets" of Voynich objects and analysis of their contextual statistics. This may be not that easy though: the most obvious candidate - the "Voynich pipes" of f69v1 lists okeod three times among otherwise unique labels, and thus is not so "perfect" a set as Voynich stars are.
  3. Mapping of different objects' occurrences together in a single map. This might be a great insight into the understanding of the Voynich narration structure, as well as of the contextual relationship of different objects. There is an experimental query at VQP which allows you to plot all labels at once, but it is too noisy, since a) it does not distinguish between labels of different kinds, and b) not all labels represent Voynich objects of the comparable rank. For example, stars and pipes are of the same rank, but f1r "section titles" are, obviously, not.

I very much suspect that ultimately the contextual analysis will reveal the picture quite different from what is usually expected from the VMS. Traditinally, for the sake of statistical analysis it has been compared with different narrations such as "On the Origin of Species" or The New Testament, but probably it is nothing of the sort, but rather some highly compacted conspectus of the "Gold-Bug"-style.

"I confess that the matter still wears a serious aspect, when regarded with a casual glance."


References

1. G. Rugg. An elegant hoax? A possible solution to the Voynich Manuscript. - Cryptologia, vol. XXVIII(1), 2004. Pp. 31-46.

2. T. Timm. How the Voynich Manuscript was created. - Rev. 2014. - arXiv:1407.6639v2.

3. B. Cham. Introduction to the curve-line system. - 2015.

4. M. Montemurro, D.Zanette. Keywords and co-occurrence patterns in the Voynich Manuscript: an information-theoretic analysis. - PloS ONE 8(6): e66344.

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  1. David Jackson · 21 October 2015, 15:37 · URL

    Hi Anton,

    On the meaning of OTOL:
    This could be considered to be a taxonomic device, possibly declaring an “essential element”, although such a device does not seem to fit into classic definitions. The five medieval elements were considered to be the building blocks of everything, it is possible that the philosophy of the VM attempts to link this element “otol” into all these different products.

    Another possibility is colour – maybe it simply means the colour of the object.

    A third possibility, of course, is that otol is a simple word just being thrown up by the generating mechanism of the random text generator.

    On the groups of Voynich stars:

    Stars were considered to be “fixed” until the Renaissance, and so stars in sky maps were generally drawn in a stylised pattern that concorded with the general illustration. It is not until almost the 17th century that illustrators came up with the technique of randomly spotting stars. The reason was the underlying philosophy of the stars – God had a finite amount of fixed points of light in the sky VS God, there’s a lot of those things up there!

    I have a suspicion that we’re actually looking at something like a lapidary here, not stars. That would explain their correlation with other parts of the manuscript, as the attributes of the stone in question would naturally combine.

  2. Anton Alipov · 20 December 2015, 18:25 · URL

    Hi David,

    Very interesting suggestion about a lapidary, thanks. I’ve added it to the post. A brief check shows that contemporary lapidaries contained some 50+ stones or minerals, which is roughly the amount of the Voynich “stars”. Although offhand I failed to detect any grouping in two distinct sets.

  3. D.N.O'Donovan · 16 July 2016, 02:37 · URL

    Anton,
    I must beg to differ from David on this.

    The horizons are rather larger than European astrology or ideas about astronomy and indications in the manuscript are that the highly technical science of non-European navigational astronomies may be more to the point. For example, that diagram showing the moon at the bottom, the Pole star at the centre, and the sun at the top is clarified by Guillaume Soler’s use of the same emblems in the same way on his carte marine. Other folios also suggest that context and the science of sidereal navigation was quite independent of the desk-bound person’s astronomy-or-astrology mindset.

    I might add that my doctoral thesis, based in this research, was written thirty years ago, and I’ve continued reading everything published on the subject since then, so with all due respect to David, I think I can fairly well recognise the different types of astronomical imagery. It may come from a globe of the usual sort, of course, but I think the language probably won’t.

  4. D.N. O'Donovan · 8 September 2016, 08:43 · URL

    Anton,
    As you know, the Voynich language is not my area, but re-reading this, I notice that you cite f.77r.

    I’m not sure how many people now adopt my conclusions about folio 77r – some sites have a had habit of taking the results of research and then using them without getting the attributions right, or even omitting them altogether – but the point is that the detail you show in Fig.1 (a) is part of a non-European, but Greek-influenced 5-elements scheme. It doesn’t include aether.

    So the label discussed is either referring to one element of that non-European five-element system, or it refers to the fundamental matter from which those elements emerged. The Greek word for it was one used for wood or a large tree branch.

    For the sake of argument, suppose the language were Greek: then you might find the same word used in one sense in relation to the heavens, another in relation to the botanical section.. and so on.

    A better example. The Bedu and Arab tradition has a false etymology for Thurayya, a name for the Pleiades’ asterism and lunar mansion, which makes it mean “Water in the ground”.

    So when labelling the star, the writer might just write “water”, and also in labelling the elements – “water”, yet the occurrence in the botanical would not refer to any star, and might have purely literal intent.

    I guess what I’m asking is whether we can equate similar forms of words with similar meaning quite so easily.

    On which theme, I might mention having recently seen the proposal that the names for stars are descriptive; ‘The Strong One’ and so on. Now this is very interesting, since in Ibn Arabi’s work we find the names of God associated with the lunar asterisms.

    I’m presently hunting again for the blog, forum or wherever where that proposal was offered, so that I can decently spread the word.

    PS – if you care to read my original analysis for folio 77r, it is here.

    I was the first person, as far as I know (the poor standard of citations being what they are in Voynich studies) to provide a detailed and documented analysis of the folio, though not the first to suggest it represented Elements (as I later learned). As ever, I added a note on that point to the relevant post.

    https://voynichimagery.wordpress.com/2012/10/25/fol-77r-five-elements/


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