11-10-2026, 02:53 AM
(This post was last modified: 11-10-2026, 06:59 AM by npcompl33t.
Edit Reason: added csv link with path values by page
)
I just shared an updated version of some page classifications I have been working on. You can read You are not allowed to view links. Register or Login to view. or view You are not allowed to view links. Register or Login to view. for more information on how these are generated. You can explore the data for this sequence specifcally on You are not allowed to view links. Register or Login to view.
Briefly:
For this theory, the actual clusters themselves are irrelevant. Instead, we just look at the MDS ( MDS is essentially PCA over this pairwise distance matrix). The graph of the 2 most significant dimensions produces a similar shape to the PCA bigram charts that have been shared on this forum before.
The Guttman Effect
Note, the distinct 'horseshoe' shape. In the above chart
This is what is known as a Guttman horseshoe. It is a You are not allowed to view links. Register or Login to view. You are not allowed to view links. Register or Login to view.that can happen with both PCA and MDS. This occurs in very specific conditions, when a sequential trend moves across some type of environmental gradient. An example is like this: if you walk two blocks north, then turn and walk 2 blocks east, you have walked 4 blocks. Yet the actual distance will be 2.83. PCA cannot recover the full distance because it attempts to fit a straight line, and one straight line cannot capture the full 4 block distance.
The most common way to check is to look at the 3rd dimension compared to the first. This 3rd dimension will normally have an S shape, but it can manifest more complex shapes depending on the dimensionality of the underlying curve. Notably, it can have a 4 factor polynomial curve, which resembles a W or M.
Lets look at this for our data:
The Bifolio Agreement
So how does this relate to manuscript chronology? Well, once you have identified a guttman horeshoe, you can correct it by applying a polynomial curve to reproduce the full nonlinear gradient. This is what that looks like:
Its immediately apparent this achieves very strong split by sections across the manuscript. The curve also splits currier A and B cleanly.
However, where this gets really interesting is when we compare the average variation of pages in the same bifolio along this curve. The 1st MDS dimension already almost perfectly orders pages by bifolio. Pages on the same bifolio's average variation on the first MDS dimension is just 2.5%, and this drops to an even lower 1.5% when applying the curve. Compare that to an average of 15-50% difference across the other MDS dimensions.
How does this connect to chronology?
Think for a second about what this is saying.
The most obvious answer is time, and in fact, time is one of the classic known drivers of a Guttman Horseshoe. For example, Flinders Petrie ordered Egyptian graves in time by their pottery styles around 1900, and in the 1970s David Kendall showed that MDS on such data produces a horseshoe whose order along the curve is the chronological order.
Therefore this pattern is not only consistent with chronology, chronology is one of the known major drivers. This, combined with the fact that the sequence has < 2% divergence along known production units, strongly suggests this is the actual chronology of the manuscript.
What is the chronology?
Note the ordering of some points can be sensitive to the exact curve being drawn. The relative ordering in between sequences is fairly reliable, but groups should be interpreted as production batches, not as an exact ordering sequence.
This shows that Herbal B was likely created after both pharma and cosmo, yet was re-ordered and interspersed within Herbal A sometime before quire numbers were added. It also implies that Stars was created in two distinct productions, before and after bio.
What else can be derived from this?
It strongly suggests that several features known to be associated with Currier A vs B splits are actually shifts in the way voynich was written, and that the entire sequence contains several such significant shifts. It is exactly this type of stepwise process that would result in a Guttman horseshoe. The next step would be to incorporate these known statistical properties alongside this chronology, to show how the voynich language appeared over time.
Links:
Briefly:
- I create a glyph level transition matrix for each page, both for individual words, and the entire line (removing spaces). I then combine these together.
- I then calculate the pairwise Jensen-Shannon Divergence between each page.
- This matrix is viewable on the web app, as a cluster-map.
For this theory, the actual clusters themselves are irrelevant. Instead, we just look at the MDS ( MDS is essentially PCA over this pairwise distance matrix). The graph of the 2 most significant dimensions produces a similar shape to the PCA bigram charts that have been shared on this forum before.
The Guttman Effect
Note, the distinct 'horseshoe' shape. In the above chart
This is what is known as a Guttman horseshoe. It is a You are not allowed to view links. Register or Login to view. You are not allowed to view links. Register or Login to view.that can happen with both PCA and MDS. This occurs in very specific conditions, when a sequential trend moves across some type of environmental gradient. An example is like this: if you walk two blocks north, then turn and walk 2 blocks east, you have walked 4 blocks. Yet the actual distance will be 2.83. PCA cannot recover the full distance because it attempts to fit a straight line, and one straight line cannot capture the full 4 block distance.
The most common way to check is to look at the 3rd dimension compared to the first. This 3rd dimension will normally have an S shape, but it can manifest more complex shapes depending on the dimensionality of the underlying curve. Notably, it can have a 4 factor polynomial curve, which resembles a W or M.
Lets look at this for our data:
The Bifolio Agreement
So how does this relate to manuscript chronology? Well, once you have identified a guttman horeshoe, you can correct it by applying a polynomial curve to reproduce the full nonlinear gradient. This is what that looks like:
Its immediately apparent this achieves very strong split by sections across the manuscript. The curve also splits currier A and B cleanly.
However, where this gets really interesting is when we compare the average variation of pages in the same bifolio along this curve. The 1st MDS dimension already almost perfectly orders pages by bifolio. Pages on the same bifolio's average variation on the first MDS dimension is just 2.5%, and this drops to an even lower 1.5% when applying the curve. Compare that to an average of 15-50% difference across the other MDS dimensions.
How does this connect to chronology?
Think for a second about what this is saying.
- The fact that pages within a bifolio have < 2% divergence across this entire sequence, shows this is measuring some production level feature
- It also forms a smooth continuous gradient across all bifolio in the manuscript.
- It is not correlated to LFD hand.
- What feature could possibly be represented by a smooth continuous gradient between production units?
The most obvious answer is time, and in fact, time is one of the classic known drivers of a Guttman Horseshoe. For example, Flinders Petrie ordered Egyptian graves in time by their pottery styles around 1900, and in the 1970s David Kendall showed that MDS on such data produces a horseshoe whose order along the curve is the chronological order.
Therefore this pattern is not only consistent with chronology, chronology is one of the known major drivers. This, combined with the fact that the sequence has < 2% divergence along known production units, strongly suggests this is the actual chronology of the manuscript.
What is the chronology?
Note the ordering of some points can be sensitive to the exact curve being drawn. The relative ordering in between sequences is fairly reliable, but groups should be interpreted as production batches, not as an exact ordering sequence.
- D25-32 is a strong candidate for the first created bifolio. It is the first in both the word level and word+line level reconstruction, and it appears 1st in 49 attempted leave-one-pool out refits.
- 2 distinct but separate production batches then produced 2 sections of early Herbal A. The exact ordering depends on the polynomial degree used, as either work, but 1a is a slightly better fit.
- A transition Herbal A section bridges the 2nd Herbal A batch with Pharma
- Several Cosmo, along with f58 came next
- Herbal B came after this, followed immediately by half of stars
- Bio follows The first stars half. The final page varies in different curves.
- T08-111 is a strong candidate for the final page of the MS. Layfield and Davis order this 2nd to last. It appears last in 46/50 leave one pool out refits.
This shows that Herbal B was likely created after both pharma and cosmo, yet was re-ordered and interspersed within Herbal A sometime before quire numbers were added. It also implies that Stars was created in two distinct productions, before and after bio.
What else can be derived from this?
It strongly suggests that several features known to be associated with Currier A vs B splits are actually shifts in the way voynich was written, and that the entire sequence contains several such significant shifts. It is exactly this type of stepwise process that would result in a Guttman horseshoe. The next step would be to incorporate these known statistical properties alongside this chronology, to show how the voynich language appeared over time.
Links:
![[+]](https://www.voynich.ninja/images/collapse_collapsed.png)
