The d'Agapayeff Cipher, continued...

In a recent post, I mentioned the idea that the d'Agapeyeff cipher might involve a diagonal transposition on the 14x14 grid cryptologists suspect it may well have been based upon. To test this out a bit, I wrote a short C++ program (which I've uploaded here) which turns the number pairs into characters (for convenience) and prints out all four diagonal transpositions (forward, reverse, forward boustrophedon, reverse boustrophedon) starting from each of the four corners.

Because the number of doubled and tripled letters is a simple measure of whether a transposition is likely to be plausible or not, I counted those up as well. The next metric to calculate would be the unique letter adjacency count (i.e. how many unique pairs of letters appear for each ordering)... but that's a task for another day.

Interestingly, transpositions starting from the top-left corner (and their reverse-order reflections in the bottom-right corner) have no triple-letters at all, as well as far fewer double-letters (9/10/11 compared to 13/14/15) than transpositions that start from the top-right. Though intriguing, I don't know if this is statistically significant: I haven't determined what the predicted doublet and triplet count would be for a totally randomised transposition, perhaps calculating that too that would be a good idea.

For any passing cryptologers, here is the ASCII version of the d'Agapeyeff cipher (as output by the C++ code) when arranged as a 14x14 grid (in numerical order but without J), followed by the 16 diagonal transpositions with their associated double & triple counts. My guess is that the top left corner reverse diagonal transposition (the second one down, starting "KBDMIDPIK...") is most likely to be the correct transposition, but we shall see (hopefully!) if this is true...

K B M P Q B Q D L D Q I P O
D I I M O N L C L L I I M B
D K N M O Q K I E N K K K S
C E E L C L K P K K D B M R
P I C M K I N L E L O P D P
D P P C M G B N B L L G L D
C K M L D N C M P L C C C Y
I L Q Q O C P O E D P E B T
B B P Q P Q I Q G K D E K F
E N B D I L M O B M D Q L S
E B D O O Q N P I Q L E G I
N N P M N D B G B E B N K R
G C M M G G N M P O K M L N
G O B M N K L D K I P L B R

*** Top left corner ***
Forward order...
KDBDIMCKIPPENMQDIEMOBCPCLONQIKPMCQLDBLMCKLKCLEBQLMIKILDE
NPQDGNPELQNBBQONBLKNIIGNDDPCCNEKKIPGCPOIQPMBLDKMOOMMOLIO
PLOBKBBMNQMQELLPMSMGDNOGDCGDRNGBPBKPCLPKNGIMDECDLMBQDEBY
DPELQKTKOBELFIKNGSPMKILLRBNR
--> number of doubles = 11, number of triples = 0
Reverse order...
KBDMIDPIKCQMNEPBOMEIDQNOLCPCDLQCMPKILCKLKCMLBDLIKIMLQBEQ
LEPNGDQPNEIINKLBNOQBBNPIKKENCCPDDNGOMKDLBMPQIOPCGBKBOLPO
ILOMMOSMPLLEQMQNMBRDGCDGONDGMPLCPKBPBGNDCEDMIGNKYBEDQBML
TKQLEPDFLEBOKSGNKIIKMPRLLNBR
--> number of doubles = 9, number of triples = 0
Simple boustrophedon (forward then reverse)...
KBDDIMPIKCPENMQBOMEIDCPCLONQDLQCMPKIBLMCKLKCLDLIKIMLQBEE
NPQDGNPELQIINKLBNOQBBNGNDDPCCNEKKIPOMKDLBMPQIOPCGOMMOLIO
PLOBKBSMPLLEQMQNMBMGDNOGDCGDRPLCPKBPBGNKNGIMDECDYBEDQBML
DPELQKTFLEBOKIKNGSIKMPLLRNBR
--> number of doubles = 10, number of triples = 0
Reverse boustrophedon (reverse then forward)...
KDBMIDCKIPQMNEPDIEMOBQNOLCPCIKPMCQLDLCKLKCMLBEBQLMIKILDQ
LEPNGDQPNENBBQONBLKNIIPIKKENCCPDDNGGCPOIQPMBLDKMOBKBOLPO
ILOMMOBMNQMQELLPMSRDGCDGONDGMNGBPBKPCLPDCEDMIGNKLMBQDEBY
TKQLEPDKOBELFSGNKIPMKIRLLBNR
--> number of doubles = 9, number of triples = 0

*** Top right corner ***
Forward order...
OPBIMSQIKRDIKMPLLKBDDDLNDPLYQCEKOGCTBLIKLLCBFQNKPELCEKSP
OQKLBLPELIMMOLNNPDDQGRBIMCIBMEKDEKNKINLKGCOGMLNLRDKEMMNP
QBQBMBDECCDCIOIEKLCIPLOQMPBOPPPMQPLNGPIDKQQIQBMKCLPDODND
IBBONGLBNDMGKEBPMNENMMNCBGOG
--> number of doubles = 14, number of triples = 2
Reverse order...
OBPSMIRKIQPMKIDDDBKLLYLPDNLDTCGOKECQFBCLLKILBSKECLEPKNQI
LEPLBLKQOPRGQDDPNNLOMMNKEDKEMBICMIBRLNLMGOCGKLNIKBMBQBQP
NMMEKDLKEIOICDCCEDPOBPMQOLPICIPGNLPQMPPKMBQIQQKDDNDODPLC
LGNOBBIKGMDNBNMPBEMMNEBCNOGG
--> number of doubles = 15, number of triples = 1
Simple boustrophedon (forward then reverse)...
OBPIMSRKIQDIKMPDDBKLLDLNDPLYTCGOKECQBLIKLLCBFSKECLEPKNQP
OQKLBLPELIRGQDDPNNLOMMBIMCIBMEKDEKNRLNLMGOCGKLNIKDKEMMNP
QBQBMBLKEIOICDCCEDCIPLOQMPBOPIPGNLPQMPPDKQQIQBMKDNDODPLC
IBBONGLKGMDNBEBPMNMMNENCBOGG
--> number of doubles = 13, number of triples = 0
Reverse boustrophedon (reverse then forward)...
OPBSMIQIKRPMKIDLLKBDDYLPDNLDQCEKOGCTFBCLLKILBQNKPELCEKSI
LEPLBLKQOPMMOLNNPDDQGRNKEDKEMBICMIBKINLKGCOGMLNLRBMBQBQP
NMMEKDDECCDCIOIEKLPOBPMQOLPICPPMQPLNGPIKMBQIQQKDCLPDODND
LGNOBBIBNDMGKNMPBEENMMBCNGOG
--> number of doubles = 14, number of triples = 0

*** Bottom right corner ***
Forward order...
RNBRLLIKMPSGNKIFLEBOKTKQLEPDYBEDQBMLDCEDMIGNKPLCPKBPBGNR
DGCDGONDGMSMPLLEQMQNMBBKBOLPOILOMMOOMKDLBMPQIOPCGPIKKENC
CPDDNGIINKLBNOQBBNQLEPNGDQPNEDLIKIMLQBELCKLKCMLBDLQCMPKI
QNOLCPCBOMEIDQMNEPPIKCMIDBDK
--> number of doubles = 11, number of triples = 0
Reverse order...
RBNLLRPMKIIKNGSKOBELFDPELQKTLMBQDEBYKNGIMDECDNGBPBKPCLPM
GDNOGDCGDRBMNQMQELLPMSOMMOLIOPLOBKBGCPOIQPMBLDKMOGNDDPCC
NEKKIPNBBQONBLKNIIENPQDGNPELQEBQLMIKILDBLMCKLKCLIKPMCQLD
CPCLONQDIEMOBPENMQCKIPDIMDBK
--> number of doubles = 9, number of triples = 0
Simple boustrophedon (forward then reverse)...
RBNRLLPMKISGNKIKOBELFTKQLEPDLMBQDEBYDCEDMIGNKNGBPBKPCLPR
DGCDGONDGMBMNQMQELLPMSBKBOLPOILOMMOGCPOIQPMBLDKMOPIKKENC
CPDDNGNBBQONBLKNIIQLEPNGDQPNEEBQLMIKILDLCKLKCMLBIKPMCQLD
QNOLCPCDIEMOBQMNEPCKIPMIDDBK
--> number of doubles = 10, number of triples = 0
Reverse boustrophedon (reverse then forward)...
RNBLLRIKMPIKNGSFLEBOKDPELQKTYBEDQBMLKNGIMDECDPLCPKBPBGNM
GDNOGDCGDRSMPLLEQMQNMBOMMOLIOPLOBKBOMKDLBMPQIOPCGGNDDPCC
NEKKIPIINKLBNOQBBNENPQDGNPELQDLIKIMLQBEBLMCKLKCLDLQCMPKI
CPCLONQBOMEIDPENMQPIKCDIMBDK
--> number of doubles = 9, number of triples = 0

*** Bottom left corner ***
Forward order...
GOGBCNMMNENMPBEKGMDNBLGNOBBIDNDODPLCKMBQIQQKDIPGNLPQMPPP
OBPMQOLPICLKEIOICDCCEDBMBQBQPNMMEKDRLNLMGOCGKLNIKNKEDKEM
BICMIBRGQDDPNNLOMMILEPLBLKQOPSKECLEPKNQFBCLLKILBTCGOKECQ
YLPDNLDDDBKLLPMKIDRKIQSMIBPO
--> number of doubles = 14, number of triples = 2
Reverse order...
GGONCBENMMEBPMNBNDMGKIBBONGLCLPDODNDDKQQIQBMKPPMQPLNGPIC
IPLOQMPBOPDECCDCIOIEKLDKEMMNPQBQBMBKINLKGCOGMLNLRBIMCIBM
EKDEKNMMOLNNPDDQGRPOQKLBLPELIQNKPELCEKSBLIKLLCBFQCEKOGCT
DLNDPLYLLKBDDDIKMPQIKRIMSPBO
--> number of doubles = 15, number of triples = 1
Simple boustrophedon (forward then reverse)...
GGOBCNENMMNMPBEBNDMGKLGNOBBICLPDODNDKMBQIQQKDPPMQPLNGPIP
OBPMQOLPICDECCDCIOIEKLBMBQBQPNMMEKDKINLKGCOGMLNLRNKEDKEM
BICMIBMMOLNNPDDQGRILEPLBLKQOPQNKPELCEKSFBCLLKILBQCEKOGCT
YLPDNLDLLKBDDPMKIDQIKRSMIPBO
--> number of doubles = 13, number of triples = 0
Reverse boustrophedon (reverse then forward)...
GOGNCBMMNEEBPMNKGMDNBIBBONGLDNDODPLCDKQQIQBMKIPGNLPQMPPC
IPLOQMPBOPLKEIOICDCCEDDKEMMNPQBQBMBRLNLMGOCGKLNIKBIMCIBM
EKDEKNRGQDDPNNLOMMPOQKLBLPELISKECLEPKNQBLIKLLCBFTCGOKECQ
DLNDPLYDDBKLLDIKMPRKIQIMSBPO
--> number of doubles = 14, number of triples = 0

The d'Agapeyeff Cipher, revisited...

I know, I did blog about this only three days ago: but science moves ever onwards, OK?

A nice email arrived from Robert Matthews, the author of an excellent page on the d'Agapeyeff Cipher: he mentioned that he had received an email in February 2006 from John Willemse in Holland, who had suggested a novel kind of transposition cipher based around a spiral:-

I'm in no way a cipher expert, but I am a very curious person and I was wondering if the positioning of the 14x14 digram table could have anything to do with a spiral. The reason I suspect this, is that a spiraling positioning of numbers have the property that each upperleft corner of such a spiral (when starting with zero in the center) is a perfect square number. I'll try to illustrate my point:

16 15 14 13 12
17 .4 .3 .2 11 ..
18 .5 .0 .1 10 ..
19 .6 .7 .8 .9 26
20 21 22 23 24 25

Starting from zero, and counting up, anti-clockwise, you will encounter a perfect square of each even number in the topleft corner. 196 is also such a number.

The '04' digram almost in the center could be a break point. If you 'break' after the zero and shift the 4 to the right, creating a new set of digrams, you end up with a set of digrams before the zero and a set after the zero. The set after the zero should probably be reversed, either the whole set or the individual digrams, to create a similar set as the first one (the digrams starting with higher digits and ending with lower digits).

You might then be able to construct a spiral like positioning, with the zero in the center or the zero obmitted. The first set might then be 'twisted' around it clockwise, and the second set anti-clockwise, possibly interweaving each other.

These are just some wild ideas, and I'm in no way capable of constructing and verifying such a table myself, but maybe it's something to investigate?

Willemse's idea is certainly interesting: but let's look again at the (derived) 14x14 layout. To recap: one of the reasons for suspecting that transposition is involved is that there are two sets of horizontal tripled letters (75 75 75 and 63 63 63), while one of the reasons for suspecting that it's not a 'matrix transpose' diagonal flip is that there are two sets of vertical tripled letters (81 81 81 and 82 82 82). That is, unless the plaintext sadistically contains a phrase like "SEPIA AARDVARK" (a phrase which, I'm delighted to note, Google believes currently appears nowhere else on the Internet).

75 62 82 85 91 62 91 64 81 64 91 74 85 84
64 74 74 82 84 83 81 63 81 81 74 74 82 62
64 75 83 82 84 91 75 74 65 83 75 75 75 93
63 65 65 81 63 81 75 85 75 75 64 62 82 92
85 74 63 82 75 74 83 81 65 81 84 85 64 85
64 85 85 63 82 72 62 83 62 81 81 72 81 64
63 75 82 81 64 83 63 82 85 81 63 63 63 04
74 81 91 91 84 63 85 84 65 64 85 65 62 94
62 62 85 91 85 91 74 91 72 75 64 65 75 71
65 83 62 64 74 81 82 84 62 82 64 91 81 93
65 62 64 84 84 91 83 85 74 91 81 65 72 74
83 83 85 82 83 64 62 72 62 65 62 83 75 92
72 63 82 82 72 72 83 82 85 84 75 82 81 83
72 84 62 82 83 75 81 64 75 74 85 81 62 92

From this, it seems that, yes, you could construct a large number of spiral transpositions without tripled letter sequences. Yet I'm not completely convinced by the idea that the 04 token is a good indicator for the centre of a spiral: from the substitution cipher angle, I'd be quite happy to tag that as a likely 'X' or 'Y' in the plaintext instead.

However, I would point out that if you examine the various diagonal transpositions of the 14x14 (i.e. reading through the 14x14 one diagonal line at a time), there is (unless I'm somehow mistaken) apparently only a single tripled letter in two of them, and that only over a line-break:-

75 62 82 85 91 62 91 64 81 64 91 74 85 84
64 74 74 82 84 83 81 63 81 81 74 74 82 62
64 75 83 82 84 91 75 74 65 83 75 75 75 93
63 65 65 81 63 81 75 85 75 75 64 62 82 92
85 74 63 82 75 74 83 81 65 81 84 85 64 85
64 85 85 63 82 72 62 83 62 81 81 72 81 64
63 75 82 81 64 83 63 82 85 81 63 63 63 04
74 81 91 91 84 63 85 84 65 64 85 65 62 94
62 62 85 91 85 91 74 91 72 75 64 65 75 71
65 83 62 64 74 81 82 84 62 82 64 91 81 93
65 62 64 84 84 91 83 85 74 91 81 65 72 74
83 83 85 82 83 64 62 72 62 65 62 83 75 92
72 63 82 82 72 72 83 82 85 84 75 82 81 83
72 84 62 82 83 75 81 64 75 74 85 81 62 92

All in all, Willemse's idea of a spiral transposition does seem intriguing: but perhaps a little more psychologically ornate than d'Agapeyeff would have considered necessary as an exercise for the reader. If I were actively looking for a solution to this cipher (which I'm not), I would instead start with the four basic diagonal transpositions of the 14x14, and see if they led anywhere interesting... you never know! :-)

The d'Agapeyeff Cipher...

Back in 1939, Alexander d'Agapeyeff wrote a tidy little book called "Codes and Ciphers" on cryptography history: though you can now buy it print-on-demand, cheap copies of the original book often come up on the various second-hand book aggregators (such as bookfinder.com), which is where I got my copy of the "Revised and reset" 1949 edition.

What is now generally understood is that d'Agapeyeff wasn't really a cryptographer per se: he had previously written a similar book on cartography for the same publisher, and so thought to tackle cryptography.

On the very last page of the text (p.144), d'Agapeyeff dropped in a little cipher challenge, saying "Here is a cryptogram upon which the reader is invited to test his skill."

75628 28591 62916 48164 91748 58464 74748 28483 81638 18174
74826 26475 83828 49175 74658 37575 75936 36565 81638 17585
75756 46282 92857 46382 75748 38165 81848 56485 64858 56382
72628 36281 81728 16463 75828 16483 63828 58163 63630 47481
91918 46385 84656 48565 62946 26285 91859 17491 72756 46575
71658 36264 74818 28462 82649 18193 65626 48484 91838 57491
81657 27483 83858 28364 62726 26562 83759 27263 82827 27283
82858 47582 81837 28462 82837 58164 75748 58162 92000

This modest little cryptogram, now known as "the d'Agapayeff Cipher", has somehow remained unbroken for 70 years, and is often to be found alongside the Voynich Manuscript on lists of cipher enigmas.

The first thing to note is that adjacent columns are formed alternately from 67890 and 12345 characters respectively: which is a huge hint that what we are looking at is (in part, at least) a grid cipher, where each pair of numbers gives a position in a grid. If so, then we can throw away the "patristrocat" spaces between the blocks of numbers and rearrange them as pairs.

75 62 82 85 91 62 91 64 81 64 91 74 85 84 64 74 74 82 84 83 81 63 81 81 74
74 82 62 64 75 83 82 84 91 75 74 65 83 75 75 75 93 63 65 65 81 63 81 75 85
75 75 64 62 82 92 85 74 63 82 75 74 83 81 65 81 84 85 64 85 64 85 85 63 82
72 62 83 62 81 81 72 81 64 63 75 82 81 64 83 63 82 85 81 63 63 63 04 74 81
91 91 84 63 85 84 65 64 85 65 62 94 62 62 85 91 85 91 74 91 72 75 64 65 75
71 65 83 62 64 74 81 82 84 62 82 64 91 81 93 65 62 64 84 84 91 83 85 74 91
81 65 72 74 83 83 85 82 83 64 62 72 62 65 62 83 75 92 72 63 82 82 72 72 83
82 85 84 75 82 81 83 72 84 62 82 83 75 81 64 75 74 85 81 62 92 00 0[0]

The first hint that the order of these might have been scrambled ('transposed') comes from the two sets of tripled letters: 75 75 75 and 63 63 63. Five centuries ago, even Cicco Simonetta and his Milanese cipher clerks knew that tripled letters are very rare (the only one in Latin is "uvula", 'little egg'). The second hint that this is a transposition cipher is the total number of characters (apart from the "00" filler at the end): 14x14. If we discard the filler & rearrange the grid we get:-

75 62 82 85 91 62 91 64 81 64 91 74 85 84
64 74 74 82 84 83 81 63 81 81 74 74 82 62
64 75 83 82 84 91 75 74 65 83 75 75 75 93
63 65 65 81 63 81 75 85 75 75 64 62 82 92
85 74 63 82 75 74 83 81 65 81 84 85 64 85
64 85 85 63 82 72 62 83 62 81 81 72 81 64
63 75 82 81 64 83 63 82 85 81 63 63 63 04
74 81 91 91 84 63 85 84 65 64 85 65 62 94
62 62 85 91 85 91 74 91 72 75 64 65 75 71
65 83 62 64 74 81 82 84 62 82 64 91 81 93
65 62 64 84 84 91 83 85 74 91 81 65 72 74
83 83 85 82 83 64 62 72 62 65 62 83 75 92
72 63 82 82 72 72 83 82 85 84 75 82 81 83
72 84 62 82 83 75 81 64 75 74 85 81 62 92

This is very probably the starting point for the real cryptography (though the presence of tripled characters in the columns implies that it probably isn't a simple "matrix-like" diagonal transposition. Essentially, it seems that we now have to solve a 14x14 transposition cipher and a 5x5 substitution cipher simultaneously, over a relatively small cryptogram - an immense number of combinations to explore.

However, we know that d'Agapeyeff wasn't a full-on cryptographer, so we should really explore the psychological angle before going crazy with an 800-year-long brute-force search. For a start, if you lay out the frequencies for the 5x5 letter grid (with 12345 on top, 67890 on the left), a pattern immediately appears:-

** .1 .2 .3 .4 .5
6. _0 17 12 16 11
7. _1 _9 _0 14 17
8. 20 17 15 11 17
9. 12 _3 _2 _1 _0
0. _0 _0 _0 _1 _0

Here, the 61 (top-left) frequency is 0, the 73 frequency is 0, and the final nine frequencies are 3, 2, 1, 0; 0, 0, 0, 1, 0. I think this points to a 5x5 mapping generated by a keyphrase, such as "Alexander d'Agapeyeff is cool" (for example). To make a keyphrase into a 5x5 alphabet, turn all Js into Is (say), remove all duplicate letters (and so it becomes ALEXNDRGPYFISCO), and then pad to the end with any unused characters in the alphabet in sequence (BHKMQTUVWZ)

* 1 2 3 4 5
6 A L E X N
7 D R G P Y
8 F I S C O
9 B H K M Q
0 T U V W Z

For a long-ish (but language-like) keyphrase, rare characters would tend to get moved to the end of the block: which is what we appear to see in the frequency counts above, suggesting that the final few letters are (for example) W X Y Z or W X Z.

Yet 61 and 73 have frequency counts of zero, which points to their being really rare letters (like Q or Z). However, if you read the frequency counts as strings, 61 62 63 = 0 17 12, while 73 74 75 = 0 14 17: which perhaps points to the first letter of the keyphrase (i.e. 61) being a rare consonant, and the second pair being Q U followed by a vowel. Might 73 74 75 76 77 be QUIET or QUITE?

I don't (of course) know: but I do strongly suspect that it might be possible for a cunning cryptographer to crack d'Agapeyeff's keyphrase quite independently of his transposition cipher. It can't be that hard, can it? ;-p

----------
Update: a follow-up post is here...

Dots for vowels, revisited...

One very early cipher involved replacing the vowels with dots. In his "Codes and Ciphers" (1939/1949) p.15, Alexander d'Agapeyeff asserts that this was a "Benedictine tradition", in that the Benedictine order of monks (of which Trithemius was later an Abbot) had long used it as a cipher. The first direct mention we have of it was in a ninth century Benedictine "Treatise of Diplomacy", where it worked like this:-

• i = .
• a = :
• e = :.
• o = ::
• u = ::.

"R:.:lly", you might well say, "wh:t : l:::d ::f b::ll::cks" (and you'd be r.ght, ::f c::::.rs:.). But for all its uselessness, this was a very long-lived idea: David Kahn's "The Codebreakers" (1967) [the 1164-page version, of course!] mentions the earlier St Boniface taking a dots-for-vowels system from England over to Germany in the eighth century (p.89), a "faint political cryptography" in Venice circa 1226, where the vowels in a few documents were replaced by "dots or crosses" (p.106), as well as vowels being enciphered in 1363 by the Archbishop of Naples, Pietro di Grazie (p.106).

However, perhaps the best story on the dots-for-vowels cipher comes from Lynn Thorndike, in his "History of Magic & Experimental Science" Volume III, pp.24-26. In 1320, a Milanese cleric called Bartholomew Canholati told the papal court at Avignon that Matteo Visconti's underlings had asked him to suffumigate a silver human statuette engraved with "Jacobus Papa Johannes" (the name of the Pope), as well as the sigil for Saturn and "the name of the spirit Amaymom" (he refused). He was then asked for some zuccum de napello (aconite), the most common poison in the Middle Ages (he refused). He was then asked to decipher some "'experiments for love and hate, and discovering thefts and the like', which were written without vowels which had been replaced by points" (he again refused). The pope thought it unwise to rely on a single witness, and sent Bartholomew back to Milan; the Viscontis claimed it was all a misunderstanding (though they tortured the cleric for a while, just to be sure); all in all, nobody comes out of the whole farrago smelling of roses.

(Incidentally, the only citation I could find on this was from 1972, when William R. Jones wrote an article on "Political Uses of Sorcery in Medieval Europe" in The Historian: clearly, this has well and truly fallen out of historical fashion.)

All of which I perhaps should have included in Chapter 12 of "The Curse of the Voynich", where I predicted that various "c / cc / ccc / cccc" patterns in Voynichese are used to cipher the plaintext vowels. After all, this would be little more than a steganographically-obscured version of the same dots-for-vowels cipher that had been in use for more than half a millennium.

As another aside, I once mentioned Amaymon as one of the four possible compass spirits on the Voynich manuscript f57v (on p.124 of my book) magic circle: on p.169 of Richard Kieckhefer's "Magic in the Middle Ages", he mentions Cecco d'Ascoli as having used N = Paymon, E = Oriens, S = Egim, and W = Amaymen (which is often written Amaymon). May not be relevant, but I thought I'd mention it, especially seeing as there's the talk on magic circles at Treadwell's next month (which I'm still looking forward to).

Finally, here's a picture of Voynichese text with some annotations of how I think it is divided up into tokens. My predictions: vowels are red, verbose pairs (which encipher a single token) are green, numbers are blue, characters or marks which are unexpected or improvised (such as the arch over the '4o' pair at bottom left, which I guess denotes a contraction between two adjacent pairs) are purple. Make of it all what you will!