Hello, everyone!

This is strange, but it seems to me that for most Codex “readers” its page numbers remain an enigma. This is strange because these numbers are perfectly clear and logical – all it takes to get them is minimal thinking!

I find it sad that only two people (seem to) have ever attempted to do such work. Allan C. Wechsler and
Ivan A. Derzhanski did it with some success in 1987 and 2004 respectively. To see their results, check
groups.google.com/group/rec...7771903c1d
and
www.math.bas.bg/~iad/serafin.html

Unfortunately, their very brief sketches are incomplete and contain some little mistakes plus one gross one (common for both).

My point here is to show that counting in Seraphinian is absolutely no rocket science!

I’ll do my best to make those numbers, which can be found at the bottom of the Codex's pages, perfectly clear. I hope I’ll be able to adequately convey the transparency of the Seraphinian numeral system (SNS).

Now let’s begin!

SNS is a non-positional system – like the well-known Roman one and unlike ours. This means, to understand a number, you don’t “read” it from left to right but rather interpret the combination of its symbols.

SNS has the base 21.

Like we have 10 symbols from 0 to 9 for counting, Seraphinians have 21 such unique symbols from 1 to 21.

For some reason, both Wechsler and Derzhanski fail to see (and this is their huge mistake!) that SNS doesn’t employ the concept of (and hence the notation for) “zero”. This is exactly the way it goes with the Roman number system. For example, though Roman numerals have the same base 10 as our daily numbers, the Roman symbols – say, 10, 50, 100 (X, L, C) – have no sign for 0!

As we don’t have distinct 21 symbols of our own fashion, I propose that we use some ad hoc notation in case we need it:

1, 2, 3, 4, 5, 6, 7, 8, 9, (10), (11), (12), (13), (14), (15), (16), (17), (18), (19), (20), (21).

For example, (14) means here number 14 represented by one single Seraphinian symbol.

These 21 symbols will be referred to as digits.

They can be found when you open either part of the Codex in the beginning. The first page number you see is 2. It’s on the left. On the right is 3. Page by page, you get to know these characters from 2 to 21. Then you notice that from 22 on, the symbols are combined with a vertical bar. The first character you see behind this bar is 1. Now you know what 1 looks like.

Be careful handling them! Since they all are not “typed” but “handwritten”, 1 often looks pretty much the same way as (20), while 2 and (16) happen to be perfect twins!

To use SNS one has to know some rules and a few EXCEPTIONS! Exceptions I’ll deal with in the end of this text.

I’ll refer to the mentioned rules as Generative Rules (GRs).

At some point it becomes obvious that there are more GRs than we can guess about from the book. Theoretically, there are infinitely many of them. All we know for sure are the first 8 GRs that enable us to construct numbers up to 189 (which is more than the number of pages in either part!).

The SNS number may consist of
a) Digits (they can be used independently);
b) Conventional signs (they can be used only in combination with digits).

Again, there are 21 digits in SNS. They are enough to count up to 21. To count on, we combine them with conventional signs according to GRs.

Thus, a GR is a rule that tells us how to arrange digits and conventional signs in a linear order!

As far as one can tell using the book, there are 3 conventional signs. (To be honest, there are 4, but one of them appears very seldom, so we’ll ignore it for the most part and deal with it separately later.) Unlike the digits (which are mostly “curves”), the conventional signs are easy to represent through typing. They are: I (a vertical bar), V and /\.

Since the base of SNS is 21, any number can be represented as:

21*N + X

Here N = any non-negative integer (0, 1, 2, 3, …),
and X = any number from 1 to 21.
Thus, if we count from 1 to 21, then N=0.
Counting from 22 to 42 makes N=1; from 43 to 63 makes N=2, etc.

Example:
80 = 21*3 + 17
Here N = 3; X = 17.

NB: 63, for example, is a multiple of 21. In a virtual 21-based number system structured like ours of the present day we would expect it to look like 21*3 + 0 = 63. This is NOT the way SNS works! In SNS, it goes 21*2 + 21 = 63, since Seraphinians have no “zero”! (To use a rough comparison, it’s as though we had to say “twenty ten” instead of “thirty”.)

Now goes the key idea.

FOR EVERY N THERE IS A UNIQUE GR.

Actually, the above statement has to be expanded:

FOR SOME N THERE ARE TWO UNIQUE GRs.

In fact, for most N in the book (from N=2 to N=6) there are 2 GRs.

Now, there’s the curious but important peculiarity of SNS. Please, keep in mind that it’s of purely conventional nature. It has nothing to do with counting as such. It only affects the notation.

Say, we have a number (with its N and its X, as you remember). If its N is less than 2 or larger than 6, then never mind.

But if its N is in the range from 2 to 6, the “16+ Rule” comes into play!!!

The “16+ Rule” says: For every number with N = 2, 3, 4, 5, 6 there are two GRs: one for all X smaller than 16, the other for all X equal to or larger than 16.

(As I already told you, we only know the numbers with N no larger than 8, so maybe the “16+ Rule” is reemployed later on – at some point outside the Codex.)

We’ll denote a GR for N as GR (N).
This means, for example, that GR (8) is a Generative Rule that shows us how to write a number represented as 21*8 + X.

GR (N-) will stand for numbers with X < 16.
GR (N+), as it’s easy to guess, will stand for numbers with X = 16 or X > 16.

For example, GR (5-) shows us the way to write out the numbers:
21*5 + 1
21*5 + 2
…
21*5 + 15.

In their turn, all the numbers
21*5 + 16
21*5 + 17
…
21*5 + 21
are constructed according to GR (5+).

Now, behold

THE TABLE OF GRs!

GR (1) I X

GR (2-) I I X

GR (2+) V I X

GR (3-) V I I X

GR (3+) /\ V I X

GR (4-) /\ V I I X

GR (4+) I /\ V I X

GR (5-) /\ V I I X X

GR (5+) I /\ V I X X

GR (6-) X I I X X

GR (6+) I X I X X

GR (7) I X X X I

GR (8) I I X X X I


Now you are probably wondering what the hell all that is supposed to mean.

So, here are some clues.

Yes, the X in the right column designates the X from 21*N + X. The symbols I, V and /\ are the conventional signs introduced in the beginning. To construct numbers, we put digits instead of X.

Here’s the coding guideline.
For example, convert number 133 into SNS.
Start by finding that 133 = 21*6 + 7.
Obviously, you’ll need to apply for either GR (6+) or GR (6-), since in 21*6 + 7 we have N = 6.
Since X = 7, which is less than 16, you’ll need GR (6-).
So, you find GR (6-) in the Table of GRs and put down:
7 I I 7 7
Voila!
Congratulations!
That’s 133 in Seraphinian!

Now let’s try it backwards.
Here’s the decoding guideline.
Choose a Seraphinian number. For example, let it be the Rosetta Stone page number, which is:
/\ V I r
(its last symbol looks pretty much like the lowercase “r”).
Now, scan the right column of the Table of GRs for this number’s look-alike. You are sure to see that your number is constructed according to GR (3+).
So, it must be of the form:
21*3 + r
You have to examine the digits to find out that “r” is (16).
Congratulations!
Now you know that the Rosetta Stone is printed on page 21*3 + 16 = 79 (of Part 2)!

However, we might have had trouble doing that, since (16) often looks exactly like 2. In our particular example it’s no problem because, minding the “16+ Rule”, we see that /\ V I r is made by a GR (N+), so it can’t have 2. Suppose we had an “r” number not affected by the “16+ Rule”. Then we would have to check out for either of the neighbor numbers. If they have 1 and 3, then “r” is 2. If they have (15) and (17), then “r” is (16).

By the way, I hope it’s absolutely clear why we ignore such thing as GR (0). This would simply mean 1, 2, 3, …, (20), (21) without any conventional signs at all.


Now it’s time for the EXCEPTIONS.
But… Before we proceed, I have to tell you a secret! Even two!

The first is: Although we don’t know how to make numbers with N larger than 8, we DO HAVE the ground to hypothesize about it!!!

How? Well, the conventional signs we deal with are juxtaposed according to certain combinational rules. We can’t say we know all those rules, but we still can guess a lot about some of them. Such rules, assuming they do exist, can be postulated in the form of restrictions.

For example, it’s obvious from the Table of GRs that no V has I next to it on the left. So, this must be a restriction: no V can (ever) have I next to it on the left and no I can (ever) have V next to it on the right. Yet, the word “ever” is not fully justified here. We are not sure if this rule – or restriction – doesn’t eventually break at some N > 8.

Some more observations/restrictions of this type:
X can have only X or I next on its left.
I can only be preceded by I or V.
V can only have V or /\ right in front of itself.

The digits, as compared to the conventional signs, possess more combinatorial freedom. That is, the conventional signs are more affected by restrictions. For example, one digit X can be repeated 3 times a row. No conventional sign can do that. Most conventional signs (except for I) can never even stand next to their copy.

Should you for any reason find the idea of restriction a bit artificial, please remember that this is exactly how the Roman numerals work. For example, any Roman numeral, say, X, is restricted to be used more than 3 times a row, so instead of XXXX we get XL.

Well, time for my other secret.

As you might have observed yourself, the digits 6 (/\) and 8 (/\ .) are the homographs of the conventional sign /\ (homograph means “written in the same way but having a different meaning”). The bottom dot next to 8 doesn’t play a big role.

So, the secret: Though /\ and /\ . are digits, they are treated as if they both were the conventional sign /\

It means that restrictions generally applied to the conventional sign /\ also apply to the digits /\ and /\ .
This leads us to…


EXCEPTION # 1

The graphical representation of (observable) numbers with X = 6 (/\) or 8 (/\ .) does not entirely rely on the Table of GRs, but obeys four restrictions.

I’ll introduce here some notation to show how really simple all this is. If something is forbidden, it will be put in parentheses and marked with an asterisk *(like this). To show what the forbidden form has to be substituted for, I’ll do this way: *(wrong) > (right).

This means, there happens a substitution or rather a transformation triggered by a certain restriction. In a narrow sense, restriction is the same as transformation, since one implies the other. But, generally speaking, that’s not always the case: some restrictions are realized in more than one transformation!

The 1st restriction (R1) states: /\ and /\ . can never be reduplicated: *(/\ /\).

This means, you CAN’T write /\ /\, even if you are required to do so by the Table of GRs. What are you supposed to do then? If the Table of GRs says you have to write /\ /\ or /\ . /\ ., then you must write /\ V and
/\ . V respectively. Or,

R1:

*(/\ /\) > (/\ V)

*(/\ . /\ .) > (/\ . V)

Logically follows…
the 2nd restriction (R2) that states: /\ and /\ . can never be triplicated: *(/\ /\ /\).

If the Table of GRs says you have to write /\ /\ /\ or /\ . /\ . /\ ., then you must write /\ V /\ and V /\ V respectively. Or,

R2:

*(/\ /\ /\) > (/\ V /\)

*(/\ . /\ . /\ .) > (V /\ V)


The 3rd restriction (R3) states: /\ can never have the conventional sign I on its right: *(/\ I).

If /\ is prescribed by the Table of GRs to be followed by the conventional sign I, this conventional sign I is substituted by the conventional sign V.

R3:

*(/\ I) > (/\ V)

Example:
Note the difference between 152 and 153
152 = 21*7 + 5 = I 5 5 5 I
153 = 21*7 + 6 = I 6 9 6 V
(Sorry, here we “cheated” and used 9 to simply put 6 upside down.)
We can reproduce this number in a true Serapninian way:
I /\ V /\ V
By R3, we place V in the end instead of I required by GR (7).

It’s worth mentioning that, semantically, the combination /\ V /\ V from the above example consists of three digits and one conventional sign (which is the final V), but, formally, they are absolutely indistinguishable!

Attention!
R3, as formulated above, holds only for final symbols (that is, when *(/\ I) has to be the last on the right)! Initial symbols (the first on the left) are transformed another way.
This is the case when one restriction is realized in 2 transformations! For the case of initial symbols let’s call it R3i. So,

R3i:

*(/\ I) > (V I)
*(/\ . I) > (V I)

The 4th restriction (R4) forbids V to have I next to it on the left: *(I V)

If I has to go before V, this I is substituted by the BRAND NEW SIGN ' /\, which is coined by means of /\ plus a left upper dot.

R4:

*(I V) > (' /\ V)

This fourth conventional sign ' /\ can, in its turn, be preceded by I.

Example (with the pseudo-Seraphinian & for 7):
175 = 21*8 + 7 = I I & & & I
176 = 21*8 + 8 = I ' /\ V /\ V I
How did we get this last number?
By R2, we can’t write /\. /\ . /\. required by GR (8), so we go V /\ V
Now, by R4, we can’t write I I V /\ V I
We have to change the second I for ' /\ which makes I ' /\ V /\ V I

Starts to look pretty much like rocket science?

Look what I suggest. All the numbers with digits 6 and 8 that can be found in the book are few enough to simply copy them! Let’s do this and see that their symbols are arranged according to GRs combined with the four restrictions:

First, let’s do it for 6 (/\):

21*1 + 6 = 27 = I /\

21*2 + 6 = 48 = I I /\

21*3 + 6 = 69 = V I I /\

21*4 + 6 = 90 = /\ V I I /\

So far it’s been as easy as a pie, uh?
Right, the algorithm is the same as for the rest of numbers!
At this point, however, the differences show up.
Next to each number below is provided its construction procedure showing the GRs plus restrictions that trigger transformation

21*5 + 6 = 111 = /\ V I I /\ V = (/\ V I I X X + R1)

21*6 + 6 = 132 = . V I I /\ V = (X I I X X + R1 + R3i + ?)

21*7 + 6 =153 = I /\ V /\ V = (I X X X I + R2 + R3)

21*8 + 6 = 174 = I I /\ V /\ V = (I I X X X I + R2 + R3)


As we can see, some extra restriction or rule we don’t know is involved in producing the N = 6 number (132). Where in the world does its front dot come from??? The reason why we can’t formulate this rule is, maybe, that we have only one number in the whole book that obeys it. One is too few for conclusions! However, please note that this leftmost bottom dot can be easily dispensed with. We can regard it as pure decoration! Check it out: without the dot, number 132 = V I I /\ V is clear to be constructed according to GR (6-), R1 and R3i.


Now let’s take 8 (/\ .):

21*1 + 8 = 29 = I /\ .

21*2 + 8 = 50 = I I /\ .

21*3 + 8 = 71 = V I I /\ .

21*4 + 8 = 92 = /\ V I I /\ .

Again! Easy so far – harder ever on!

21*5 + 8 = 113 = /\ V I I /\ . V = (/\ V I I X X + R1)

21*6 + 8 = 134 = V I I /\ . V = (X I I X X + R1 + R3i)

21*7 + 8 = 155 = ' /\ V /\ V I = (I X X X I + R2 + R4)

21*8 + 8 = 176 = I ' /\ V /\ V I = (I I X X X I + R2 + R4)

Well, listed above are all such numbers encountered in the Codex’s page numbering!


EXCEPTION # 2

Good news, everyone!
No more exceptions of THAT kind!!!

This one just says there appears for a moment a number succession irregularity.
It happens in both parts, with one little difference though.
The segment “…169, 170, 171, 172, 173, 174…” is for some reason transformed into “…169, 169a, 170, 171, 172, 174…”
The idea is that we add a new extra number 169a and drop out 173.
169a goes as I I 3 1 4 I.
As you see, this number is constructed according to GR (8) like its neighbors, but the collection of digits is weird.

The mentioned difference is that only in Part 1, number 170 that follows 169a is written upside down.


EXCEPTION # 3

In both parts, 178 is also dropped out.


EXCEPTION # 4

…that looks more like a printing failure!
In my Rizzoli 2006 edition, in part 2, the pages run “…, 175, 176, 177, 176, 177, 179,…”.
In other words, it’s as though the spread with 176 on the left and 177 on the right was (mis)printed twice. This concerns only the page numbers – the text and pictures go all right.


SOME CONJECTURES

Now let me hypothesize a little. This is just a fantasy, so it’s up to you how seriously you take it. Anyway, what could some further numbers possibly look like? I propose

GR (9) V I X X X I (this one is quite logical!)
GR (10) /\ V I X X
GR (11) /\ V I X X X



Well, that’s all I had to say.
At least some little fraction of the Codex’s content is no longer a mystery!
The rest is yet to be discovered!

Best regards,
Yenisey Avdeyev