Tools for Discovering New Sator Squares

What is the Sator Square?

The Sator Square is a famous "2D Palindrome" in Latin that was widespread through the ancient world.

Its 5 Latin words in a grid read the same in any direction. Try it!

SATOR
AREPO
TENET
OPERA
ROTAS

• Left-to-right then top-to-bottom (normal English order): Sator Arepo tenet opera rotas.
• Top-to-bottom then left-to-right: Sator Arepo tenet opera rotas.
• Right-to-left then bottom-to-top: Sator Arepo tenet opera rotas.
• Bottom-to-top then right-to-left: Sator Arepo tenet opera rotas.

If written out as a 1D sentence, it is also a palindrome.

For the careful reader who realizes that there are four more possibilities (e.g., left-to-right then bottom-to-top), the reverse word order ("Rotas opera tenet Arepo Sator") means the same thing in Latin because the word order doesn't really matter in this regard.

The Sator Square was reproduced all over Europe for hundreds of years and its cryptic meaning was embued with mystical significance because of the sheer awesomeness of its palindromic properties, which led many to believe that the phrase must mean something deep.

As for how to translate it fittingly, you can read various opinions online, but I'm going with "Arepo the Sower holds the Wheel Work." In both pagan and Christian milieux this sort of translation tended to see the phrase as indicating that God (or Christ as "the Sower", or the logos spermatikos of Stoic philosophy) governed the world (by having hold of its "wheels" = inner workings).

Word mysticism aside, it's been a long time since we've had another Sator-like square, and they are not easy to make up with pencil and paper and sheer cleverness.

Given the enormous popularity of the first Sator Square, wouldn't it be nice to find more? Now you can!

Tools Required: An algorithm and a dictionary

Algorithm

The algorithm is not too hard to discover. To make the Sator Square work, the 5x5 matrix has the same characters reflected across both diagonals. This fundamental property generalizes to Sator Squares of any size.

For odd sizes (5x5, 7x7) we require a palindromic word in the center row, and then pairs of words such that one is the reverse of the other for all the other rows. We shall call such pairs palindromic pairs. For even sizes, we only require palindromic pairs of words.

As my daughter observed, the requirement that all palindromic pairs be comprised of two meaningful words is actually more rigorous than the original Sator Square since "Arepo" means nothing in Latin, and by sheer presumption "Arepo" was taken as a mysterious proper name for the sake of an intelligible reading. (So I provide an optional way likewise to soften this requirement for one palindromic pair if you wish.)

The rest of the algorithm comes down to finding combinations of palindromes and palindromic pairs that satisfy our further requirement about how the characters are the same reflected across the diagonals.

For a 5x5 matrix, if we label the coordinates row-first then column like so:

(0,0)   (0,1)   (0,2)   (0,3)   (0,4)
(1,0)   (1,1)   (1,2)   (1,3)   (1,4)
(2,0)   (2,1)   (2,2)   (2,3)   (2,4)
(3,0)   (3,1)   (3,2)   (3,3)   (3,4)
(4,0)   (4,1)   (4,2)   (4,3)   (4,4)

We require that the character C(0,1) = C(1,0), C(0,2)=C(2,0), C(1,2)=C(2,1), etc. for one diagonal, and also that C(0,3)=(1,4), C(0,2)=C(2,4), C(1,2)=C(2,3), etc. for the other diagonal. Closer consideration reveals that because of symmetry, we don't have to check all these conditions.

For the 5x5 matrix, we proceed as follows:

• Comb a text file dictionary for words that are 5 characters long and put them in the words vector
• Search the words vector, and create a palindromes vector of all palindromic words (for the middle row of the square)
• Search the words vector, and create a pairs vector for the palindromic pairs needed for the other rows. This task has N-squared time complexity, but it doesn't take too long on a modern computer even for a large dictionary.

If we go with the stricter requirement that all palindromic pairs are meaningful words in the dictionary, then next we do:

• For every word in the pairs vector (=candidate for Row 0 word), find every word in the pairs vector such that C(0,1)=C(1,0) and C(0,3)=C(1,4), which makes it a candidate for a Row 1 word.
• For every combination we find above, search the palindromes vector for a word such that C(0,2)=C(2,0) and C(1,2)=C(2,1).
• If we a palindrome that satisfies these requirements, we've got a Sator Square! Save the trio to the results vector. Otherwise, move on.

If we relax the requirement that the second/fourth word of the Sator Square not be a palindromic pair, but we permit one word of the pair to mean nothing, then first step in the immediately preceding list becomes:

• For every word in the pairs vector (=candidate for Row 0 word), find every word in the words vector such that C(0,1)=C(1,0) and C(0,3)=C(1,4), which makes it a candidate for a Row 1 word.

Our approach can be generalized in a similar fashion for Sator Squares of larger size.

Dictionary Challenges: Getting All the Latin Words!

If one wishes to work in English, one can download spell-check dictionaries of English which have all the valid forms of an English root word as individal entries in the plaintext file. These plaintext files are easy to parse. I have saved several to the repo. The smaller ones do not have enough words to yield any 5x5 Sator Square.

Sizeable English spell-checking plaintext dictionaries exist (e.g., 130K individual words) in part because English morphology is relatively simple. Consider the word "show." For the noun, there's simply "show" and "shows" for all forms. The same word is also a verb, but its morphology is complete with "show," "shows," "showed" and "shown" -- and two of these four words are already in the dictionary under the noun's forms.

Latin is a different story! Most nouns decline into 10 different forms based on their number and the their case, and the verbs conjugate into hundreds of forms based on person, number, tense, voice and mood, plus participial forms. There are vastly more Latin words to sift! How many more? Compared to the 130K entries for an extensive English dictionary with some very obscure words, consider 7.2M entries.

How to get a list of all possible Latin words

I started with the Hunspell Latin dictionary, which is impressively large in terms of its number of root words. You can download it at The Free Office Website or from Titus Wormer's Github. (Hunspell is commonly used for spellchecking in Linux applications like LibreOffice.)

Hunspell dictionaries do not store all possible forms of a word as individual entries. Rather, they consist of a pair of files: a dictionary file (.dic) which has all the root words, one to a line, and followed by a slash (/) and a series of characters which are flags for their morphology rules. A separate file (.aff) contains all the morphology rules. This provides for considerable compaction. The Ubuntu Hunspell manual page provides details about how all this works.

For our use, however, we need to unpack that collection into every conceivable form of every Latin word. Thankfully, hunspell provides the ability to do so for any pair of .dic and .aff files with its unmunch utility found in the Hunspell Github repo.

There is a second problem, however, which arises from the fact that these are spell-checking dictionaries and Latin permits the writing of the vowel u with the same character as the consonant v. Thus ubique can be written vbiqve. While such usage reflects the writing of antiquity, I find it deeply annoying (so did Emperor Claudius, who tried to discontinue the practice) and moreover it produces duplicates of what is the same word, simply spelled with two different characters. (The Hunspell dictionary did not do likewise with i and j, which are similarly interchangeable in later Latin).

So first I wrote a brief program to cull all the morphology rules in the .aff file that used v in place of u (e.g., changing a -us or -vs ending to -vm rather than -um for second declension nominative masculine to accusative case). A few rules still survived so I deleted what I saw remaining by hand. Next, by inspection the .dic file was easy to remedy. Its first half had all the root words with the u spelling, and the second half simply repeated all these with the v-for-u spelling. I deleted this half. The culled .aff and .dic files are available in my repository. I further eliminated the end of the .dic file which contained abbreviations and Roman numerals since these are useless for our purposes.

Then I set unmunch to work unpacking all the forms. The result? A massive list. A few other abbreviations remained from elsewhere but they don't cause much trouble as the period at the end means there's not a palindromic pair matching it, since words don't start off with periods. But I noticed it still had some duplicates (not due to the v-for-u spelling rules -- I'm not sure what causes them) so I wrote a simply program to reduce the list to unique entries. The result? A word list of 7,278,827 unique Latin words! I have uploaded it to my repository in .zip format to save space. (It's 13.8MB zipped, 104MB unzipped!) Here's a little graph that compares how many more Latin words there are than English:

I used this Latin word list as the dictionary for the NewSatorSquare program which implements the algorithm above. I got about 4000 results. Now all that remains is to comb through the squares myself for ones that actually mean something in Latin and are not just an interlocking grid of word salad!

As a side note: Another feature of the Hunspell dictionary is support of ligatures for diphthongs (e.g., ae written as one character). These I did not cull simply out of personal preference. They might make for some interesting options. At the same time, one could easily write a simple program to filter these out of the dictionary list too.

Further extensions

The code can be expanded to include algorithms for larger size Sator Squares (e.g., 6x6 or 7x7). While a larger square requires more conditions are met to find a set of words that fits, in Latin at least, there many more words of length 7 or more than in English, as the chart above attests. So perhaps larger squares can be found -- and maybe some meaningful expressions among them. I don't know if the ancients ever produced a larger square than 5x5.

But why stop there? It would also be fun to do something more radical. What about a three-dimensional palindrome? Words in a cube that read the same left-to-right, front-to-back, top-to-bottom, reflect about the four diagonals of the cube? Or a four-dimensional one, a Sator hypercube, just to make sure it hadn't been done before in antiquity. :-)