All the SATOR squares in English, with code

Using code for recreational word play is very fun. Having fun with a housemate we drunkenly built a list of all 4- and 5- sided SATOR squares.

These squares are special because they read the same left-to-right, top-to-bottom, and both directions in reverse. The famous one is an ancient one from Latin

SATOR
AREPO
TENET
OPERA
ROTAS

In the ancient world these were used both as magic spells and possibly the first memes. They pop up all over Western history, and into today: the movie Tenet is a reference to this square.

Here are the 70 5-letter squares:

‘assam’, ‘shama’, ‘sagas’, ‘amahs’, ‘massa’
‘assam’, ‘shama’, ‘samas’, ‘amahs’, ‘massa’
‘assam’, ‘shaya’, ‘sagas’, ‘ayahs’, ‘massa’
‘assam’, ‘shaya’, ‘samas’, ‘ayahs’, ‘massa’
‘asses’, ‘slive’, ‘simis’, ‘evils’, ‘sessa’
‘asses’, ‘slive’, ‘siris’, ‘evils’, ‘sessa’
‘asses’, ‘state’, ‘sagas’, ‘etats’, ‘sessa’
‘asses’, ‘state’, ‘samas’, ‘etats’, ‘sessa’
‘asses’, ‘stime’, ‘simis’, ’emits’, ‘sessa’
‘asses’, ‘stime’, ‘siris’, ’emits’, ‘sessa’
‘asses’, ‘swone’, ‘solos’, ‘enows’, ‘sessa’
‘ayahs’, ‘yrneh’, ‘anana’, ‘henry’, ‘shaya’
‘cares’, ‘amene’, ‘refer’, ‘enema’, ‘serac’
‘dedal’, ‘enema’, ‘deked’, ‘amene’, ‘laded’
‘dedal’, ‘enema’, ‘deled’, ‘amene’, ‘laded’
‘dedal’, ‘enema’, ‘dered’, ‘amene’, ‘laded’
‘dedal’, ‘enema’, ‘dewed’, ‘amene’, ‘laded’
‘derat’, ‘enema’, ‘refer’, ‘amene’, ‘tared’
‘gater’, ‘amene’, ‘tenet’, ‘enema’, ‘retag’
‘gnats’, ‘nonet’, ‘anana’, ‘tenon’, ‘stang’
‘hales’, ‘amene’, ‘lemel’, ‘enema’, ‘selah’
‘hales’, ‘amene’, ‘level’, ‘enema’, ‘selah’
‘laded’, ‘amene’, ‘deked’, ‘enema’, ‘dedal’
‘laded’, ‘amene’, ‘deled’, ‘enema’, ‘dedal’
‘laded’, ‘amene’, ‘dered’, ‘enema’, ‘dedal’
‘laded’, ‘amene’, ‘dewed’, ‘enema’, ‘dedal’
‘lares’, ‘amene’, ‘refer’, ‘enema’, ‘seral’
‘massa’, ‘amahs’, ‘sagas’, ‘shama’, ‘assam’
‘massa’, ‘amahs’, ‘samas’, ‘shama’, ‘assam’
‘massa’, ‘ayahs’, ‘sagas’, ‘shaya’, ‘assam’
‘massa’, ‘ayahs’, ‘samas’, ‘shaya’, ‘assam’
‘pelas’, ‘enema’, ‘lemel’, ‘amene’, ‘salep’
‘pelas’, ‘enema’, ‘level’, ‘amene’, ‘salep’
‘resat’, ‘enema’, ‘sedes’, ‘amene’, ‘taser’
‘resat’, ‘enema’, ‘seles’, ‘amene’, ‘taser’
‘resat’, ‘enema’, ‘semes’, ‘amene’, ‘taser’
‘resat’, ‘enema’, ‘seres’, ‘amene’, ‘taser’
‘resat’, ‘enema’, ‘sexes’, ‘amene’, ‘taser’
‘retag’, ‘enema’, ‘tenet’, ‘amene’, ‘gater’
‘salep’, ‘amene’, ‘lemel’, ‘enema’, ‘pelas’
‘salep’, ‘amene’, ‘level’, ‘enema’, ‘pelas’
‘selah’, ‘enema’, ‘lemel’, ‘amene’, ‘hales’
‘selah’, ‘enema’, ‘level’, ‘amene’, ‘hales’
‘serac’, ‘enema’, ‘refer’, ‘amene’, ‘cares’
‘seral’, ‘enema’, ‘refer’, ‘amene’, ‘lares’
‘sesey’, ‘edile’, ‘simis’, ‘elide’, ‘yeses’
‘sesey’, ‘edile’, ‘siris’, ‘elide’, ‘yeses’
‘sesey’, ‘elide’, ‘simis’, ‘edile’, ‘yeses’
‘sesey’, ‘elide’, ‘siris’, ‘edile’, ‘yeses’
‘sessa’, ’emits’, ‘simis’, ‘stime’, ‘asses’
‘sessa’, ’emits’, ‘siris’, ‘stime’, ‘asses’
‘sessa’, ‘enows’, ‘solos’, ‘swone’, ‘asses’
‘sessa’, ‘etats’, ‘sagas’, ‘state’, ‘asses’
‘sessa’, ‘etats’, ‘samas’, ‘state’, ‘asses’
‘sessa’, ‘evils’, ‘simis’, ‘slive’, ‘asses’
‘sessa’, ‘evils’, ‘siris’, ‘slive’, ‘asses’
‘shaya’, ‘henry’, ‘anana’, ‘yrneh’, ‘ayahs’
‘stang’, ‘tenon’, ‘anana’, ‘nonet’, ‘gnats’
‘start’, ’tiler’, ‘alula’, ‘relit’, ‘trats’
‘tared’, ‘amene’, ‘refer’, ‘enema’, ‘derat’
‘taser’, ‘amene’, ‘sedes’, ‘enema’, ‘resat’
‘taser’, ‘amene’, ‘seles’, ‘enema’, ‘resat’
‘taser’, ‘amene’, ‘semes’, ‘enema’, ‘resat’
‘taser’, ‘amene’, ‘seres’, ‘enema’, ‘resat’
‘taser’, ‘amene’, ‘sexes’, ‘enema’, ‘resat’
‘trats’, ‘relit’, ‘alula’, ’tiler’, ‘start’
‘yeses’, ‘edile’, ‘simis’, ‘elide’, ‘sesey’
‘yeses’, ‘edile’, ‘siris’, ‘elide’, ‘sesey’
‘yeses’, ‘elide’, ‘simis’, ‘edile’, ‘sesey’
‘yeses’, ‘elide’, ‘siris’, ‘edile’, ‘sesey’

To generate them yourself (and the fours), here is code that you can run by pressing Play.
https://colab.research.google.com/drive/14gaONdrLuxc3Pzz6M8y1RWpTLIdLY7H0?usp=sharing
For words we used the official Scrabble list. The tests are hard to read but they check the symmetries of the square.

The interesting findings are that

• there are 70 5-letter ones in English,
• 494 in 4 letters,
• none use only familiar words,
• few make technically readable sentences,
• we did surprisingly well building 4-letter ones by hand without the help of code, but
• building 5 letter ones by hand is very very hard
• They are counter-intuitive and having code made it a lot easier to think about them and understand the constraints they have to satisfy

The basic rules in building them are that

• all n words have to be n letters long,
• each should be reversible (form a word in both directions),
  • if there is a middle word (for 3- and 5- and other odd lengths), it should be a palindrome (e.g. “TENET”; palindromes are a special case of reversible words), and
• at least one should begin with a vowel
  • in english the only vowels that appeared in legal 5-letter vowel-ended words in our SATOR squares were a and e, with a’s accounting for the majority.

The next challenge would be to build a SATOR cube (filled or hollow—n slices of cube or a cube with one square on each face). Probably there are none in 5 letters (if there are any, I’d guess there is just one), a couple in 4 letters, and several in 3 letters, with filled obviously more rare than hollow.

Another challenge would be to find words that I want to include that aren’t on the Scrabble list and see if they change anything.