Are there any sentences that could be written on a cylinder and merge their finish into their start, such that they form a valid sentence no matter where the reader begins?
I'm assuming that the reader is guaranteed to start at a word boundary. There are some trivial solutions
because any one-word sentence (e.g., "No!") obviously works
and some slightly less trivial solutions
from two-word sentences (e.g., "I am" / "am I?", if you don't mind the punctuation issue.
Another variety of triviality
comes from sentences where all the words are the same. For instance, any number of copies of the word "buffalo" forms a sentence! "Buffalo" is a plural noun (the animal), an attributive adjective ("from Buffalo"), and a transitive verb (roughly synonymous with "harass"), so e.g. "buffalo Buffalo buffalo buffalo buffalo buffalo" can be parsed as "those buffalo that Buffalo buffalo buffalo, buffalo buffalo"; that is "those cattle that Texan cattle harass, harass cattle". So if we write our words all in capitals, we can put any number of instances of BUFFALO around our cylinder.
A bit less trivially,
"train school kid" = "teach child who goes to school", "school kid train" = "give education to railway train full of children", and "kid train school" = "play a joke on place where people learn about trains".
Same sort of structure but maybe a little less contrived:
"roast sheep skin" = "put lamb's outer layer in oven", "sheep skin roast" = "the lambs remove the outer layer from something that's been roasted", and "skin roast sheep" is "take off the outer layer from a lamb that's been in the oven".
We can extend this (content warning: one of the meanings is very unpleasant):
"roast black sheep skin", "black sheep skin roast" and "skin roast black sheep" are all just like the foregoing but with an adjective qualifying "sheep" and (unfortunately) "sheep skin roast black" is about some sheep piling further abuse on the victim of a particularly gruesome racist atrocity.
One more word:
prototype machine breaks separate aircraft
machine breaks separate aircraft prototype
breaks separate aircraft prototype machine
separate aircraft prototype machine breaks
aircraft prototype machine breaks separate
where the main difficulty is that "separate" is a verb in the third and fifth versions. (Transitive in #3 and intransitive in #5.)
I am allowed
proper names as verbs?
yes? ok then :-)
Dice rob fools(Some fools gamble on some sort of dice game, and lose; by metonymy, we can say that dice rob them.)
Rob fools dice(Rob, on the other hand is clever, and wins; we can say he fools dice.)
Fools dice Rob(Fools get jealous, gang up, and cut him to pieces. Poor Rob!)
I guess the moral of the story is nobody likes a smarty pants :-) please don't hurt me