An autogram (Greek: αὐτός = self, γράμμα = letter) is a sentence that describes itself in the sense of providing an inventory of its own characters. They were invented by Lee Sallows, who also coined the word 'autogram'. An essential feature is the use of full cardinal number names such as "one", "two", etc, in recording character counts. Autograms are also called 'self-enumerating' or 'self-documenting' sentences. Often only letter counts are recorded while punctuation signs are ignored, as in this example:
This sentence employs two a’s, two c’s, two d’s, twenty-eight e’s, five f’s, three g’s, eight h’s, eleven i’s, three l’s, two m’s, thirteen n’s, nine o’s, two p’s, five r’s, twenty-five s’s, twenty-three t’s, six v’s, ten w’s, two x’s, five y’s, and one z.
Only the fool would take trouble to verify that his sentence was composed of ten a’s, three b’s, four c’s, four d’s, forty-six e’s, sixteen f’s, four g’s, thirteen h’s, fifteeen i’s, two k’s, nine l’s, four m’s, twenty-five n’s, twenty-four o’s, five p’s, sixteeen r’s, forty-one s’s, thirty-seven t’s, ten u’s, eight v’s, eight w’s, four x’s, eleven y’s, twenty-seven commas, twenty-three apostrophes, seven hyphens and, last but not least, a single !
The task of producing an autogram is perplexing because the object to be described cannot be known until its description is first complete.
A type of autogram that has attracted special interest is the pangram, a self-enumerating sentence in which every letter of the alphabet occurs at least once. Certain letters do not appear in either of the two autograms above, which are therefore not pangrams. The first ever self-enumerating pangram appeared in a Dutch newspaper and was composed by Rudy Kousbroek. Sallows, who lives in the Netherlands, was challenged by Kousbroek to produce a self-enumerating 'translation' of this pangram into English—an impossible-seeming task. This prompted Sallows to construct an electronic Pangram Machine. Eventually the machine succeeded, producing the example below which was published in Scientific American in October 1984:
This pangram contains four as, one b, two cs, one d, thirty es, six fs, five gs, seven hs, eleven is, one j, one k, two ls, two ms, eighteen ns, fifteen os, two ps, one q, five rs, twenty-seven ss, eighteen ts, two us, seven vs, eight ws, two xs, three ys, & one z.
The infinitely long autogram below was created by J. K. Aronson of Oxford, England and is the basis of Aronson's sequence:
T is the first, fourth, eleventh, sixteenth, ... letter in this sentence, not counting spaces or commas.
Autograms exist that exhibit extra self-descriptive features. Besides counting each letter, here the total number of letters appearing is also named:
This sentence contains one hundred and ninety-seven letters: four a's, one b, three c's, five d's, thirty-four e's, seven f's, one g, six h's, twelve i's, three l's, twenty-six n's, ten o's, ten r's, twenty-nine s's, nineteen t's, six u's, seven v's, four w's, four x's, five y's, and one z.
Just as an autogram is a sentence that describes itself, so there exist closed chains of sentences each of which describes its predecessor in the chain. Viewed thus, an autogram is such a chain of length 1. Here follows a chain of length 2:
The right-hand sentence contains four a's, one b, three c's, three d's, thirty-nine e's, ten f's, one g, eight h's, eight i's, one j, one k, four l's, one m, twenty-three n's, fifteen o's, one p, one q, nine r's, twenty-three s's, twenty-one t's, four u's, seven v's, six w's, two x's, five y's, and one z.
The left-hand sentence contains four a's, one b, three c's, three d's, thirty-five e's, seven f's, four g's, eleven h's, eleven i's, one j, one k, one l, one m, twenty-six n's, fifteen o's, one p, one q, ten r's, twenty-three s's, twenty-two t's, four u's, three v's, five w's, two x's, five y's, and one z.
Closely related to autograms are 'reflexicons' (short for "reflexive lexicon"). A reflexicon is a self-descriptive word list that describes its own letter frequencies. The constraints on reflexicons are much tighter than on autograms because the freedom to choose alternative words such as "contains", "comprises", "employs", and so on, is lost. Thus, according to Sallows, there exist two and only two English reflexicons:
fifteen e's, seven f's, four g's, six h's, eight i's, four n's, five o's, six r's, eighteen s's, eight t's, four u's, three v's, two w's, three x's.
sixteen e's, five f's, three g's, six h's nine i's, five n's, four o's, six r's, eighteen s's, eight t's, three u's, three v's, two w's, four x's.