Isomorphic Words ("cryptograms")

Word:
Exact isomorphism: no yes
Language: English Swedish
Print mappings: Yes No

This program shows the isomorphic words (sometimes also called "cryptograms") for a given input word. It is done by comparing the "word structure" of the target word and the structure of the words in a word list (English or Swedish). The program was presented in my Swedish blog post Isomorfa ord (Isomorphic Words).

The basic principle is that two words are isomorphic if they have the same lists of character occurrences. Example: The word kjellerstrand (my last name) has the character occurrence list [1, 1, 1, 1, 1, 1, 1, 2, 2, 2], which means that there are three characters which occurs 2 times (the characters e, l, and r), and seven characters which has one occurrence (the rest).

Using an English word list with about 128000 words, there are quite a few words (453) that are isomorphic to kjellerstrand, for example: appreciations, astonishingly and wonderfulness.

The case of characters matters, i.e. the word isomorphic (two lower case i) has not the same word structure as Isomorphic (one capital I).

Note that the order or position of the characters are not important here. But see exact isomorphism below, where the exact position is important.

Exact isomorphism

The basic principle stated above, is that the order of the characters are not important. But the option exact isomorphism makes it possible to compare words where the exact positions of the characters should be compared. Two words with the same structure in this stricter sense, I call exact isomorphic.

Example: The word abba consists of two different characters ("a" and "b") where the first and last character are the same, and the two mid characters are the same. There are 8 words in the word list with this form, i.e. which are exact isomorphic to abba, is: anna, noon, otto.

The word structure is quite different from "plain" isomorphism. For abba it is [0,1,1,0], which means that the two first characters (index 0 and 1, starting from 0) is different, then the same characters as index 1 (the second character), and at the end the same character as the first. The exact isomorph structure of kjellerstrand is [0, 1, 2, 3, 3, 2, 6, 7, 8, 6, 10, 11, 12]; though there are no exact isomorphism for this word in the word list.

With this option there is much less probability of isomorphisms between words. By the way, the words probability and overwriting are exact isomorphic. And there are 5 exact isomorphisms to isomorphic: economized, economizer, economizes, and eliminated, and eliminates.

Print mappings

This option states if the mapping between the characters of two isomorphic words should be shown. The default is to show the mapping. See below for some examples of the mappings.

If the word consists of only unique characters, however, no mapping will never be shown, since all characters in the word are interchangeable. In these cases, all the exact isomorphic words are also "plain" isomorphic words.

Output of the program

If there are any isomorphic words, they will be shown in a list along with a mapping between the characters from the input word to the isomorphic word.

Example of a mapping: The word hakank is isomorphic to visits. It will be show in this way: The two lines with -> is the characters for mapping the input word (hakank) to the other word (visits). The exact interpretation depends of it is an exact isomorphism or not. Here is shown where there are not search for exact isomorphism.

For not exact isomorphisms, the characters left of the -> are interchangeable with each other, as well as the characters on the right. The example above means that either "h" and "n" can be mapped to either "t" and "v", and "a" and "k" can be mapped to either "i" or "s". The notation of [..] is somewhat related to character classes in regular expressions.

However, if it was a search for exact isomorphism the characters are not interchangeable with each other. The word andersson is exact isomorphic to unwritten, and is represented like this: This means that "a" (left side) is mapped to "e" (right side), "e" (left side) is mapped to "i" (right side),... and "s" to "t".

Some notes


Implementational note: The program was coded in Python, and do not contain a single regular expression. A CLI version of the program is available: isomorphic_words.py.


Back to my homepage
Back to my other "useless" programs
Created by Hakan Kjellerstrand hakank@gmail.com