Pangrammatic window
Encyclopedia
A pangram
matic window is a stretch of naturally occurring text that contains all the letters in the alphabet.
's 1912 book In the Courts of Memory: "I sang, and thought I sang very well; but he just looked up into my face with a very quizzical expression, and said, 'How long have you been singing, Mademoiselle?'" at 56 letters. A 47-letter example has, however, been found on the internet: "JoBlo's movie review of The Yards
: Mark Wahlberg, Joaquin Phoenix, Charlize Theron...".
Some estimates can be made using the frequencies of the letters. Assuming that the incidence of each letter in a passage is independent of the incidence of the other letters, and provided m is much larger than 26, the probability that a sequence of length m will contain all 26 letters is approximately
P(a)P(b)...P(y)P(z), where P(letter) = 1 − (1 − p(letter))m and p(letter) is the frequency of the letter as a fraction (e.g. 5% is 0.05). Inputting the letter frequencies
for the English language, the probability that a 1,700-letter sequence will contain all 26 letters is about 50%. At 1,000, there is about a 19.5% chance, and at 2,500, there is about a 73% chance.
For example, the shortest pangrammatic window in Around the World in Eighty Days, by Jules Verne
, is 155 letters: "Once a man approached and spoke to him, and the detective merely replied by shaking his head. Thus the night passed. At dawn, the half-extinguished disc of the sun rose above a misty horizon; but it was now possible to recognise objects two miles off. Phileas Fogg and the squad had gone southward; in the south all was still vacancy."
The shortest pangrammatic window in the United States Declaration of Independence
, a significantly shorter work, is 592 letters:
"We hold these truths to be self-evident, that all men are created equal, that they are endowed, by their Creator, with certain unalienable Rights, that among these are Life, Liberty, and the pursuit of Happiness. That to secure these rights, Governments are instituted among Men, deriving their just powers from the consent of the governed, That whenever any Form of Government becomes destructive of these ends, it is the Right of the People to alter or abolish it, and to institute new Government, laying its foundation on such principles, and organizing its powers in such form, as to them shall seem most likely to effect their Safety and Happiness. Prudence, indeed, will dictate that Governments long established should not be changed for light and transient causes; and accordingly all experience hath shewn, that mankind are more disposed to suffer, while evils are sufferable, than to right themselves by abolishing the forms to which they are accustomed."
Pangram
A pangram , or holoalphabetic sentence, is a sentence using every letter of the alphabet at least once. Pangrams have been used to display typefaces, test equipment, and develop skills in handwriting, calligraphy, and keyboarding...
matic window is a stretch of naturally occurring text that contains all the letters in the alphabet.
Shortest examples
Until recently, the shortest known window was found in Lillie de Hegermann-LindencroneLillie de Hegermann-Lindencrone
Lillie De Hegermann-Lindencrone, , born Lillie Greenough in Cambridge, Massachusetts and later known as Lillie Moulton, was a trained singer, and latterly a diplomat's wife noted for publishing in 1913 a book of letters describing The Sunny Side of Diplomatic Life.As a child she developed the...
's 1912 book In the Courts of Memory: "I sang, and thought I sang very well; but he just looked up into my face with a very quizzical expression, and said, 'How long have you been singing, Mademoiselle?'" at 56 letters. A 47-letter example has, however, been found on the internet: "JoBlo's movie review of The Yards
The Yards
The Yards is a 2000 American crime film with Mark Wahlberg, James Caan, Joaquin Phoenix, and Charlize Theron, written and directed by James Gray...
: Mark Wahlberg, Joaquin Phoenix, Charlize Theron...".
Probability of occurrence
Generally, according to the law of probability, the shorter the work, the longer the minimal pangrammatic window (if any) will be.Some estimates can be made using the frequencies of the letters. Assuming that the incidence of each letter in a passage is independent of the incidence of the other letters, and provided m is much larger than 26, the probability that a sequence of length m will contain all 26 letters is approximately
P(a)P(b)...P(y)P(z), where P(letter) = 1 − (1 − p(letter))m and p(letter) is the frequency of the letter as a fraction (e.g. 5% is 0.05). Inputting the letter frequencies
Letter frequencies
The frequency of letters in text has often been studied for use in cryptography, and frequency analysis in particular. No exact letter frequency distribution underlies a given language, since all writers write slightly differently. Linotype machines sorted the letters' frequencies as etaoin shrdlu...
for the English language, the probability that a 1,700-letter sequence will contain all 26 letters is about 50%. At 1,000, there is about a 19.5% chance, and at 2,500, there is about a 73% chance.
For example, the shortest pangrammatic window in Around the World in Eighty Days, by Jules Verne
Jules Verne
Jules Gabriel Verne was a French author who pioneered the science fiction genre. He is best known for his novels Twenty Thousand Leagues Under the Sea , A Journey to the Center of the Earth , and Around the World in Eighty Days...
, is 155 letters: "Once a man approached and spoke to him, and the detective merely replied by shaking his head. Thus the night passed. At dawn, the half-extinguished disc of the sun rose above a misty horizon; but it was now possible to recognise objects two miles off. Phileas Fogg and the squad had gone southward; in the south all was still vacancy."
The shortest pangrammatic window in the United States Declaration of Independence
United States Declaration of Independence
The Declaration of Independence was a statement adopted by the Continental Congress on July 4, 1776, which announced that the thirteen American colonies then at war with Great Britain regarded themselves as independent states, and no longer a part of the British Empire. John Adams put forth a...
, a significantly shorter work, is 592 letters:
"We hold these truths to be self-evident, that all men are created equal, that they are endowed, by their Creator, with certain unalienable Rights, that among these are Life, Liberty, and the pursuit of Happiness. That to secure these rights, Governments are instituted among Men, deriving their just powers from the consent of the governed, That whenever any Form of Government becomes destructive of these ends, it is the Right of the People to alter or abolish it, and to institute new Government, laying its foundation on such principles, and organizing its powers in such form, as to them shall seem most likely to effect their Safety and Happiness. Prudence, indeed, will dictate that Governments long established should not be changed for light and transient causes; and accordingly all experience hath shewn, that mankind are more disposed to suffer, while evils are sufferable, than to right themselves by abolishing the forms to which they are accustomed."