Seasons Greetings from Bananagrammer

Impossible objects. Impossible words.

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Impossible objects are drawings of apparently three-dimensional objects which look correct when their individual parts are examined, but when you look at the object as a whole, it turns out to be not realizable. One of the most famous examples was created by D. H. Schuster and published in a psychology journal in 1964. The paper was titled "A New Ambiguous Figure: A Three-Stick Clevis", and said figure looks like this:

As emphasized by the colored background, the top of the object resembles the upper part of a stirrup (which has the basic form of what is known as a "clevis") and the bottom of the object looks like three parallel rods. Somewhere in between lies the ambiguity that destroys the three-dimensionality. Martin Gardner referred to such drawings as "undecidable figures".

The three-stick clevis has since gone by many other names: blivet, devil's tuning fork, widget, and poiuyt.

Mad Magazine used "poiuyt" as the name for the above blivet when they featured it on their March 1965 cover. The difficulty you may be having in deciding how to pronounce "poiuyt" is due to its unusual origin. The word "QWERTY" was formed by starting from the left side of the top row of a typewriter and taking the first six letters. Applying the same technique to the other end of the keyboard you get the looking glass version of "QWERTY"... "poiuyt".

Just as perspective drawing was invented to allow us to make two-dimensional depictions of three-dimensional things, spelling was invented to allow transcription of spoken language. And just as we can draw objects that are logically inconsistent, so can we write combinations of letters that correspond to no spoken word.

"Poiuyt" has no apparent, standard, or authoritative pronunciation. Dictionaries ignore it. If the reader will indulge me, I will nominate it as our first impossible word.

Another candidate for impossibility is "balge" (as in "balge yellow"), a term that is listed in Merriam-Webster's Third New International Dictionary as having no known pronunciation and no known origin (as if it spontaneously generated on a piece of paper on some lexicographer's desk). "Balge yellow" has been defined as "a brilliant yellow color" and "sunflower yellow", so at least that part of its wordhood is known.

A 1976 survey of color names by the National Bureau of Standards identified

balge yellow as the color pictured here, which also goes by names such as "jonquil" and "Naples yellow". These redundant names may explain why "balge" use ended.

Even though no one seems to know how to pronounce "balge", it doesn't feel undecidable in the way that "poiuyt" does, probably due to the latter's discombobulating four consecutive vowels.

My third nomination for impossible word is YHWH which is the English version of the Hebrew word: יהוה. Controversy surrounds this word. It is used throughout the original Hebrew texts of the Old Testament as the primary name for God. Some pronounce it as "Jehovah" or "Yahweh". Since ancient Hebrew lacked was written without vowels, the correct pronunciation of יהוה is not known. There is a strong taboo against speaking this name in Judaism, so it may be that whatever correct pronunciation might have existed has disappeared due to lack of use. Some believe that the pronunciation is a secret preserved by only a few people in each generation. What I like best about this word is that there is another name you can use when talking about it: "the Tetragammaton" (from the Greek for "having four letters"). The undecidability of YHWH's pronunciation is in an entirely different class than that of poiuyt, but maybe a property of impossible words is that they are all impossible in their own ways. This one seems to be more of an arms-crossed, exasperated "Tetragammaton, you're impossible!" way.

I feel obligated to mention one word that I thought would be impossible but has turned out not to be: Mxyzptlk. Mister Mxyzptlk is a mischievous prank-playing imp from the fifth dimension who occasionally visits Earth to wreak havoc until Superman deals with him. The gimmick was that the only way to send Mister Mxyzptlk back home was to trick him into saying his name backwards.

While Mxyzptlk has been pronounced in a variety of ways throughout the years, allegedly the DC Comics editor gave an authoritative pronunciation early on: "mix-yez-PIT-elick". But I suppose that one could claim that it is in Mxyzptlk's trickster nature that the pronunciation of his name refuses to be nailed down.


Then there are heteronyms which are words that are spelled the same way but pronounced differently:

"bass" can rhyme with "glass" or "space".

"wind" can be pronounced with a short I (like the thing that blows) or a long I (the verb that describes forming a ball of yarn).

They're better classified as ambiguous than outright impossible.

As heteronyms change pronunciation based on the context they are used in, they are analogous to the Necker Cube:

Rather than representing a figure that has no sensible three-dimensional realization, the Necker cube confounds the viewer because it has more than one realization. Most people initially see it as a wire-frame cube, viewed from the top, with the lower-left square as the front face. After studying the figure for some time, it may seem to suddenly shift to a cube seen from above with the upper-right square as the front. I find that I can switch between the viewpoints by focussing on a face that appears to be at the back of the cube which seems to cause it to pull forward.


The impossible word is an exceedingly rare thing because we tend to make up pronunciations for words, even if we have to break the laws of phonics. (Doing so yields ultraphonic words (words outside the range of normal phonics), such as Big Bird's pronunciation of ABCDEFGHIJKLMNOPQRSTUVWXYZ as a single long word by sneaking vowel sounds into strings of consonants like JKLMN.)


Spelling a word is a reductive, lossy process. Accents, tones, sarcasm, are all generally omitted. English orthography, in particular, requires collapsing the full spoken word into a few characters, introducing considerable ambiguity, but from this ambiguity is born many good things, like puns and poiuyts.

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Further reading:

• On balge: According to an 1875 Bulletin from the National Association of Wool Manufacturers, balge yellow was "generally employed on cassimere for vestings". Google Books also has the recipe for dying wool balge yellow.

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A word about the very cool font used for the IMPOSSIBLE graphic above: “ISOSIBILIA Typography Designed by Rodrigo Fuenzalida for Neo2. - [Back to footnote reference]

How Scrabble dictionaries are made

A web site called Word Buff has an interview with Darryl Francis on the making of the British Scrabble tournament word lists. From his self-description, Francis sounds like a cool guy with interests in wordplay and language. He writes articles for Word Ways, a magazine of recreational linguistics (which I can recommend if you enjoy wordplay). And he cites Martin Gardner's Scientific American columns as a major influence.

Darryl Francis and Allan Simmons are the Dictionary Committee for WESPA, the World English Scrabble Players Association. They've basically been in charge of the British Scrabble word list since its inception.

The British Scrabble tournament word list (previously called SOWPODS and now apparently "CSW" as an abbreviation for Collins Official Scrabble Words) is formed by taking all the words in the most current American Scrabble tournament word list and adding in any valid words from the Chambers Dictionary and the Collins English Dictionary. Francis's interpretation of what constitutes a valid word is given in his response to a question about whether he would ever exclude words that satisfy all the rules:

Let's go back to a group of dictionary entries I mentioned earlier - the internet domain names for countries. There's around 200 of these, running from AC (Ascension Island) to ZW (Zimbabwe). They appear to satisfy the criteria for acceptability of words.

They're not dictionary-listed with an initial capital letter, nor a hyphen nor an apostrophe. They're not marked as abbreviations, they're not marked as foreign. On what basis should they not be allowed as two-letter words?

My answer to this question is that a) these two-letter abbreviations (called "country code top-level domains") are proper nouns, and b) they are in fact abbreviations, whether the dictionary says so or not.

Francis goes on to say:

Yet to allow a sudden influx of two-letter words, most of which are unpronounceable and not recognisable to the man in the street, would be to upset the fine balance that already exists with two-letter words.

Two-letter words are so key to the game that to double their number overnight would almost certainly provoke an outcry from Scrabble players - and probably the media, too.

I could portray this as a question of how to balance strict rule-following with common sense. Ultimately the Dictionary Committee chose not to include all those country codes, so they do use some common sense in their decisions. And they do have to make many difficult judgment calls. But it seems like they only rejected these country codes because there are so many of them and because Scrabble players would be upset by their inclusion.

To me, this demonstrates the subtle biases that have crept into the system to make official Scrabble dictionaries (unsurprisingly) give tournament Scrabble players what they want. As I understand it, what a plurality of them want is a word list that retains the words they have spent so much time memorizing, while occasionally adding handfuls of new words that increase Scrabble scores and make the game easier and more fun for them.

And this is perfectly fine, so long as these Scrabble word lists aren't misappropriated as authoritative sources for other games...

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Further reading:
• Well, that about wraps it up for the Official Scrabble Players Dictionary
• All of my dictionary rants

The 27th Letter of the Alphabet

Rogues, to speak thus irreverently of the alphabet, I shall live to see you glad to serve old Q — to curl the wig of great S — adjust the dot of little i — stand behind the chair of X. Y. Z. — wear the livery of Etcetera — and ride behind the sulky of And-by-itself-and.

From Act I of Charles Lamb's Mr. H

If you were a schoolchild in the 19th century, the alphabet that you learned would have had 27 letters: all 26 letters of our current alphabet, plus the ampersand symbol.

A	B	C	D	E
F	G	H	I	J
K	L	M	N	O
P	Q	R	S	T
U	V	W	X	Y
	Z		&

Due to the awkwardness of ending a recitation of the alphabet with "W X Y Z and", it was traditional to instead say "W X Y and Z, and per se and", where per se, Latin for "by itself", means that &, standing by itself, represents "and". (Words with one-letter spellings, like A or I, were often orally spelt as "A per se" or "A per se A".) It was this process of alphabet recitation, and hurried enunciations of "and per se and" which spawned a variety of names for the & symbol which ultimately converged on "ampersand". & had been part of the alphabet going back to the days of Old English.

Other than standing for the conjunction "and", the ampersand also sometimes appears in the abbreviation &c, for et cetera. This is due to the origins of the & symbol in the first century A.D., when the Romans would write et (Latin for "and") in cursive in a run-together fashion which became a stand-alone written symbol.

So why do we no longer consider & to be part of the alphabet?

The leading theory is that it's because of that alphabet song, the one that goes

A B C D E F G
H I J K LMNOP
Q R S, T U V
W X, Y and Z.
Now I know my A B Cs.
Next time won't you sing with me?

Many incorrectly believe that this is based on a tune by Mozart. While Mozart wrote variations on this theme at the age of 25 [see Köchel listing K. 265], the original melody that inspired him was a French folk song called "Ah! vous dirai-je, Maman" which eventually served as the music for Twinkle, Twinkle, Little Star. In 1835, the alphabet song was copyrighted under the name "The A.B.C., a German air with variations for the flute with an easy accompaniment for the piano forte", so it does seem like Mozart was responsible for popularizing the melody.

It turns out that that song has influence beyond the ousting of the ampersand. Historically, it has been mainly in the U.S. that Z has been pronounced zee; pretty much everywhere else they say zed. But a quick look at the rhyming scheme of the alphabet song, shows that the zee pronunciation works better. And apparently a lot of children who learn English outside of the U.S. are still exposed to this alphabet song through American children's programming, like Sesame Street. Teachers in England reportedly have to correct kindergarteners who enter school singing the alphabet song in an American accent, right down to the zee.

I leave you with this quote from Steven Wright:

Why is the alphabet in that order? Is it because of that song? The guy who wrote that song wrote everything.

PAX, the Omegathon, and novelty in video games

I love books and documentaries that examine quirky subcultures. The book Word Freak provides a fascinating look inside the world of tournament Scrabble. Murderball was a great film about the players of wheelchair rugby. My favorite quirky documentary though is The King of Kong: A Fistful of Quarters which dramatizes the competition between two players for the high score in the classic arcade game Donkey Kong.

I've just found an amazing article online called PAX Primer which is the perfect introduction to the quirky subculture that is the Penny Arcade Expo. It covers the origins of the Expo (in case you ever wanted to know how a web comic can spawn a convention dedicated to video games and board games), the growth of video games, and their transition from fringe to mainstream culture.

Earlier this year, I posted about PAX because Bananagrams was an event in the PAX East Omegathon. It turns out that the Omegathon organizers decided to feature Bananagrams in the west coast PAX Omegathon as well. The article says that in the convention program, Bananagrams is described as "like Scrabble, only not boring and for old people".

It later goes on to discuss a few computer games which I might opine are "like video games, only not boring and for old people". The new wave of computer games does not suck you into endless repetition.

Portal is a game where you solve puzzles by shooting two holes on different walls, ceilings, or other surfaces in your environment. These "portals" are connected (as if by a wormhole), so whatever goes in one, comes out the other, with the same momentum. I recently started playing this game and can not get enough of it.

Braid is an even stranger game in which the player gets to control the flow of time. The selling points of the game are listed on the game's web site:

• Every puzzle in Braid is unique. There is no filler.
• Braid treats your time and attention as precious.
• Braid does everything it can to give you a mind-expanding experience.

Braid's programmer, Jonathan Blow, self-financed the game as he coded it over three years as a statement about how video games could and should be different.

Braid does not look like any other computer game. The artwork is great. It was done by the artist behind the surreal web comic A Lesson Is Learned But The Damage Is Irreversible. Braid also does not sound like any other computer game. Its atmospheric music helped to win me over.

With the success of these games, even more ambitious games are in the works, on topics such as non-Euclidean geometry (Antichamber) and four-dimensional space (Miegakure).


In a world where video games have become mainstream, it makes sense for a niche to develop for games that emphasize originality. I am glad that quirky subcultures exist to sustain this kind of bold experimentation.

Why words are the lengths they are

Some words are long and others are short. What determines how long a particular word should be? If you look at some long words (like "serendipity", "pandemonium", and "hypothesis") and some short words ("my", "in", and "of"), you might come to the conclusion that short words are short because they are used frequently while long words can afford to be long because they come up rarely. This idea was first proposed by a Harvard linguist named George Zipf in 1936.

Researchers at the MIT Department of Brain and Cognitive Sciences took a fresh look at this question and came up with a new theory. They present their results in a paper titled (spoilers!) "Word lengths are optimized for efficient communication".

How much information is conveyed by a word? Consider the sentence that starts "After I got home, I walked the...". If I finish the sentence as "I walked the dog", the extra word "dog" doesn't convey much information because it's probably one of the words your brain was expecting. More surprising would have been "I walked the cat" or "I walked the bulldozer", "I walked the quasar" or "I walked the plank". It is the amount of surprise that researchers are equating with the information contained in a word. Consequently the information content of a word depends on the context that it appears in. [For those who want a more quantitative explanation, the information contribution from a particular context (like, "I walked the...") is -log(p), where p is the probability that the word appears at the end of that phrase and where log() is the natural logarithm function. To get the total information for a word like "dog", you just sum -p log(p) over all the contexts that "dog" appears in.]

Ideally what the researchers would have liked to examine is the relationship between how long it takes to say words and how much information they convey, but it was easier (and, they argue, an adequate approximation) to use the number of letters in a word in place of its utterance duration. But later, they went back and ran the same tests (for a few languages) using number of syllables instead of number of letters, and the results were the same.

To calculate the relationship between word length and frequency, the researchers used the same N-gram data set that Google used in its N-gram viewer. This figure from the paper summarizes their findings:

The plot on the left shows word length versus word use frequency, with frequency decreasing from left to right. (Here the data has been divided into large groups of words ("bins") and the average lengths and frequency have been used.) For the first few points (high-frequency words like "the"), the slope of the line is strong, but then it quickly flattens out, indicating that for low-frequency words, the frequency of the word doesn't change the length very much.

The plot on the right shows average word length versus the information content of the word. Here, the line starts off jagged but then becomes strongly-sloped and very straight. This tells us that how much information a word carries is indeed a good predictor of how long the word will be.

The researchers also cite other work that has shown that, when speaking, people will speak more information-dense syllables more slowly than less information-dense syllables. (If you've ever listened to the synthesized voice of something like a GPS, you'll be familiar with the jerkiness of the pronunciation that sounds like it is speaking some syllables too slowly and others too quickly.)

It would seem that a corollary to this principle is that as a word becomes more common (or more precisely, loses information density), it experiences a linguistic force, pushing it toward a shorter form. This shortening process is called phonetic erosion. Examples of the resulting shortenings (also called clippings) are "refrigerator" becoming "fridge", "going to" becoming "gonna", and "cabriolet" being completely replaced by "cab". Here are a few other terms that have evolved much shorter forms:

• advertisement → ad
• caravan → van
• examination → exam
• gasoline → gas
• gymnasium → gym
• influenza → flu
• public house → pub

So, essentially, the researchers found that the old idea that word length is based mainly on frequency of word usage (short words are used often while long words are used rarely) does a poor job of explaining why words are the lengths they are. The amount of information in a word (averaged over the various contexts that it is used in) is a far better predictor for how long the word will be. The only exception to this is the 5% to 20% of words that are the least informative (generally short, high-frequency words like "the" and "and").

This result holds, not just for English, but also for the other ten languages that they examined (Czech, Dutch, French, German, Italian, Polish, Portuguese, Romanian, Spanish, and Swedish).

The basic idea that I take away from this work is that there is some maximum rate that our brains can understand incoming speech, and that our speech patterns reformulate what we are saying to evenly distribute information over time. It makes me wonder whether pausing for effect is taking advantage of this fact. Similarly, when I say a word slowly to emphasize it, maybe I am just slowing it down to suggest that it contains a lot of information.

Epilogue: In case you were wondering, the actual ending to the sentence that started "After I got home, I walked the..." was "...tightrope.".

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Further reading:
• "Word lengths are optimized for efficient communication" - the actual paper, in PDF
• "Zen" and the art of Google N-gram Viewing

The Bananagrammer Equation

A warning to regular readers: This post is not about games nor about words. It is about math and bananas.

Recently the "Batman Equation" has been memetically propagating around the Internet.

The equation represents the outline of the Batman logo. It is apparently the work of a user on Reddit.

I liked it enough to try to make my own. There are a few tricks to this process. First break the shape up into curves that you can easily write equations for, of the form f(x,y)=0.

Then, to make the curves stop at the desired end points, add in terms like the ones you see under the square roots. They evaluate to either 1 or -1, depending upon the grid position; when this value is negative, the square root is no longer real, and the plotting program will not plot anything. Finally, multiply all the equations together, and you get one big long equation:

(This is a cleaned-up and slightly approximated version of the equation I used for plotting.)

The final plot looks like this:

It is remarkable how well a single ellipse traces out the outer edge of a banana silhouette. I checked a couple of other bananas, and they also have this property. I finally went to a grocery store and sifted through all their bananas to find the least elliptical one I could:

From the red sample points along the edge, I found that even this banana was almost well-approximated by an ellipse.

It is the first three data points that make this an exception to Bananagrammer's First Law of Bananas:

The outer edge of the longitudinal section of a banana follows an elliptical path, with the banana's stem being roughly on the end of the ellipse's long axis.

The question to ask at this point is, "Why are bananas shaped the way they are?". The simple answer is that when a bunch of bananas start growing on a tree, they are initially pointing more down than up. As they become larger, they curve up toward the sun. A banana's exact shape will therefore depend on where it is with respect to its neighbors.

A full explanation of why bananas are so elliptical will require more investigation. People who want to give me research grants are welcome to do so. Actually, everybody is welcome to do so. To everyone else, tune in next week. Same Banana-time, same Banana-channel!