Taxi-Cab City: A Travelogue

I have been told that there is a city hidden in a valley of the Andes mountains by the name of Catcaxitybi (by their language cat-ca-chee-tih-bee) that is laid out in a perfect square. And though most modern conveniences are absent they have managed to port a good number of cars over the intervening mountains. But because cars are still quite scarce it is decreed that those cars that run shall be taxi-cabs. Perhaps for this reason, but possibly not, they have developed a subtle form of geometry. Pythagoras and his right triangles intrude into their calculations not at all. Of Euclid’s cowl they have never heard, though there is in that city a popular headdress that appears to be made of cloth in the approximate shape of the cowl called a cloules cwido pronounced clue-less-kwid-doe.

Euclid’s cowl gives a demonstration of Pythagoras’ theorem by showing that the colored triangles all have the same area.

Oddly enough the word for stranger in the native tongue is cwido.

Indeed, the hypotenuse is such a strange notion to them that there is a legend told about the town wherein it is said that a scientific team set out in search of this hidden city to correct this most grievous fault but turned up missing (as the locals say). By this they mean that the geometers were lost in the jungle. A search party was formed in ancient native costume and sent to mislead the strangers back to civilization and its geometry.

The city is 25 blocks by 25 blocks. The streets running north and south are numbered First Avenue to Twenty-sixth Avenue going from west to east. The streets running east and west are called A Street to Z Street going from south to north.

Two different equilateral triangles in taxi-cab geometry

Communication in the city is carried out by the use of homing pigeons called Leoc saw shireft pronounced Lok-saw-sure-eft. These birds apparently do not care for the ways of crows and their methods of flying but strictly follow the street routes to their proper destinations.


The buildings are tall and all of regulation height, width and length sporting luscious tropical gardens atop each one. There are strange gardens with overhanging vines, or curiously those that look like the grounds of an English Manor house. Often there are beautiful gardens full of orchids but then there are also many vegetable gardens providing produce for the town folk. Some buildings are covered with apple orchards altogether giving the city a distinctive look from above or rather a look indistinguishable from the surrounding jungle.

A view from above

The Catcaxitybians play a curious game called chorecks reshecs (chow-rakes-ray-shiks) played on a board of sixty four squares. There are eight white pieces usually of the same fashion and eight black pieces facing each other from opposite ends of the board. A chess board and pawns or checkers can be used to play the game. A piece may move a distance one in a single direction or a distance 2 if it changes direction. A piece must be taken if it is on an adjacent block to any opposite piece. The object of the game is to be rid of your pieces first. The city boasts of many world champions in the game even though it is the only place of my acquaintance that suffers to play such an unorthodox game wherein the loser is forced to win. It is a tradition to recite the maximum distance between your pieces and your opponents pieces before each move with a time a penalty for incorrect recitations. In friendly games with foreigners this custom is usually abandoned. When the last piece is taken the winner says: ihndet belf. The word is in an ancient tongue which no one here seems to know.

White to play and win.

Each city block is exactly one tenth of a melotikre (mel-low-teek-rah). The fares are fixed at a price of xiced frepi ( chee-sid freh-pee) per block in the local currency. A frep is about one quarter of a dollar at current exchange rates, though the exchange rates are determined by the locals and probably have nothing to do with actual values of the currency. I asked my native friend for one good reason that this should be so. He replied that the locals justify this strange practice by pointing out the curious fact that international exchange rates have nothing to do with the relative values of the currencies involved. I told him that it was a reason but not a good one. There are no meters in the taxi-cabs as all the drivers are good at mental arithmetic and have an innate sense of geometry, such as it is in Catcaxitybi. Thus if you get in a taxi-cab at Fifth and D and wish to go to Thirteenth and R the driver will immediately quote the price of 22 xiced frepi.


I have listed below the entrance locations for several buildings in the city. If you can compute the fares mentally, you could become a taxi-cab driver in the city. I am afraid I failed the entrance exam in miserable fashion for I am too slow and must use my fingers to count.

Shroe Eapou - Tenth and G;


Ranem Trod - Twelveth and Q;


Dacherlat - Twentieth and B;


Mafer Karstrem - Thirteenth and N

Perhaps you noticed that there are several anagram phrases in the description of the taxi-cab city above. The anagrams are associated to the words in the native language.

Can you decipher each anagram?

Can you do taxicab geometry in your head?

Can you find another kind of equilateral triangle with side length 6?

What does a square look like? What does a circle look like?

Is it possible to fit a square peg in a circular hole in this geometry?