Taxicab Geometry: An Adventure in Non-Euclidean Geometry

by Eugene F. Krause

This entertaining, stimulating textbook offers anyone familiar with Euclidean geometry — undergraduate math students, advanced high school students, and puzzle fans of any age — an opportunity to explore taxicab geometry, a simple, non-Euclidean system that helps put Euclidean geometry in sharper perspective.
In taxicab geometry, the shortest distance between two points is not a straight line. Distance is not measured as the crow flies, but as a taxicab travels the "grid" of the city street, from block to block, vertically and horizontally, until the destination is reached. Because of this non-Euclidean method of measuring distance, some familiar geometric figures are transmitted: for example, circles become squares.
However, taxicab geometry has important practical applications. As Professor Krause points out, "While Euclidean geometry appears to be a good model of the 'natural' world, taxicab geometry is a better model of the artificial urban world that man has built."
As a result, the book is replete with practical applications of this non-Euclidean system to urban geometry and urban planning — from deciding the optimum location for a factory or a phone booth, to determining the most efficient routes for a mass transit system.
The underlying emphasis throughout this unique, challenging textbook is on how mathematicians think, and how they apply an apparently theoretical system to the solution of real-world problems.

Genres: Mathematics, Textbooks, Science, Nonfiction

96 pages, Paperback

First published January 1, 1975

Reviews

[Chris Esposo · Rating 3 out of 5]

This was a fun little book. To be clear, I have not completed all of the pen-and-paper exercises, just about 1/3 of them, though I've completely read the exposition. The book goes through many elementary properties of the taxicab Manhattan/metric/norm, as well as to define analog properties and geometric constructions of the norm vis-a-vis the standard Euclidean norm. This is all done at the classical level of the old theory of equations (pre linear & modern algebra), though the book was written during the 60s I believe. This is done to maximize the readability of the material to the largest audience.

The book is very easy to read, and a good puzzler. I could imagine kids doing this on a long car ride. For the more mature audience, they will know this norm as L1 (especially if you work within the worlds of statistics and/or machine learning), and so a quick easy tour & exercise on this object should be a worthy way to spend some extra time (if you have the extra time), especially given that most of the books you probably have read would consider this level of detailed exposition "beneath" your level. However, given the diversity of people moving into the field, I feel like that high-nosed attitude is counterproductive. Also, there are people who are lazy, and maybe haven't gone about and done the exploration on their own volition. This is a good motivator, quick guide to get those people started.

I would give it more stars if there were more content, what content there is I am satisfied as an ancillary text. Get this book as close to free as possible, otherwise, it can be skipped until an attractive price presents itself (close to 99 cents or lower by my reckoning).

[Benjamin · Rating 3 out of 5]

It is a delightful little book. I would have liked to see more emphasis on proof and justification.

my favorite quote: "The process of setting up a mathematical model of a real situation nearly always involves making simplifying assumptions."

[Pat · Rating 4 out of 5]

This is the type of mathematics that children should be introduced to rather than focusing on tedious and uninteresting relics of algebra.

The books very easy to read and full of constructive exercises which are fun and simple to complete. I would say its aimed at a younger or lay audience as a first introduction to non-Euclidean Geometry and basic notions of distance.

[Maurizio Codogno · Rating 3 out of 5]

Il sottotitolo di questo libretto è "An Adventure in Non-Euclidean Geometry", ma la geometria del taxi è di tipo completamente diverso. Infatti il piano non è continuo ma discreto, come fosse un foglio di carta quadrettata; i punti sono gli incroci di righe e colonne, e ci si può muovere soltanto lungo di esse, un po' come l'immaginario collettivo pensa succeda a Torino.
Succedono cose un po' strane: ad esempio, un "cerchio" assomiglia a un quadrato con i vertici che indicano i quattro punti cardinali, come il segnale di diritto di precedenza, e pi greco = 4, il che semplifica sicuramente la vita per ricordarsene il valore; in compenso le altre coniche sono un po' strane.
Il testo è sicuramente per giovani studenti: non solo è categorizzato come "Juvenile literature" ma la maggior parte del testo è composto da esercizi, dove la teoria viene per così dire "recuperata". Un buon acquisto per chi ama divertirsi con i pensieri "cosa succederebbe se..."

[Bob Finch · Rating 3 out of 5]

I read this charming little book almost 20 years ago, and it has stuck with me since. An elementary introduction to non-euclidean geometry from an intuitive "space". It is not deep, but it is enjoyable and, for me, memorable.