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Mathematical Snapshots

Hugo Steinhaus
Dover Publications
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Underwood Dudley
, on

Hugo Steinhaus (1887–1972) was a powerful Polish mathematician (e.g., the Banach-Steinhaus theorem). This book, meant for the general reader, was written in 1937 and originally published in Polish as Kalejdoskop Matematyczny. Steinhaus said “My purpose is neither to teach, in the usual sense of the word, nor to amuse the reader with some charades.” He wanted to show what it meant to be a mathematician: “That was how I conceived this book, in which the sketches, diagrams, and photographs provide a direct language and allow proofs to be avoided or at least reduced to a minimum.”

Sketches, diagrams, and photographs there are in plenty, 391 of them in 295 pages, more pictures than text. Since geometry lends itself to pictures, there is quite a bit of it in the contents, but much else as well. Steinhaus proceeds in stream-of-consciousness mode, with one topic suggesting another. For example, chapter 6 starts with a linkage that produces straight lines, followed by a method to determine automatically the centroid of a rod (the rod comes up maybe because the linkage consists of rods). Then comes an illustration of how a straightedge-and-compass construction can be done without a straightedge (see, who needs rods?), the circles in that construction suggest mentioning an approximation to \(\pi\) and then the circles of Apollonius, followed by the statement that any three domains — Wisconsin, Indiana, and Missouri in the picture — can be simultaneously halved by a circle, which generalizes to the ham sandwich theorem that one cut can divide any sandwich made of bread, butter, and ham into two parts with equal amounts of ingredients in each. This is all in six pages. Later come cycloids, curves of constant width, the conchoid of Nicomedes, Pascal’s limaçon, a discussion of perspective, and on and on we go. The last paragraph of the chapter is

“A very simple optical fallacy hinders our estimating the distance of a horizontal wire without visible poles supporting it. The reason is the homogeneity of the wire. If there were a red spot on it, we should direct both axes of vision on the spot and immediately get the perception of distance; if there is no such mark, it is sufficient to incline the head on one shoulder. (Why?)

Then it’s off to the next chapter that, if you are like me, you will find equally delightful.

The contents of the book have not dated. (Though mathematics is timeless, this is not always the case. There are books that induce the thought “Why did anyone ever bother with that?” It never arises from Steinhaus’s book.) Also, there is no book like it. It is not that it was not worth imitating, it is that no one was up to it. Evidently, you had to be a Steinhaus to produce it. It’s had a long life. Long may it continue.

Woody Dudley has been a member of the MAA for sixty years and hopes that that continues too.


"Triangles, Squares, and Games"
2. "Rectangles, Numbers, and Tunes"
3. "Weighing, Measuring, and Fair Division"
4. "Tessellations, Mixing of Liquids, Measuring Areas and Lengths"
5. "Shortest Paths, Locating Schools, and Pursuing Ships"
6. "Straight Lines, Circles, Symmetry, and Optical Illusions"
7. "Cubes, Spiders, Honeycombs, and Bricks"
8. "Platonic Solids, Crystals, Bee's Heads, and Soap"
9. "Soap-Bubbles, Earth and Moon, Maps, and Dates"
10. "Squirrels, Screws, Candles, Tunes, and Shadows"
11. "Surfaces Made of Straight Lines, the Chain, the Toycart, and the Minimal Surface"
12. "Platonic Bodies Again, Crossing Bridges, Tying Knots, Coloring Maps, and Combing Hair"
13. "Board of Fortune, Frogs, Freshmen, and Sunflowers"