09-01-2026, 03:38 PM
Mathematical Surprises
Book:Mathematical Surprises
Author: Mordechai Ben-Ari
Publication date: 2022
Publisher: Springer Nature Switzerland AG
Subject: Recreational/elementary mathematics, geometry, algebra, combinatorics
Level: Advanced secondary-school mathematics and above
Open Access: Yes — Creative Commons Attribution 4.0 (CC BY 4.0).
Summary
Mathematical Surprises is a collection of 16 largely independent chapters devoted to mathematical results that are unexpected, elegant, or surprisingly accessible. Ben-Ari deliberately chooses topics that normally do not appear in school or introductory university textbooks but whose proofs can still be understood with a strong secondary-school background. The book is not intended as a conventional textbook; rather, it is designed for mathematical enrichment, particularly for secondary-school students, college seminars, teachers, and interested readers willing to work through sometimes lengthy proofs.
A major theme is that apparently simple mathematical questions can hide deep structures. The book begins with classical straightedge-and-compass constructions, explaining why Euclid's collapsing compass is just as powerful as a modern fixed compass and emphasizing the danger of trusting geometric diagrams without proof. It then examines the famous impossible classical problems of trisecting an arbitrary angle and squaring the circle, while also showing alternative instruments that can accomplish these constructions. Ramanujan's remarkably accurate geometric approximations to $\pi$ are another example of the sort of result Ben-Ari considers mathematically surprising.
The scope then expands beyond classical geometry. The reader encounters the five- and six-color theorems, graph theory, the art-gallery or museum-guarding problem, unusual applications of mathematical induction, Fibonacci and Fermat numbers, the Josephus problem, Po-Shen Loh's approach to quadratic equations, Ramsey theory, Pythagorean triples, SAT solving and Langford's problem. One particularly attractive example is the museum theorem: what initially looks like a geometry problem is solved elegantly by translating it into a graph-coloring problem.
Several chapters explore the unexpectedly rich mathematics of origami. Ben-Ari presents seven axioms of mathematical origami, Lill's method and the Beloch fold, and demonstrates that origami can perform constructions impossible with an ordinary straightedge and compass—including angle trisection, doubling the cube and constructing a regular nonagon. This is one of the subjects that most surprised the author himself; he explains that he had originally doubted whether origami contained serious mathematics at all.
Perhaps the book's most striking results concern the apparent limitations of classical geometric instruments. The Mohr–Mascheroni theorem says that every construction possible with straightedge and compass can actually be performed with a compass alone. Mohr discovered the result in 1672 and Mascheroni independently proved it in 1797. Even more surprisingly, the Poncelet–Steiner theorem shows that a straightedge alone is sufficient if just one fixed circle is already provided—the circle may have essentially arbitrary position and radius.
Another unexpected chapter asks whether two triangles having the same perimeter and the same area must be congruent. They need not be: for example, triangles with sides $(17,25,28)$ and $(20,21,29)$ both have perimeter $70$ and area $210$. The analysis eventually connects an elementary-looking geometry problem to elliptic curves, illustrating the book's recurring theme that simple questions can lead to unexpectedly sophisticated mathematics.
The culmination is Gauss's celebrated proof that a regular heptadecagon (17-gon) can be constructed with straightedge and compass. Gauss expresses the necessary quantities using arithmetic operations and repeated square roots, linking geometric constructibility to algebra and the roots of polynomials. The book supplements Gauss's algebraic argument with an explicit geometric construction and related constructions of the regular pentagon.
What makes the book interesting
The unifying idea is not a single branch of mathematics but the experience of mathematical surprise: familiar assumptions turn out to be false, apparently impossible constructions become possible when the rules change slightly, elementary questions connect to unexpected areas of mathematics, and difficult-looking theorems sometimes possess elegant proofs. The author deliberately selects results that are rarely encountered in normal curricula while remaining accessible through algebra, Euclidean and analytic geometry, and trigonometry.
Key takeaways
BOOK DOWNLOAD
Book:Mathematical Surprises
Author: Mordechai Ben-Ari
Publication date: 2022
Publisher: Springer Nature Switzerland AG
Subject: Recreational/elementary mathematics, geometry, algebra, combinatorics
Level: Advanced secondary-school mathematics and above
Open Access: Yes — Creative Commons Attribution 4.0 (CC BY 4.0).
Summary
Mathematical Surprises is a collection of 16 largely independent chapters devoted to mathematical results that are unexpected, elegant, or surprisingly accessible. Ben-Ari deliberately chooses topics that normally do not appear in school or introductory university textbooks but whose proofs can still be understood with a strong secondary-school background. The book is not intended as a conventional textbook; rather, it is designed for mathematical enrichment, particularly for secondary-school students, college seminars, teachers, and interested readers willing to work through sometimes lengthy proofs.
A major theme is that apparently simple mathematical questions can hide deep structures. The book begins with classical straightedge-and-compass constructions, explaining why Euclid's collapsing compass is just as powerful as a modern fixed compass and emphasizing the danger of trusting geometric diagrams without proof. It then examines the famous impossible classical problems of trisecting an arbitrary angle and squaring the circle, while also showing alternative instruments that can accomplish these constructions. Ramanujan's remarkably accurate geometric approximations to $\pi$ are another example of the sort of result Ben-Ari considers mathematically surprising.
The scope then expands beyond classical geometry. The reader encounters the five- and six-color theorems, graph theory, the art-gallery or museum-guarding problem, unusual applications of mathematical induction, Fibonacci and Fermat numbers, the Josephus problem, Po-Shen Loh's approach to quadratic equations, Ramsey theory, Pythagorean triples, SAT solving and Langford's problem. One particularly attractive example is the museum theorem: what initially looks like a geometry problem is solved elegantly by translating it into a graph-coloring problem.
Several chapters explore the unexpectedly rich mathematics of origami. Ben-Ari presents seven axioms of mathematical origami, Lill's method and the Beloch fold, and demonstrates that origami can perform constructions impossible with an ordinary straightedge and compass—including angle trisection, doubling the cube and constructing a regular nonagon. This is one of the subjects that most surprised the author himself; he explains that he had originally doubted whether origami contained serious mathematics at all.
Perhaps the book's most striking results concern the apparent limitations of classical geometric instruments. The Mohr–Mascheroni theorem says that every construction possible with straightedge and compass can actually be performed with a compass alone. Mohr discovered the result in 1672 and Mascheroni independently proved it in 1797. Even more surprisingly, the Poncelet–Steiner theorem shows that a straightedge alone is sufficient if just one fixed circle is already provided—the circle may have essentially arbitrary position and radius.
Another unexpected chapter asks whether two triangles having the same perimeter and the same area must be congruent. They need not be: for example, triangles with sides $(17,25,28)$ and $(20,21,29)$ both have perimeter $70$ and area $210$. The analysis eventually connects an elementary-looking geometry problem to elliptic curves, illustrating the book's recurring theme that simple questions can lead to unexpectedly sophisticated mathematics.
The culmination is Gauss's celebrated proof that a regular heptadecagon (17-gon) can be constructed with straightedge and compass. Gauss expresses the necessary quantities using arithmetic operations and repeated square roots, linking geometric constructibility to algebra and the roots of polynomials. The book supplements Gauss's algebraic argument with an explicit geometric construction and related constructions of the regular pentagon.
What makes the book interesting
The unifying idea is not a single branch of mathematics but the experience of mathematical surprise: familiar assumptions turn out to be false, apparently impossible constructions become possible when the rules change slightly, elementary questions connect to unexpected areas of mathematics, and difficult-looking theorems sometimes possess elegant proofs. The author deliberately selects results that are rarely encountered in normal curricula while remaining accessible through algebra, Euclidean and analytic geometry, and trigonometry.
Key takeaways
- Elementary mathematics can produce genuinely deep and surprising results without requiring advanced university machinery.
- Geometric construction problems reveal profound connections between geometry and algebra.
- Origami is mathematically more powerful than straightedge-and-compass geometry for certain constructions.
- The Mohr–Mascheroni, Poncelet–Steiner and Gauss heptadecagon results are among the book's standout surprises.
- The book is particularly suitable for mathematics teachers, strong secondary-school students, competition-oriented students and mathematically curious readers rather than someone looking for a standard course textbook.
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