A POPULAR MATHEMATICS BOOK
A classic soccer ball always has twelve. The reason is Euler’s formula. The same formula explains why you cannot comb a hairy ball flat, and how ants could find out whether the Earth is a sphere or a doughnut without ever leaving the ground.
For high-school students and adults who are curious about mathematics.
THE BOOK
Topology and the Hidden Rules of Shape
The book starts by counting the patches on a soccer ball and ends with theorems that link the shape of a surface to its curvature.
From vertices, edges and faces it moves on to curved surfaces, to peaks and passes in a mountain landscape, to vector fields and their zeros, and to maps that must leave some point in place. In each of these places the same number turns up: V − E + F.
It is written for high-school students and adults. Pictures, questions and thought experiments show where the theorems come from.
Free preview: “Why This Book?” and contents (PDF) ↓FROM THE BOOK
A classic ball is sewn from pentagons and hexagons. The number of hexagons can vary. The number of pentagons cannot.
Suppose the ball is made of pentagons and hexagons only, with three patches meeting at every corner. Count the corners, seams and patches and put them into Euler’s formula: the hexagons cancel out, and what is left says that there are 12 pentagons.
V − E + F = 2 ⇒ P = 12
The ants never leave the ground and never see their planet from outside. Can they still tell a sphere from a doughnut?
Yes. They can divide their world into triangular plots and count the vertices, edges and faces. However the plots are drawn, V − E + F comes out as 2 on a sphere and 0 on a torus.
On a sphere, some hair always sticks up. On the surface of a doughnut, you can comb everything flat.
A sphere has no continuous tangent vector field that is nonzero everywhere; a torus has one. By the Poincaré–Hopf theorem the zeros of such a field, counted with their indices, add up to V − E + F: 2 for the sphere, 0 for the torus.
MACIEJ STAROSTKA / AUTHORABOUT THE AUTHOR
Maciej Starostka is an Associate Professor of Mathematics at Gdańsk University of Technology. His research is in topology and Hamiltonian dynamics, and he teaches topology to university students.
The book was first published in Polish as Dlaczego piłka ma 12 pięciokątów?
“What I would most like you to carry away from each chapter is an image and a question.”From the introduction
WHERE TO BUY
The English edition is available in paperback on Amazon. The Polish edition is available in print and as an e-book.
Read the free preview first ↓You buy directly from the bookstore. Prices and availability are on its website.
CONTACT
For review copies, interviews or talks:
maciejstarostka@gmail.com ↗