The Wonderful and Fantastic World of Mathematics

The Wonderful and Fantastic World of Mathematics

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Heinrich Serre

Content

Introduction

Math everywhere

A Dodecahedron in the Sea

Spread of epidemics

Circles in the streets

Sunflowers and rabbits

The traffic tickets

Symmetries in multiplication tables

The time of the traffic

Escher's art

The Spirograph

Ingenuity problems

The eight employees

The Josephus Problem

3x8 board

The frog game

Counter tower

Nine counters

A five-pointed star

The jars

Water, electricity and gas for three houses

The width of the river

Blue and yellow

Regards

Bibliography

Introduction

There is no doubt that mathematics is still the subject that society least prefers, there is still a great animus for it. Much has to do with the way it has been taught, but parents also support this situation by telling their children how complicated the subject is.

I, unlike most people, think math is actually very easy and simple, and I'll explain why. Like everyone else, I've faced some very complicated problems that have taken me weeks, months and even years to solve. At this point, you might think that this shows how difficult math is, but it doesn't, because when I manage to solve a very complicated problem, I realize that it wasn't really difficult, just that I hadn't understood some detail correctly, or that I didn't understand the situation a little more. But that's not all, that peak moment, when I find the solution, is a moment of the most incredible, it gives you a feeling of peace and also of immense joy.

This is a bit of what I want to convey with you, so I have divided the book into two parts. The first part is about mathematical popularization, to show you how everywhere mathematics is, even where one least thinks about it. The second part is about problems and games in which I include the solution. I try to show you how the process of building mathematical knowledge is, to make you feel that pleasant sensation that solving a problem gives.

I hope you like the topics and problems I have chosen and that in the future I can write a second part deepening several ideas that I comment here.

Heinrich Serre

February 2020.

I walk infinitely

until we reach your warm light.

For my beloved family

Math everywhere

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For a person who devotes his life to mathematics, it is even idle to talk about how this area impacts on people's lives, yet for most people it seems that mathematics is only useful for torturing them during their time at school.

During the more than twenty years that I have taught both at the university level and at the high school level, I have constantly heard the question, "Where will this help me? Of course, students think that one day they will go to the store and ask for kilograms of apples or something like that. In some cases, when they already study a university career, they discover the importance of mathematics, they realize that certain things would not be possible, things that are present in our daily lives.

This first part of the book is intended to show you, how math is present in our daily lives, even if we don't realize it. I hope to surprise you, and even more, to stimulate you to try to find new things in mathematics, to broaden your horizons in this important area of study, without which, you can be sure, the future of humanity would be disastrous.

A Dodecahedron in the Sea

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Maybe everyone has in mind what a phytoplankton is, yes, that character in the SpongeBob cartoon. 

Among these beings are special ones called coccolithophores, which have played a unique dual role in the ocean's carbon cycle for some 220 million years. These beings convert the inorganic carbon dissolved in seawater into organic carbon through photosynthesis, but that is not all, they use the dissolved inorganic carbon to produce a mineral shell consisting of coccolithophores, overlapping plates of calcium carbon.

Well, this section is about one of these tiny beings, called Braarudosphaera Bigelowii whose fossil dates back a little over a hundred million years.  But before presenting it, let's remember a very interesting mathematical figure, the dodecahedron.

Most probably you heard this name once in your life, either in your basic education, or in the middle level, although if you don't remember it anymore, don't worry, now we will see it again.

A dodecahedron is a polyhedron (a three-dimensional geometric figure whose faces are flat and have a finite volume) with twelve faces, either convex or concave. Dodecahedron faces are polygons with eleven sides or less, and if these faces are also regular pentagons (whose sides measure the same thing) then the dodecahedron is said to be regular.

In the following picture we have a regular dodecahedron, because each of the twelve faces is a regular pentagon:

Going back to our phytoplankton, this being is very tiny and to measure it we have to use microns, a unit of measurement that is equal to one millionth of a meter in length. To understand what this measurement is, we must observe that a human hair has an average of fifty microns in its cross section. Well, the Braarudosphaera Bigelowii measures at