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Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations

Simple applications of Continued Fractions and an elementary result on Heron’s algorithm

Joint Mathematics Meetings

AMS Special Session on Continued Fractions

Dr. Antonino Leonardis

Former Ph.D. student at Scuola Normale Superiore, Pisa

Italy

Jan 7th 2017

Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItaly Simple applications of Continued Fractions

Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations

Table of contents

1 Introduction

2 Geometric illusions

3 Pseudo-Pythagoric triangle

4 Heron’s algorithm

5 Generalizations

Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItaly Simple applications of Continued Fractions

Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations

Moving on to...

1 Introduction

2 Geometric illusions

3 Pseudo-Pythagoric triangle

4 Heron’s algorithm

5 Generalizations

Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItaly Simple applications of Continued Fractions

Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations

Introduction Aim of the talk

This talk is addressed to: Experts in continued fractions, as they will explore educational ways to apply C.F.’s properties, and see a precise result on their connection with Heron’s algorithm.

Newcomers in this topic, as they will see a lot of connections with other elementary areas of algebra and geometry.

Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations

Introduction About continued fractions

We briefly recall the main concept: continued fractions.

What goes under the name of “continued fraction”? A continued fraction is of course a fraction, but which involves nested fractions which are continued possibly indefinitely.

Example (Fraction) 7 5

Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations

Introduction About continued fractions

We briefly recall the main concept: continued fractions.

What goes under the name of “continued fraction”? A continued fraction is of course a fraction, but which involves nested fractions which are continued possibly indefinitely.

Example (Nested fractions) 1− 25 1− 47

Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations

Introduction About continued fractions

We briefly recall the main concept: continued fractions.

What goes under the name of “continued fraction”? A continued fraction is of course a fraction, but which involves nested fractions which are continued possibly indefinitely.

Example (Nested fractions) 1−

1− 45 1− 12

1− 1− 67 1− 34

Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations

Introduction About continued fractions

A continued fraction, then, is something like:

Typical continued fraction

n0 + 1

n1 + 1n2+ 1n3+···

where in our context the ni are rational integers.

Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations

Introduction About continued fractions

A continued fraction, then, is something like:

Example

1 + 1 2 + 13+ 14+···

Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations

Introduction About continued fractions

or, in a more general fashion, something like:

Typical continued fraction

q0 + p1

q1 + p2q2+ p3q3+···

where in our context the pi and the qi are rational integers.

Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations

Introduction About continued fractions

or, in a more general fashion, something like:

Example

1 + 2 3 + 45+ 67+···

Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations

Introduction About continued fractions

When the truncations of the nested fractions (called approximants) have a limit, we can associate the expansion to a real number, similarly to usual decimal expansions. While periodic decimal expansions are associated to rational numbers, in the case of continued fractions a periodicity in the “digits” relates to a quadratic irrational, i.e. the solution of a quadratic equation with integral entries?.

?This is widely known as the Lagrange’s theorem for continued fractions. Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItaly Simple applications of Continued Fractions

Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations

Moving on to...

1 Introduction

2 Geometric illusions

3 Pseudo-Pythagoric triangle

4 Heron’s algorithm

5 Generalizations

Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations

Geometric illusions Fibonacci illusion

Let’s begin with the classical Fibonacci illusion:

(a) Decomposition of an 8× 13 rectangle apparently using 4 triangles and two 5× 5 squares.

(b) Decomposition similar to the former one, but a 5× 5 square is replaced by a 3× 8 rectangle.

Figure: Fibonacci illusion

Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations

Geometric illusions Fibonacci illusion: explanation

Where did the “missing square” go?

The trick is that the four points on the diagonal are not aligned but form a parallelogram with area exactly 1. Thus, some small fragments with total measure 1 are cut or added to adjust the geometrical figures, and all of these sum up for the “missing square”. On the other side, by the formula for the area of a triangle, the distance of the two inner vertices from the diagonal line is very small? and as a consequence we have an illusory alignment.

?It is exactly the reciprocal of the length of the diagonal segment. Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItaly Simple applications of Continued Fractions

Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations

Geometric illusions Fibonacci illusion: explanation

Where did the “missing square” go? The trick is that the four points on the diagonal are not aligned but form a parallelogram with area exactly 1.

Thus, some small fragments with total measure 1 are cut or added to adjust the geometrical figures, and all of these sum up for the “missing square”. On the other side, by the formula for the area of a triangle, the distance of the two inner vertices from the diagonal line is very small? and as a consequence we have an illusory alignment.

?It is exactly the reciprocal of the length of the diagonal segment. Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItaly Simple applications of Continued Fractions

Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations

Geometric illusions Fibonacci illusion: explanation

Where did the “missing square” go? The trick is that the four points on the diagonal are not aligned but form a parallelogram with area exactly 1. Thus, some small fragments with total measure 1 are cut or added to adjust the geometrical figures, and all of these sum up for the “missing square”.

On the other side, by the formula for the area of a triangle, the distance of the two inner vertices from the diagonal line is very small? and as a consequence we have an illusory align