Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Simple applications of Continued Fractions andan 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, PisaItalySimple 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, PisaItalySimple 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, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
IntroductionAim of the talk
This talk is addressed to:Experts in continued fractions,as they will explore educational ways to apply C.F.’s properties, andsee 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 ofalgebra and geometry.
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
IntroductionAbout 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 involvesnested fractions which are continued possibly indefinitely.
Example (Fraction)75
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
IntroductionAbout 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 involvesnested fractions which are continued possibly indefinitely.
Example (Nested fractions)1− 2
51− 4
7
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
IntroductionAbout 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 involvesnested fractions which are continued possibly indefinitely.
Example (Nested fractions)1−
1− 45
1− 12
1−1− 6
71− 3
4
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
IntroductionAbout continued fractions
A continued fraction, then, is something like:
Typical continued fraction
n0 + 1n1 + 1
n2+ 1n3+···
where in our context the ni are rational integers.
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
IntroductionAbout continued fractions
A continued fraction, then, is something like:
Example
1 + 12 + 1
3+ 14+···
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
IntroductionAbout continued fractions
or, in a more general fashion, something like:
Typical continued fraction
q0 + p1q1 + p2
q2+ p3q3+···
where in our context the pi and the qi are rational integers.
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
IntroductionAbout continued fractions
or, in a more general fashion, something like:
Example
1 + 23 + 4
5+ 67+···
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
IntroductionAbout continued fractions
When the truncations of the nested fractions (calledapproximants) have a limit, we can associate the expansion toa real number, similarly to usual decimal expansions.While periodic decimal expansions are associated to rationalnumbers, in the case of continued fractions a periodicity in the“digits” relates to a quadratic irrational, i.e. the solution of aquadratic 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, PisaItalySimple 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, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Geometric illusionsFibonacci illusion
Let’s begin with the classical Fibonacci illusion:
(a) Decomposition of an 8× 13rectangle apparently using 4triangles and two 5× 5 squares.
(b) Decomposition similar to theformer one, but a 5× 5 square isreplaced by a 3× 8 rectangle.
Figure: Fibonacci illusion
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Geometric illusionsFibonacci illusion: explanation
Where did the “missing square” go?
The trick is that the four points on the diagonal are notaligned but form a parallelogram with area exactly 1.Thus, some small fragments with total measure 1 are cut oradded to adjust the geometrical figures, and all of these sumup for the “missing square”.On the other side, by the formula for the area of a triangle, thedistance of the two inner vertices from the diagonal line is verysmall? 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, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Geometric illusionsFibonacci illusion: explanation
Where did the “missing square” go?The trick is that the four points on the diagonal are notaligned but form a parallelogram with area exactly 1.
Thus, some small fragments with total measure 1 are cut oradded to adjust the geometrical figures, and all of these sumup for the “missing square”.On the other side, by the formula for the area of a triangle, thedistance of the two inner vertices from the diagonal line is verysmall? 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, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Geometric illusionsFibonacci illusion: explanation
Where did the “missing square” go?The trick is that the four points on the diagonal are notaligned but form a parallelogram with area exactly 1.Thus, some small fragments with total measure 1 are cut oradded to adjust the geometrical figures, and all of these sumup for the “missing square”.
On the other side, by the formula for the area of a triangle, thedistance of the two inner vertices from the diagonal line is verysmall? 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, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Geometric illusionsFibonacci illusion: explanation
Where did the “missing square” go?The trick is that the four points on the diagonal are notaligned but form a parallelogram with area exactly 1.Thus, some small fragments with total measure 1 are cut oradded to adjust the geometrical figures, and all of these sumup for the “missing square”.On the other side, by the formula for the area of a triangle, thedistance of the two inner vertices from the diagonal line is verysmall? 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, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Geometric illusionsMatricial methods
We recall that the area of a parallelogram is given by thedeterminant of a 2× 2 matrix.
In our case, Fibonacci numbers approximate a geometricprogression with golden ratio, thus the matrix(Fn+1 FnFn Fn−1
)has determinant ±1.
Indeed, when we put numerators and denominators of twofractions in the four entries of a matrix, their difference is givenby the matrix determinant divided by the product ofdenominators; thus, it is smallest when determinant is ±1.
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Geometric illusionsMatricial methods
We recall that the area of a parallelogram is given by thedeterminant of a 2× 2 matrix.In our case, Fibonacci numbers approximate a geometricprogression with golden ratio, thus the matrix(Fn+1 FnFn Fn−1
)has determinant ±1.
Indeed, when we put numerators and denominators of twofractions in the four entries of a matrix, their difference is givenby the matrix determinant divided by the product ofdenominators; thus, it is smallest when determinant is ±1.
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Geometric illusionsMatricial methods
We recall that the area of a parallelogram is given by thedeterminant of a 2× 2 matrix.In our case, Fibonacci numbers approximate a geometricprogression with golden ratio, thus the matrix(Fn+1 FnFn Fn−1
)has determinant ±1.
Indeed, when we put numerators and denominators of twofractions in the four entries of a matrix, their difference is givenby the matrix determinant divided by the product ofdenominators; thus, it is smallest when determinant is ±1.
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Geometric illusionsMatricial methods
The former approximating result, which belongs to the branchof mathematics called diophantine approximation, is strictlyrelated to continued fractions.
The entries ni in a continued fraction are classically associated
to matrices ni =(ni 11 0
), which give projective transforms
on P1(Q) corresponding to LFTs.Thus, a continued fraction is obtained from the limit of ahomogeneous matricial product.Each of these matrices has determinant −1, thus the bestapproximations are taken truncating this product.
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Geometric illusionsMatricial methods
The former approximating result, which belongs to the branchof mathematics called diophantine approximation, is strictlyrelated to continued fractions.The entries ni in a continued fraction are classically associated
to matrices ni =(ni 11 0
), which give projective transforms
on P1(Q) corresponding to LFTs.
Thus, a continued fraction is obtained from the limit of ahomogeneous matricial product.Each of these matrices has determinant −1, thus the bestapproximations are taken truncating this product.
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Geometric illusionsMatricial methods
The former approximating result, which belongs to the branchof mathematics called diophantine approximation, is strictlyrelated to continued fractions.The entries ni in a continued fraction are classically associated
to matrices ni =(ni 11 0
), which give projective transforms
on P1(Q) corresponding to LFTs.Thus, a continued fraction is obtained from the limit of ahomogeneous matricial product.
Each of these matrices has determinant −1, thus the bestapproximations are taken truncating this product.
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Geometric illusionsMatricial methods
The former approximating result, which belongs to the branchof mathematics called diophantine approximation, is strictlyrelated to continued fractions.The entries ni in a continued fraction are classically associated
to matrices ni =(ni 11 0
), which give projective transforms
on P1(Q) corresponding to LFTs.Thus, a continued fraction is obtained from the limit of ahomogeneous matricial product.Each of these matrices has determinant −1, thus the bestapproximations are taken truncating this product.
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Geometric illusionsFibonacci illusion: properties of the Golden ratio
RemarkWe will now see that the Fibonacci matrix is obtained fromcontinued fractions, explaining immediately the presented illusoryresult.
Golden ratio λ satisfies the following golden proportion:
λ : 1 = (λ+ 1) : λ
thus:λ = λ+ 1
λ= 1 + 1
λ= 1 + 1
1 + 1λ
= · · ·
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Geometric illusionsFibonacci illusion: properties of the Golden ratio
RemarkWe will now see that the Fibonacci matrix is obtained fromcontinued fractions, explaining immediately the presented illusoryresult.
Golden ratio λ satisfies the following golden proportion:
λ : 1 = (λ+ 1) : λ
thus:λ = λ+ 1
λ= 1 + 1
λ= 1 + 1
1 + 1λ
= · · ·
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Geometric illusionsFibonacci illusion: properties of the Golden ratio
RemarkWe will now see that the Fibonacci matrix is obtained fromcontinued fractions, explaining immediately the presented illusoryresult.
Golden ratio λ satisfies the following golden proportion:
λ : 1 = (λ+ 1) : λ
thus:λ = λ+ 1
λ= 1 + 1
λ= 1 + 1
1 + 1λ
= · · ·
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Geometric illusionsFibonacci illusion: properties of the Golden ratio
We have then the following continued fraction expansion:λ = [1, 1, 1, . . .] = [1]
and the approximants have exactly triples of subsequent Fibonaccinumbers as coefficients:
[1]→ 1 =(
1 11 0
); [1, 1, 1, 1]→ 14 =
(5 33 2
);
[1, 1]→ 1 · 1 =(
2 11 1
); [1, 1, 1, 1, 1]→ 15 =
(8 55 3
);
[1, 1, 1]→ 13 =(
3 22 1
); [1, 1, 1, 1, 1, 1]→ 16 =
(13 88 5
).
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Geometric illusionsFibonacci illusion: properties of the Golden ratio
We have then the following continued fraction expansion:λ = [1, 1, 1, . . .] = [1]
and the approximants have exactly triples of subsequent Fibonaccinumbers as coefficients:
[1]→ 1 =(
1 11 0
); [1, 1, 1, 1]→ 14 =
(5 33 2
);
[1, 1]→ 1 · 1 =(
2 11 1
); [1, 1, 1, 1, 1]→ 15 =
(8 55 3
);
[1, 1, 1]→ 13 =(
3 22 1
); [1, 1, 1, 1, 1, 1]→ 16 =
(13 88 5
).
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Geometric illusionsPapersheet illusion
A normal sheet of paper can be cut in two equal pieces whichare similar to the original sheet.
For instance, an A3 paper can be cut into two A4 sheets.Thus, the shorter side is middle proportional between the longerone and its half, so that the ratio between sides is exactly
√2.
We now use this ratio in order to get the same geometricillusion for a different type of rectangle.We will call this one “papersheet illusion”, from the justmentioned proportion between sides.
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Geometric illusionsPapersheet illusion
A normal sheet of paper can be cut in two equal pieces whichare similar to the original sheet.For instance, an A3 paper can be cut into two A4 sheets.
Thus, the shorter side is middle proportional between the longerone and its half, so that the ratio between sides is exactly
√2.
We now use this ratio in order to get the same geometricillusion for a different type of rectangle.We will call this one “papersheet illusion”, from the justmentioned proportion between sides.
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Geometric illusionsPapersheet illusion
A normal sheet of paper can be cut in two equal pieces whichare similar to the original sheet.For instance, an A3 paper can be cut into two A4 sheets.Thus, the shorter side is middle proportional between the longerone and its half, so that the ratio between sides is exactly
√2.
We now use this ratio in order to get the same geometricillusion for a different type of rectangle.We will call this one “papersheet illusion”, from the justmentioned proportion between sides.
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Geometric illusionsPapersheet illusion
A normal sheet of paper can be cut in two equal pieces whichare similar to the original sheet.For instance, an A3 paper can be cut into two A4 sheets.Thus, the shorter side is middle proportional between the longerone and its half, so that the ratio between sides is exactly
√2.
We now use this ratio in order to get the same geometricillusion for a different type of rectangle.
We will call this one “papersheet illusion”, from the justmentioned proportion between sides.
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Geometric illusionsPapersheet illusion
A normal sheet of paper can be cut in two equal pieces whichare similar to the original sheet.For instance, an A3 paper can be cut into two A4 sheets.Thus, the shorter side is middle proportional between the longerone and its half, so that the ratio between sides is exactly
√2.
We now use this ratio in order to get the same geometricillusion for a different type of rectangle.We will call this one “papersheet illusion”, from the justmentioned proportion between sides.
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Geometric illusionsPapersheet illusion
We take the following papersheet-shaped rectangle:
(a) Decomposition of a 12× 17rectangle apparently using 4triangles and two 7× 7 squares.
(b) Decomposition similar to theformer one, but a 7× 7 square isreplaced by a 5× 10 rectangle.
Figure: Papersheet illusion
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Geometric illusionsPapersheet illusion: explanation
In this case ratios 1/1, 3/2, 7/5, 17/12, 41/29 are increasinglybetter approximations of
√2.
They are indeed given by the continued fraction expansion of√2 = [1, 2, 2, . . .] = [1, 2]:
[1]→ 1 =(
1 11 0
); [1, 2, 2]→ 1 · 22 =
(7 35 2
);
[1, 2]→ 1 · 2 =(
3 12 1
); [1, 2, 2, 2]→ 1 · 23 =
(17 712 5
).
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Geometric illusionsOther illusions
As expected, the illusion can be adjusted for any 2× 2 matrixwith determinant ±1, specifically for the ones arising in anycontinued fraction expansion.
We close this section with another illusion obtaineddecomposing (improperly) the big rectangle in four triangles.The trick is still the same: the two inner vertices are notexactly on the diagonal, but form a very thin parallelogramwith the endpoints of the diagonal itself.In this illusion, it is way easier to see the importance of theparallelogram.
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Geometric illusionsOther illusions
As expected, the illusion can be adjusted for any 2× 2 matrixwith determinant ±1, specifically for the ones arising in anycontinued fraction expansion.We close this section with another illusion obtaineddecomposing (improperly) the big rectangle in four triangles.
The trick is still the same: the two inner vertices are notexactly on the diagonal, but form a very thin parallelogramwith the endpoints of the diagonal itself.In this illusion, it is way easier to see the importance of theparallelogram.
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Geometric illusionsOther illusions
As expected, the illusion can be adjusted for any 2× 2 matrixwith determinant ±1, specifically for the ones arising in anycontinued fraction expansion.We close this section with another illusion obtaineddecomposing (improperly) the big rectangle in four triangles.The trick is still the same: the two inner vertices are notexactly on the diagonal, but form a very thin parallelogramwith the endpoints of the diagonal itself.
In this illusion, it is way easier to see the importance of theparallelogram.
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Geometric illusionsOther illusions
As expected, the illusion can be adjusted for any 2× 2 matrixwith determinant ±1, specifically for the ones arising in anycontinued fraction expansion.We close this section with another illusion obtaineddecomposing (improperly) the big rectangle in four triangles.The trick is still the same: the two inner vertices are notexactly on the diagonal, but form a very thin parallelogramwith the endpoints of the diagonal itself.In this illusion, it is way easier to see the importance of theparallelogram.
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Geometric illusionsFour triangles variation
(a) The area of the four trianglesseems to be 5× 8 + 5× 13 = 105but the whole rectangle has area8× 13 = 104.
(b) The area of the four trianglesseems to be 7× 12 + 7× 17 = 203but the whole rectangle has area12× 17 = 204.
Figure: Triangles illusion
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple 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, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Pseudo-Pythagoric trianglesIncommensurability
Pythagorean disciples were startled by the following:
FactAn isosceles right triangle has incommensurable sides.
Indeed, it is well known that, if the proportion were rational, itwould contradict the following elementary:
Theorem (of the reduced fraction)
Any fraction is equivalent to exactly one reduced fraction.
In our case, the resulting fraction n/d would satisfy 2|(n, d).
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Pseudo-Pythagoric trianglesIncommensurability
Pythagorean disciples were startled by the following:
FactAn isosceles right triangle has incommensurable sides.
Indeed, it is well known that, if the proportion were rational, itwould contradict the following elementary:
Theorem (of the reduced fraction)
Any fraction is equivalent to exactly one reduced fraction.
In our case, the resulting fraction n/d would satisfy 2|(n, d).
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Pseudo-Pythagoric trianglesIncommensurability
Pythagorean disciples were startled by the following:
FactAn isosceles right triangle has incommensurable sides.
Indeed, it is well known that, if the proportion were rational, itwould contradict the following elementary:
Theorem (of the reduced fraction)
Any fraction is equivalent to exactly one reduced fraction.
In our case, the resulting fraction n/d would satisfy 2|(n, d).
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Pseudo-Pythagoric trianglesApproximated solution
As Pythagoreans did with music tones, we will find now anapproximated solution using our diophantine approximationmethods.
We begin by giving the following:
DefinitionWe will denote by Pseudo-Pythagoric triangle a right trianglewith integral sides which is almost isosceles, i.e. whose catheti areconsecutive integers.
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Pseudo-Pythagoric trianglesApproximated solution
As Pythagoreans did with music tones, we will find now anapproximated solution using our diophantine approximationmethods.We begin by giving the following:
DefinitionWe will denote by Pseudo-Pythagoric triangle a right trianglewith integral sides which is almost isosceles, i.e. whose catheti areconsecutive integers.
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Pseudo-Pythagoric trianglesApproximated solution
Thus we need to solve the following Pell’s equation:
Pseudo-Pythagoric condition
l2 + (l + 1)2 = d2.
It is immediate to check that it’s equivalent to the following:
Pseudo-Pythagoric condition
(2l + 1)2 + 1 = 2d2
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Pseudo-Pythagoric trianglesApproximated solution
Thus we need to solve the following Pell’s equation:
Pseudo-Pythagoric condition
l2 + (l + 1)2 = d2.
It is immediate to check that it’s equivalent to the following:
Pseudo-Pythagoric condition
(2l + 1)2 + 1 = 2d2
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Pseudo-Pythagoric trianglesApproximated solution
Now solving is straightforward, with the substitution k = 2l+ 1:
(2l+1)2+1 = 2d2 → k2+1 = 2d2 → k2−2d2 = −1.
Odd powers of 1 +√
2, written in the form k + d√
2, give allthe solutions, having exactly norm −1 in the number fieldQ[√
2]:
k = 1, d = 1 → 12 + 02 = 12
k = 7, d = 5 → 32 + 42 = 52
k = 41, d = 29 → 202 + 212 = 292
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Pseudo-Pythagoric trianglesFinal remarks
We notice that the couples of integers we have just listed givenumerator and denominator (k/d) of some of the continuedfraction approximations of
√2 that we considered before.
Indeed, as discussed by the author in his previous works,continued fractions are strictly related to Pell’s equations.
RemarkAnother type of approximation can be built by looking forisosceles right triangles with integral catheti and “almost integral”hypothenuse; in this case a cathetus length d from continuedfraction approximations k/d satisfies 2d2 = k2 ± 1, thus gives anhypothenuse of length approximately k.
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Pseudo-Pythagoric trianglesFinal remarks
We notice that the couples of integers we have just listed givenumerator and denominator (k/d) of some of the continuedfraction approximations of
√2 that we considered before.
Indeed, as discussed by the author in his previous works,continued fractions are strictly related to Pell’s equations.
RemarkAnother type of approximation can be built by looking forisosceles right triangles with integral catheti and “almost integral”hypothenuse; in this case a cathetus length d from continuedfraction approximations k/d satisfies 2d2 = k2 ± 1, thus gives anhypothenuse of length approximately k.
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Pseudo-Pythagoric trianglesFinal remarks
We notice that the couples of integers we have just listed givenumerator and denominator (k/d) of some of the continuedfraction approximations of
√2 that we considered before.
Indeed, as discussed by the author in his previous works,continued fractions are strictly related to Pell’s equations.
RemarkAnother type of approximation can be built by looking forisosceles right triangles with integral catheti and “almost integral”hypothenuse; in this case a cathetus length d from continuedfraction approximations k/d satisfies 2d2 = k2 ± 1, thus gives anhypothenuse of length approximately k.
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple 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, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Heron’s algorithmDefinition
Algorithm (Heron)
Consider an absolute rational number x ∈ Qa such that its squareroot is irrational. Let a0 ∈ Qa be an approximation of
√x s.t.
a20 > x. Then the sequences defined as follows:
bi = x
ai; ai+1 = ai + bi
2
are rational-valued, strictly monotone and having both limit√x;
more precisely, the ai decrease from above and the bi increase frombelow?.
?The sequence of intervals Ii = [bi, ai] is also known under the name of“chinese boxes” for the real number
√x, implying their nesting structure.
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Heron’s algorithmDefinition: remarks
Remark (#1)
We may take also a0 <√x and obtain the same sequences as
taking a′0 = x/a0 >√x, with just the first terms a0 and b0
switching places.
Remark (#2)
The name “Heron’s algorithm” is commonly associated to thefirst sequence {ai}i∈N only. Moreover, this sequence is just a specialcase of the more general Newton’s method for approximatingpolynomial roots (in this case, the roots of f(X) = X2 − x).
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Heron’s algorithmDefinition: remarks
Remark (#1)
We may take also a0 <√x and obtain the same sequences as
taking a′0 = x/a0 >√x, with just the first terms a0 and b0
switching places.
Remark (#2)
The name “Heron’s algorithm” is commonly associated to thefirst sequence {ai}i∈N only. Moreover, this sequence is just a specialcase of the more general Newton’s method for approximatingpolynomial roots (in this case, the roots of f(X) = X2 − x).
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Heron’s algorithmMain theorem
Main theoremSuppose
√x has continued fraction expansion [n0, n1, n2, . . .] with
period of length 1 or 2. Set a0 = b√xc = n0 and apply on it the
Heron’s algorithm obtaining a sequence {a0, a1, . . .}.
Then ai is the 2i-th approximant via the continued fraction.
Vice versa, if the continued fraction has period length greater than2, the Heron’s algorithm does not give the same sequence ofapproximants.
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Heron’s algorithmMain theorem: proof
Proof (Sketch)
The number y = a0 +√x has the same continued fraction as
√x
except for the first term, and is moreover purely periodic. Thus, inthis case:
y = 2a0 + 1t+ 1
y
.
This can easily be solved for t obtaining t = 2a0x−a2
0, which gives
immediately a1 = a0 + 1t . For the general ai, the proof relies on the
algebraic properties of matrix powers(2a0 · t
)2i−1
related? to y.
?Alternatively, one can consider(a0 · t · a0 · 0
)2i−1relating to the improper
continued fraction expansion√
x = [a0, t, a0, 0].Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Heron’s algorithmMain theorem: examples
Example (√
2)
We have:a0 = 1;a1 = 3/2 = [1, 2];a2 = 17/12 = [1, 2, 2, 2];
and so on. Indeed, 1 +√
2 = [2] as already seen.The theorem holds because the continued fraction has periodlength 1.
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Heron’s algorithmMain theorem: examples
Example (√
6)
We have:a0 = 2;a1 = 5/2 = [2, 2];a2 = 49/20 = [2, 2, 4, 2];
and so on. Indeed, 2 +√
6 = [4, 2].The theorem holds because the continued fraction has periodlength 2.
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Heron’s algorithmMain theorem: examples
Example (√
7)
We have:a0 = 2;a1 = 11/4 = [2, 1, 3];a2 = 233/88 = [2, 1, 1, 1, 5, 5];
and so on. Indeed, 2 +√
7 = [4, 1, 1, 1].The theorem does not hold because the continued fraction hasperiod > 2, confirming the necessity of the condition on the period.
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple 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, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Generalizationsn-continued fractions: generalization of the main theorem
The main theorem also holds for a generalization of continuedfractions (the “n-continued fractions”) which has the propertyto converge in p-adic fields too (for every prime divisor p|n),as discussed in a still unpublished work of the author.
The “Newton’s method” has a p-adic analogue, which isexactly the fundamental Hensel’s lemma, named after themost important contributor to this mathematical topic.
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Generalizationsn-continued fractions: generalization of the main theorem
The main theorem also holds for a generalization of continuedfractions (the “n-continued fractions”) which has the propertyto converge in p-adic fields too (for every prime divisor p|n),as discussed in a still unpublished work of the author.The “Newton’s method” has a p-adic analogue, which isexactly the fundamental Hensel’s lemma, named after themost important contributor to this mathematical topic.
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Generalizationsn-continued fractions: definition
The definition of n-continued fractions uses the generalizedform:
q0 + p1q1 + p2
q2+ p3q3+···
with conditions ∀i ∈ N: pi = n, qi > n, (qi, n) = 1.
In this case, one can check easily that the associated matriceshave as determinant a power of (−n).Consequently, the difference between two subsequentapproximants has numerator ± a power of n (thus,infinitesimal) and denominator coprime with n (thus, withzero-valuation in all p-adic fields); this gives immediately theexistence of the limit, ensuring that the one just given is indeeda good definition.
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Generalizationsn-continued fractions: definition
The definition of n-continued fractions uses the generalizedform:
q0 + p1q1 + p2
q2+ p3q3+···
with conditions ∀i ∈ N: pi = n, qi > n, (qi, n) = 1.In this case, one can check easily that the associated matriceshave as determinant a power of (−n).
Consequently, the difference between two subsequentapproximants has numerator ± a power of n (thus,infinitesimal) and denominator coprime with n (thus, withzero-valuation in all p-adic fields); this gives immediately theexistence of the limit, ensuring that the one just given is indeeda good definition.
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Generalizationsn-continued fractions: definition
The definition of n-continued fractions uses the generalizedform:
q0 + p1q1 + p2
q2+ p3q3+···
with conditions ∀i ∈ N: pi = n, qi > n, (qi, n) = 1.In this case, one can check easily that the associated matriceshave as determinant a power of (−n).Consequently, the difference between two subsequentapproximants has numerator ± a power of n (thus,infinitesimal) and denominator coprime with n (thus, withzero-valuation in all p-adic fields); this gives immediately theexistence of the limit, ensuring that the one just given is indeeda good definition.
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Moving on to......the conclusion
Thanks for the attention!Antonino Leonardis
You will soon find this presentationon my website: uz.sns.it/~antonino
Questions?I’ll be glad to answer.
For further questions e-mail me at: [email protected]
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions
Introduction Geometric illusions Pseudo-Pythagoric triangle Heron’s algorithm Generalizations
Moving on to......the conclusion
Thanks for the attention!Antonino Leonardis
You will soon find this presentationon my website: uz.sns.it/~antonino
Questions?I’ll be glad to answer.
For further questions e-mail me at: [email protected]
Dr. Antonino Leonardis Former Ph.D. student at Scuola Normale Superiore, PisaItalySimple applications of Continued Fractions