Vanishing cell and the Fibonacci numbers

A square $8\times 8$ can be cut into four parts from which a rectangle $5\times 13$ of area $65$ is formed!

Another well-known geometric sophism: a right triangle with cathetuses of $5$ and $13$ is being cut into four parts, which make up the same right triangle, but with one empty cell!

But wait, the area of the figure is equal to the sum of the areas of the parts of which it is made up. So it can’t change when you rearrange it. What is the discrep­ancy?

The expla­na­tion of the paradox in both cases is essen­tially the same — the pieces being consid­ered are not the ones described. In the “Chess­board Paradox”, presented by chess player and puzzle author Samuel Loyd in the mid XIX century at the Chess Congress, the square is trans­posed not into a rectangle, but into a rectangle without an elon­gated, almost imper­cep­tible to the eye, paral­lel­o­gram of a unit area (stretched along the diag­onal of the rectangle). In the triangle paradox, invented by Martin Gardner in the middle of the XX century, both hypotenuses (of the orig­inal triangle and the resulting triangle) are not actu­ally straight lines: the figure made up of them is also a paral­lel­o­gram of unit area.

To make it easier to see this paral­lel­o­gram, let us look at an analogue of a smaller Gardner triangle — with sides of $3$ and $5$.

All the vertices of all the parts lie in the nodes of the square grid. And it is easy to make sure that the borders of the parts do not add up in a straight line, but form the sides of a paral­lel­o­gram (with vertices in the nodes) by counting the slope of each segment by the cells. In the rectangle $5\times 13$, in the yellow triangle the ratio of the cathetuses is $\tg \alpha=\dfrac{3}{8}$, and for the blue trapezium the tangent of the “same angle” is $\dfrac{2}{5}$. For the triangle sophism: in Gardner’s variant $\dfrac{2}{5}\ne \dfrac{3}{8}$, in the smaller version $\dfrac{1}{2}\ne \dfrac{2}{3}$. In all cases the sides of the paral­lel­o­gram are, as they should be, pair­wise equal and parallel. The vertices of the paral­lel­o­gram lie in the nodes of the grid, but there are no nodes inside the paral­lel­o­gram. Which, however, is not surprising, if we remember that the area is equal to one and the Pick’s formula.

Having solved the incon­sis­tency, let us think how to construct such sophisms. We may notice that the numbers encoun­tered, $1,$ $2,$ $3,$ $5,$ $8,$ $13$, are the begin­ning of the famous sequence of Fibonacci numbers

$\{1,$ $1,$ $2,$ $3,$ $5,$ $8,$ $13,$ $21,$ $34,$ $55,$ $89,\ \dotso\}.$

This sequence is given by the recur­rence rela­tion $$F_n=F_{n-1}+F_{n-2}$$ and a pair of prime numbers $F_0=1$, $F_1=1$.

There are many inter­esting rela­tions between the Fibonacci numbers. In the year $1680$ a French astronomer of Italian origin, Giovanni Domenico Cassini, who discov­ered the satel­lites of Saturn and a gap in its rings, noticed this rela­tion: $F_{n+1}\cdot F_{n-1}-F_n^2=(-1)^n$, which now goes by his name.

When $n=6$ the equa­tion obtained is $5\cdot 13-8^2=1$ — the familiar numbers from “The Chess­board Paradox” and the familiar increase by one! But to take a Fibonacci number with an odd number as a side of the square is impos­sible — the parts can be added to the corre­sponding rectangle only with an overlap (the inter­sec­tion is still the same paral­lel­o­gram of unit area).

In the “Gardner triangle” the cathetates of the small (real and not changing area) trian­gles are Fibonacci numbers: the red one has 3 and 8, the yellow one has 2 and 5. Corre­spond­ingly, the sides of the rectangle (in which the increase per cell occurs) are: before rearrange, 3 and 5, and after rearrange, 2 and 8. The increase in the area of the rectangle per cell provides the ratio by four consec­u­tive Fibonacci numbers: $F_{n}\cdot F_{n-3}-F_{n-1}\cdot F_{n-2}=(-1)^n$, which can be obtained from the Cassini rela­tion and the recur­rence rela­tion. For $n=6$ the resulting equality is $8\cdot 2-5\cdot 3=1$, on which the “Gardner triangle” relies.

Thus sophisms are built on the fact that the sizes of figures and parts of which they are composed are several consec­u­tive Fibonacci numbers. On the basis of these rela­tions we can construct similar sophisms for figures of larger sizes. In Loyd’s variant one should not forget about parity, and in Gardner’s variant, if one wants to make the square gather into a square cell, one should increase the number of parts of which the basic rectangle is made up.

For Fibonacci numbers there is also an explicit, not recur­ring, assign­ment, called the Binet formula: $$F_n=\frac{1}{\sqrt{5}}\Big(\frac{1+\sqrt{5}}{2}\Big)^n-\frac{1}{\sqrt{5}}\Big(\frac{1-\sqrt{5}}{2}\Big)^n.$$ Note that although Fibonacci numbers are inte­gers, there is an irra­tional number in the formula for them — the golden ratio $\phi=\dfrac{1+\sqrt5}2$.

The first paren­thesis in the Binet formula equals nearly $1{,}618$, and the second paren­thesis is a nega­tive number and modulo less than one (approx­i­mately $-0{,}618$). So the Fibonacci numbers are rapidly increasing, or, more precisely, $F_n\sim\dfrac1{\sqrt5}\varphi^n$. This explains why as the slit in the form of a paral­lel­o­gram gets narrower and narrower and more diffi­cult to notice (compare for example the small and large “Gardner trian­gles”). Indeed, the slopes of different segments in sophisms have the form $\dfrac{F_{n+2}}{F_n}$, and as $n$ grows these rela­tions become closer and closer to $\phi^2$ and prac­ti­cally indis­tin­guish­able.

The unit paral­lel­o­gram and its diag­onal are the objects of a beau­tiful science begun by Hermann Minkowski — geom­etry of numbers. More precisely, geometric inter­pre­ta­tion of chain frac­tions.

The figure shows the $x=\dfrac{1+\sqrt{5}}2y$ line and marks the nodes of the grid closest to it. Their coor­di­nates are adja­cent Fibonacci numbers, and the points them­selves, jumping one after another above and below the line, approach it. And the rela­tions of neigh­bouring Fibonacci numbers give in some sense the best rational approx­i­ma­tions to the Golden Section... The reader, intrigued, may refer to the brochure “Chain Frac­tions” by Vladimir Igore­vich Arnold.

It may seem just a trick, just a picture, walking around the Internet… But they contain so much non-trivial math­e­matics!