Real Tornado

Year: 2009 Authors: Akio Hizume; Yoshikazu Yamagishi

Core claim

Every suitable real number R yields a unique tornado pattern determined by consecutive convergents of its continued fraction expansion, and conversely tornadoes characterize such convergents.

Topics

continued fractions, spiral triangular patterns, convergents, similar triangles

Domains

number theory, geometry, continued fractions, approximation theory, mathematical visualization, pattern design, generative art

Methods

continued fraction expansion, geometric construction, law of sines, theorem proof

Media

plane coordinates, line segments, triangle sequences, figures

Paper text

The text below is the locally extracted OCR/Markdown version of the paper. Raw PDF files remain local and are not published here.

Bridges 2009: Mathematics, Music, Art, Architecture, Culture

Real Tornado

Akio Hizume, Yoshikazu Yamagishi Department of Applied Mathematics and Informatics Ryukoku University Seta, Otsu, Shiga, Japan

E-mail: akio@starcage.org yg@rins.ryukoku.ac.jp

Abstract

The continued fraction expansion of a real number R > 0 generates a family of spiral triangular patterns, called “tornadoes.” Each tornado consists of similar triangles, any two of which are non-congruent.

Basic Operation

Let R > 0 and 0 < s < 1 . In the plane, the sequence of points $V(j)=(s^j\cos 2\pi jR,s^j\sin 2\pi jR)$ for $j=0,1,\dots$ , which we call the ‘vertices’, naturally converges to the origin. Fix an integer k > 0 , which is called the ‘modulo’ or the ‘step size’, and join the vertex $V(j)$ with $V(j+k)$ by the line segment $\overline{V(j)V(j+k)}$ for $j\ge 0$ .

Fibonacci Tornado

The Fibonacci numbers $f_n$ are defined by $f_1=f_2=1$ and $f_n=f_{n-2}+f_{n-1}$ , n > 2 . In the previous paper [2], we showed that if $k=f_{n-1}$ and $R=\tau$ , where $\tau =(1+\sqrt{5})/2$ is the golden ratio, there exists a 0 < s < 1 such that the vertex $V(j+f_{n+2})$ lands on the line segment $\overline{V(j+f_{n+1})V(j+f_n)}$ for each $j\ge 0$ . By the Basic Operation above, we obtain the spiral pattern of similar triangles as shown in Figure 1 ( $k=2$ ), which is called a “tornado”. As $k$ gets larger, we could see that the tornado comes out like a blooming flower, while the argument $jR$ of each vertex $V(j)$ remains unchanged.

Remark that the well-known spirals as in Figure 2 are different from our tornadoes because they have congruent triangles.

Figure 1: Fibonacci Tornado. $[\tau ,3,5]$

Hizume and Yamagishi

Figure 2: A Non-Fibonacci Tornado.

Figure 3: Continued Fraction and Convergents

Real Tornado

A generic real number $R$ also generates a family of tornadoes. As is well-known (see [1]), the continued fraction expansion of $R$ as in the Figure 3 is defined by R = C_0 + \varepsilon_0, 0 \leq \varepsilon_0 < 1 , and $1/{\epsilon }_n=C_{n+1}+{\epsilon }_{n+1}$ , 0 \leq \varepsilon_{n+1} < 1 for $n\ge 0$ , where $C_n$ are called the partial denominators. If $R$ is rational, it is related to the Euclidean algorithm and stops when ${\epsilon }_n=0$ . The $n$ -th convergent $p_n/q_n$ is defined by $p_0=C_0$ , $q_0=1$ , $p_1=C_1{p}_0+1$ , $q_1=C_1$ , and $p_{n+1}=C_{n+1}{p}_n+p_{n-1}$ , $q_{n+1}=C_{n+1}{q}_n+q_{n-1}$ for n > 0 . It is known that $p_n/q_n$ are the best approximations of $R$ , where

For example, the convergents of $R=\sqrt{3}$ are $1/1,2/1,5/3,7/4,19/11,26/15,71/41,\dots$ . The denominators $q_n$ and $q_{n+1}$ are coprime.

Choose any pair of consecutive convergents $p_n/q_n$ and $p_{n+1}/q_{n+1}$ , and denote by $q=q_n$ and $q^{\prime }=q_{n+1}$ . Define the step size by $k=q^{\prime }-q$ . Then there exists a unique 0 < s < 1 such that under the Basic Operation the vertex $V(j+q+q^{\prime })$ lands on the segment $V(j+q)V(j+q^{\prime })$ and we obtain a spiral pattern named as the tornado $[R,q,q^{\prime }]$ , consisting of similar triangles $T_j=\Delta V(j)V(j+q)V(j+q^{\prime })$ for $j\ge 0$ . Figure 4 presents the tornadoes $[R,q,q^{\prime }]=[\sqrt{3},3,4]$ and $[\sqrt{3},4,11]$ .

The basic idea of the Real Tornado was originally published in Japanese in [3]. Here we show how to

find a 0 < s < 1 . Denote the

length of the three edges of $T_j$ by

Figure 4: $\sqrt{3}$ Tornado.

Real Tornado

By Figure 5 we can see that

The three angles of $T_j$ are

or

where the signs are chosen to satisfy that $\sin \varphi ,\sin \delta$ and $\sin \theta$ are all positive. The law of sines is expressed by

and we obtain the equation

It is easy to see that this equation has a unique solution 0 < s < 1.

Figure 5: Principle.

Additional Results

Conversely, we can also prove that any possible tornado $[R,q,q^{\prime }]$ with $q,q^{\prime }$ positive is related to the continued fraction expansion of $R$.

Theorem: Let $R$ be a real number and $q,q^{\prime }$ positive integers. There exists a tornado $[R,q,q^{\prime }]$ if and only if $R$ has a convergent $\frac{p_n}{q_n}$ and an (intermediate) convergent $\frac{c{p}_n+p_{n-1}}{c{q}_n+q_{n-1}}$, 0 < c \leq C_{n+1}, where we denote by $p=p_n,q=q_n,p^{\prime }=c{p}_n+p_{n+1}$ and $q^{\prime }=c{q}_n+q_{n+1}$, such that

(1) R is distinct from $\frac{p}{q}$ and $\frac{p^{\prime }}{q^{\prime }}$, that is, \frac{p}{q} < R < \frac{p'}{q'} or \frac{p'}{q'} < R < \frac{p}{q}, and (2) \left|\{qR\} - \{q'R\}\right| > 1/2, where 0 \leq \{x\} = x - [x] < 1 denotes the fractional part.

See [4] for the proof and further discussions. Note that the golden ratio $\tau$ is a special irrational number which has no intermediate convergents.

Acknowledgements

The authors would like to thank the reviewers for their helpful comments and suggestions. They suggested to consider the equation $z^{q+k}=\alpha z^k+(1-\alpha )$ with 0 < \alpha < 1 given, where $q$ and $k$ are relatively prime. By experiments, they claim that the tornado $[R,q,q+k]$ is obtained by using the root

Hizume and Yamagishi

$z=s{e}^{2\pi iR}\ne 1$ of the largest magnitude. Note that in our setting above, the ratio $\alpha$ tends to 0 or 1 as $R$ approaches to $p/q$ or $p^{\prime }/q^{\prime }$ respectively.

References

[1] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, fifth edition, Oxford, 1979. [2] Akio Hizume, Real Tornado, MANIFOLD #17, pp. 8-11. 2008. (in Japanese) [3] Akio Hizume, Fibonacci Tornado, Bridges Proceedings, pp. 485-486. 2008. [4] Akio Hizume and Yoshikazu Yamagishi, Monohedral similarity tilings, in preparation.

[7/12, 5, 7]

[1.45, 9, 11]

[0.28, 3, 4]

[1.7, 3, 7]

[7/12, 2, 5]

[13/9, 2, 7]

[0.31, 1, 3]

[0.84, 1, 6]

[0.43, 2, 7] Figure 6: Real Tornado Samples.