Celebratio Mathematica

D. Fuchs and S. Tabachnikov: “Invariants of Legendrian and transverse knots in the standard contact space,” Topology 36 : 5 (1997), pp. 1025–1053. MR 1445553 Zbl 0904.57006 article

D. Fuchs and S. Tabachnikov: “More on paperfolding,” Am. Math. Monthly 106 : 1 (January 1999), pp. 27–35. MR 1674137 Zbl 1037.53501 article

D. B. Fuchs and M. B. Fuchs: “The arithmetic of binomial coefficients,” pp. 1–12 in Kvant selecta: Algebra and analysis, I. Edited by S. Tabachnikov. Mathematical World 14. American Mathematical Society (Providence, RI), 1999. English translation of Russian original published in Kvant 1970:6 (1970). MR 1727594 incollection

Every student knows the formulas \begin{align*} (1+x)^2 &= 1 + 2x + x^2, \\ (1+x)^3 &= 1 + 3x + 3x^2 + x^3. \end{align*} The numbers \( (1,\,2,\,1) \), \( (1,\,3,\,3,\,1) \), as well as numbers obtained in an analogous way by raising \( (1+x) \) to the fourth power, the fifth power, and so on, are called binomial coefficients. This article deals with various properties of binomial coefficients. In the first section we lay out the “general theory”: Many of the theorems we prove here used to be part of the school curriculum. In the second section we will show very easy way to find the remainder when a binomial coefficient is divided by a prime number. The third, and concluding, section deals with certain remarkable properties of binomial coefficients. The main assertions in this section are formulated as hypotheses.

Serge Lvovich Tabachnikov	Related
M. B. Fuchs	Related

D. B. Fuchs and M. B. Fuchs: “On best approximations, I,” pp. 27–35 in Kvant selecta: Algebra and analysis, I. Edited by S. Tabachnikov. Mathematical World 14. American Mathematical Society (Providence, RI), 1999. English translation of Russian original published in Kvant 1971:6 (1971). MR 1727597 incollection

How can one find the best rational approximation for an irrational number? For example, which approximation of the number \( \sqrt{2} \) is the best: \( \frac{3}{2} \), \( \frac{7}{5} \) or \( 1.41 \)? The answer seems to be fairly simple: The smaller the error, the better the approximation. But this is not the whole story, since we write \( \pi \approx 3.14 \) even though we might know five or more decimal digits. It stands to reason that when choosing an approximation we want to decrease not only the error but also the denominator: The smaller the denominator, the less awkward the fraction and the easier to manage it (to store, substitute in formulas, etc.).

In this article we examine the problem of how to meet these two contradictory demands, of finding a rational approximation for a given irrational number that is as far as possible simultaneously the most precise and least awkward.

Serge Lvovich Tabachnikov	Related
M. B. Fuchs	Related

D. B. Fuchs and M. B. Fuchs: “On best approximations, II,” pp. 37–47 in Kvant selecta: Algebra and analysis, I. Edited by S. Tabachnikov. Mathematical World 14. American Mathematical Society (Providence, RI), 1999. English translation of Russian original published in Kvant 1971:11 (1971). MR 1727598 incollection

It is inconvenient to work with irrational numbers. Rational numbers are much easier to work with. Hence the problem of rational approximations to irrational numbers and their efficiency is most important. In Part I of our article we compared different approximations to the number \( \alpha \) and proved that for any irrational \( \alpha \) and arbitrarily large \( N \) there exist infinitely many rational approximations \( p/q \) such that \[ q\Bigl|\alpha - \frac{p}{q}\Bigr| < \frac{1}{N} .\] This, second part of the article deals with the subtler question of whether for any \( \alpha \) there exist infinitely many approximations \( p/q \) such that \[ q^2\Bigl|\alpha - \frac{p}{q}\Bigr| \] is less than a given number. It turns out that this number cannot be arbitrary, in accordance with the Hurwitz–Borel theorem, which states that for any \( \alpha \) there exist infinitely many different approximations \( p/q \) such that \[ q^2\Bigl|\alpha - \frac{p}{q}\Bigr| < \frac{1}{\sqrt{5}} .\] Note that the number \( \sqrt{5} \) cannot be replaced by a larger number. Before reading any further, the reader is advised to have another look at the previous part of the article and also to read the article by N. M. Baskin in Kvant 1970, no. 8.

Serge Lvovich Tabachnikov	Related
M. B. Fuchs	Related

D. B. Fuchs and M. B. Fuchs: “Rational approximations and transcendence,” pp. 65–69 in Kvant selecta: Algebra and analysis, I. Edited by S. Tabachnikov. Mathematical World 14. American Mathematical Society (Providence, RI), 1999. English translation of Russian original published in Kvant 1973:12 (1973). MR 1727601 incollection

Serge Lvovich Tabachnikov	Related
M. B. Fuchs	Related

D. B. Fuchs: “On the removal of parentheses, on Euler, Gauss, and MacDonald, and on missed opportunities,” pp. 39–49 in Kvant selecta: Algebra and analysis, II. Edited by S. Tabachnikov. Mathematical World 15. American Mathematical Society (Providence, RI), 1999. English translation of Russian original published in Kvant 1981:8 (1981). A version of this also appeared in A mathematical omnibus (2007). MR 1728803 incollection

Leonhard Euler	Related
Ian McDonald	Related
Carl Friedrich Gauss	Related
Serge Lvovich Tabachnikov	Related

Differential topology, infinite-dimensional Lie algebras, and applications: D. B. Fuchs’ 60th anniversary collection. Edited by A. Astashkevich and S. Tabachnikov. AMS Translations. Series 2 194. American Mathematical Society (Providence, RI), 1999. MR 1729355 Zbl 0921.00044 book

Alexander B. Astashkevich	Related
Serge Lvovich Tabachnikov	Related

Mathematical omnibus: Thirty lectures on classic mathematics. Edited by D. Fuchs and S. Tabachnikov. American Mathematical Society (Providence, RI), 2007. A German translation was published in 2011. MR 2350979 Zbl 1318.00004 book

D. Fuchs and S. Tabachnikov: “Self-dual polygons and self-dual curves,” Funct. Anal. Other Math. 2 : 2–4 (2009), pp. 203–220. MR 2506116 Zbl 1180.51012 ArXiv 0707.1048 article

D. Davis, D. Fuchs, and S. Tabachnikov: “Periodic trajectories in the regular pentagon,” Mosc. Math. J. 11 : 3 (July–September 2011), pp. 439–461. To the memory of V. I. Arnold. MR 2894424 Zbl 1276.37033 ArXiv 1102.1005 article

Vladimir Igorevich Arnold	Related
Diana Davis	Related
Serge Lvovich Tabachnikov	Related

D. B. Fuchs and S. Tabachnikov: Ein Schaubild der Mathematik: 30 Vorlesungen über klassische Mathematik [Mathematical omnibus: Thirty lectures on classic mathematics]. Springer (Berlin), 2011. German translation of 2007 English original. Zbl 1211.00003 book

D. Fuchs, Y. Ilyashenko, B. Khesin, V. Vassiliev, and H. Hofer: “Memories of Vladimir Arnold,” Notices Am. Math. Soc. 59 : 4 (April 2012), pp. 482–502. Coordinating editors were Boris Khesin and Serge Tabchnikov. MR 2951953 Zbl 1284.37002 article

Helmut Hofer	Related
Vladimir Igorevich Arnold	Related
Victor Anatolyevich Vassiliev	Related
Yulij Sergeevich Ilyashenko	Related
Boris A. Khesin	Related
Serge Lvovich Tabachnikov	Related

D. Fuchs and S. Tabachnikov: “Periodic trajectories in the regular pentagon, II,” Mosc. Math. J. 13 : 1 (January–March 2013), pp. 19–32. MR 3112214 Zbl 1347.37066 ArXiv 1201.0026 article

D. Fuchs: “Dima Arnold in my life,” pp. 133–140 in Arnold: Swimming against the tide. Edited by B. A. Khesin and S. L. Tabachnikov. American Mathematical Society (Providence, RI), 2014. incollection

Vladimir Igorevich Arnold	Related
Boris A. Khesin	Related
Serge Lvovich Tabachnikov	Related

D. Fuchs and S. Tabachnikov: “On Lagrangian tangent sweeps and Lagrangian outer billiards,” Geom. Dedicata 182 (2016), pp. 203–213. MR 3500383 Zbl 1351.37161 ArXiv 1502.03177 article

Given a Lagrangian submanifold in linear symplectic space, its tangent sweep is the union of its (affine) tangent spaces, and its tangent cluster is the result of parallel translating these spaces so that the foot point of each tangent space becomes the origin. This defines a multivalued map from the tangent sweep to the tangent cluster, and we show that this map is a local symplectomorphism (a well known fact, in dimension two). We define and study the outer billiard correspondence associated with a Lagrangian submanifold. Two points are in this correspondence if they belong to the same tangent space and are symmetric with respect to its foot point. We show that this outer billiard correspondence is symplectic and establish the existence of its periodic orbits. This generalizes the well studied outer billiard map in dimension two.

M. Arnold, D. Fuchs, I. Izmestiev, S. Tabachnikov, and E. Tsukerman: “Iterating evolutes and involutes,” Discrete Comput. Geom. 58 : 1 (2017), pp. 80–143. MR 3658331 Zbl 1381.51007 ArXiv 1510.07742 article

This paper concerns iterations of two classical geometric constructions, the evolutes and involutes of plane curves, and their discretizations: evolutes and involutes of plane polygons. In the continuous case, our main result is that the iterated involutes of closed locally convex curves with rotation number one (possibly, with cusps) converge to their curvature centers (Steiner points), and their limit shapes are hypocycloids, generically, astroids. As a consequence, among such curves only the hypocycloids are homothetic to their evolutes. The bulk of the paper concerns two kinds of discretizations of these constructions: the curves are replaced by polygons, and the evolutes are formed by the circumcenters of the triples of consecutive vertices (\( \mathcal{P} \)-evolutes), or by the incenters of the triples of consecutive sides (\( \mathcal{A} \)-evolutes). For equiangular polygons, the theory is parallel to the continuous case: we define discrete hypocycloids (equiangular polygons whose sides are tangent to hypocycloids) and a discrete Steiner point. The space of polygons is a vector bundle over the space of the side directions; our main result here is that both kinds of evolutes define vector bundle morphisms. In the case of \( \mathcal{P} \)-evolutes, the induced map of the base is 4-periodic, and the dynamics reduces to the linear maps on the fibers. We prove that the spectra of these linear maps are symmetric with respect to the origin. The asymptotic dynamics of linear maps is determined by their eigenvalues with the maximum modulus, and we show that all types of behavior can occur: in particular, hyperbolic, when this eigenvalue is real, and elliptic, when it is complex. We also study \( \mathcal{P} \)- and \( \mathcal{A} \)-involutes and prove that the side directions of iterated \( \mathcal{P} \)-involutes of polygons with odd number of sides behave ergodically; this generalizes well-known results concerning iterations of the construction of the pedal triangle. In addition to the theoretical study, we performed numerous computer experiments; some of the observations remain unexplained.

Maxim D. Arnold	Related
Emmanuel Tsukerman	Related
Serge Lvovich Tabachnikov	Related
Ivan Veniaminovich Izmestiev	Related

D. Fuchs and S. Tabachnikov: “Iterating evolutes of spacial polygons and of spacial curves,” Mosc. Math. J. 17 : 4 (October–December 2017), pp. 667–689. MR 3734657 Zbl 1422.53005 ArXiv 1611.08836 article

The evolute of a smooth curve in an \( m \)-dimensional Euclidean space is the locus of centers of its osculating spheres, and the evolute of a spatial polygon is the polygon whose consecutive vertices are the centers of the spheres through the consecutive \( (m{+}1) \)-tuples of vertices of the original polygon. We study the iterations of these evolute transformations. This work continues the recent study of similar problems in dimension two. Here is a sampler of our results.

The set of \( n \)-gons with fixed directions of the sides, considered up to parallel translation, is an \( (n{-}m) \)-dimensional vector space, and the second evolute transformation is a linear map of this space. If \( n=m+2 \), then the second evolute is homothetic to the original polygon, and if \( n=m+3 \), then the first and the third evolutes are homothetic. In general, each eigenvalue of the second evolute map has double multiplicity. We also study curves, with cusps, in 3-dimensional Euclidean space and their evolutes. We provide continuous analogs of the results obtained for polygons, and present a class of curves which are homothetic to their second evolutes; these curves are spatial analogs of the classical hypocycloids.

M. Arnold, D. Fuchs, I. Izmestiev, and S. Tabachnikov: Cross-ratio dynamics on ideal polygons. Preprint, December 2018. ArXiv 1812.05337 techreport

Two ideal polygons, \( (p_1,\dots,p_n) \) and \( (q_1,\dots,q_n) \), in the hyperbolic plane or in hyperbolic space are said to be \( \alpha \)-related if the cross-ratio \[ [p_i,p_{i+1},q_i,q_{i+1}] = \alpha \] for all \( i \) (the vertices lie on the projective line, real or complex, respectively). For example, if \( \alpha = -1 \), the respective sides of the two polygons are orthogonal. This relation extends to twisted ideal polygons, that is, polygons with monodromy, and it descends to the moduli space of Möbius-equivalent polygons. We prove that this relation, which is, generically, a \( 2{-}2 \) map, is completely integrable in the sense of Liouville. We describe integrals and invariant Poisson structures, and show that these relations, with different values of the constants \( \alpha \), commute, in an appropriate sense. We investigate the case of small-gons, describe the exceptional ideal polygons, that possess infinitely many \( \alpha \)-related polygons, and study the ideal polygons that are \( \alpha \)-related to themselves (with a cyclic shift of the indices).

Maxim D. Arnold	Related
Serge Lvovich Tabachnikov	Related
Ivan Veniaminovich Izmestiev	Related

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