Three-gap theorem

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In mathematics, the three-gap theorem, three-distance theorem, or Steinhaus conjecture states that if one places n points on a circle, at angles of θ, 2θ, 3θ ... from the starting point, then there will be at most three distinct distances between pairs of points in adjacent positions around the circle. When there are three distances, the largest of the three always equals the sum of the other two.[1] Unless θ is a rational multiple of π, there will also be at least two distinct distances.

This result was conjectured by Hugo Steinhaus, and proved in the 1950s by Vera T. Sós,  [hu], and Stanisław Świerczkowski; more proofs were added by others later. Applications of the three-gap theorem include the study of plant growth and musical tuning systems, and the theory of light reflection within a mirrored square.

Applications[]

Plant growth[]

End-on view of a plant stem in which consecutive leaves are separated by the golden angle

In phyllotaxis, the theory of plant growth, it has been observed that each successive leaf on the stems of many plants is turned from the previous leaf by the golden angle, approximately 137.5°. It has been suggested that this angle maximizes the sun-collecting power of the plant's leaves.[2] If one looks end-on at a plant stem that has grown in this way, there will be at most three distinct angles between two leaves that are consecutive in the cyclic order given by this end-on view.[3]

For example, in the figure, the largest of these three angles occurs three times, between the leaves numbered 3 and 6, between leaves 4 and 7, and between leaves 5 and 8. The second-largest angle occurs five times, between leaves 6 and 1, 9 and 4, 7 and 2, 10 and 5, and 8 and 3. And the smallest angle occurs only twice, between leaves 1 and 9 and between leaves 2 and 10. The phenomenon of having three types of distinct gaps depends only on fact that the growth pattern uses a constant rotation angle, and not on the relation of this angle to the golden ratio; the same phenomenon would happen for any other rotation angle, and not just for the golden angle.[3]

Music theory[]

A geometric view of the tones of the Pythagorean tuning as points on a circle, showing the Pythagorean comma (the gap between the first and last points of the path) as the amount by which this tuning system fails to close up to a regular dodecagram. The edges between the points of the circle are the perfect fifths from which this tuning system is constructed.

In music theory, the three-gap theorem implies that if a tuning system is generated by some number of consecutive multiples of a given musical interval, reduced to a cyclic sequence by considering two tones to be equivalent when they differ by whole numbers of octaves, then there are at most three different intervals between consecutive tones of the scale.[4][5] For instance, the Pythagorean tuning is constructed in this way from multiples of a perfect fifth. It has only two distinct intervals between its tones, representing two different types of semitones,[6] but if it were extended by one more step then the sequence of intervals between its tones would include a third, much shorter interval, the Pythagorean comma.[7]

Mirrored reflection[]

The three-gap theorem also has applications to Sturmian words, infinite sequences of two symbols (for instance, "H" and "V") that can be used to describe the sequence of horizontal and vertical reflections of a light ray within a mirrored square, starting along a line of irrational slope, or equivalently the sequence of horizontal and vertical lines of the integer grid crossed by the starting line. For any positive integer n, such a sequence has exactly n + 1 distinct consecutive subsequences of length n. The three-gap theorem implies that these n + 1 subsequences occur with at most three distinct frequencies. If there are three frequencies, then the largest frequency must equal the sum of the other two. The proof involves partitioning the y-intercepts of the starting lines (modulo 1) into n + 1 subintervals within which the initial n elements of the sequence are the same, and applying the three-gap theorem to this partition.[8]

History and proof[]

The three-gap theorem was conjectured by Hugo Steinhaus, and its first proofs were published in the late 1950s by Vera T. Sós,[9]  [hu],[10] and Stanisław Świerczkowski.[11] Several later proofs have also been published.[12][13][14][15][16]

The following simple proof is due to Frank Liang. Define a gap to be an arc A of the circle that extends between two consecutive points of the given set, and define a gap to be rigid if the arc A + θ of the same length, obtained by rotating A by an angle of θ, is not another gap. If A is not rigid, meaning that A + θ is another gap, its endpoints are farther along in the sequence in which the points were placed. Repeatedly adding θ will produce equal-length gaps that are farther and farther along this sequence, which cannot continue indefinitely unless the sequence of angles repeats, having only one gap. Therefore, if there is more than one length of gap, then every gap has the same length as a rigid gap. However, the only ways for a gap A to be rigid are for one of its two endpoints to be the last point in the placement sequence (so that the corresponding endpoint of A + θ is missing from the given points) or for one of the given points to land within A + θ, preventing it from being a gap. A point can only land within A + θ if it is the first point in the placement ordering, because otherwise its predecessor in the sequence would land within A, contradicting the assumption that A is a gap. So there can be at most three rigid gaps, the two on either side of the last point and the one in which the predecessor of the first point (if it were part of the sequence) would land. Because there are at most three rigid gaps, there are at most three lengths of gaps.[17][18]

The same proof additionally shows that, when there are exactly three gap lengths, the longest gap length is the sum of the other two. For, in this case, the rotated copy A + θ that has the first point in it is partitioned by that point into two smaller gaps, which must be the other two gaps.[17][18]

A closely related but earlier theorem, also called the three-gap theorem, is that if A is any arc of the circle, then the integer sequence of multiples of θ that land in A has at most three gaps between sequence values. Again, if there are three gaps then one is the sum of the other two.[19][20]

See also[]

References[]

  1. ^ Allouche, Jean-Paul; Shallit, Jeffrey (2003), "2.6 The Three-Distance Theorem", Automatic Sequences: Theory, Applications, Generalizations, Cambridge University Press, pp. 53–55, ISBN 9780521823326
  2. ^ Adam, John A. (2011), A Mathematical Nature Walk, Princeton University Press, pp. 35–41, ISBN 9781400832903
  3. ^ a b van Ravenstein, Tony (1987), "Number sequences and phyllotaxis", Bulletin of the Australian Mathematical Society, 36 (2): 333, doi:10.1017/s0004972700026605
  4. ^ Carey, Norman (2007), "Coherence and sameness in well-formed and pairwise well-formed scales", Journal of Mathematics and Music, 1 (2): 79–98, doi:10.1080/17459730701376743
  5. ^ Narushima, Terumi (2017), Microtonality and the Tuning Systems of Erv Wilson: Mapping the Harmonic Spectrum, Routledge Studies in Music Theory, Routledge, pp. 90–91, ISBN 9781317513421
  6. ^ Strohm, Reinhard; Blackburn, Bonnie J., eds. (2001), Music as Concept and Practice in the Late Middle Ages, Volume 3, Part 1, New Oxford history of music, Oxford University Press, p. 252, ISBN 9780198162056
  7. ^ Benson, Donald C. (2003), A Smoother Pebble: Mathematical Explorations, Oxford University Press, p. 51, ISBN 9780198032977
  8. ^ Lothaire, M. (2002), "Sturmian Words", Algebraic Combinatorics on Words, Cambridge: Cambridge University Press, ISBN 978-0-521-81220-7, Zbl 1001.68093
  9. ^ Sós, V. T. (1958), "On the distribution mod 1 of the sequence ", Ann. Univ. Sci. Budapest, Eötvös Sect. Math., 1: 127–134
  10. ^ Surányi, J. (1958), "Über die Anordnung der Vielfachen einer reelen Zahl mod 1", Ann. Univ. Sci. Budapest, Eötvös Sect. Math., 1: 107–111
  11. ^ Świerczkowski, S. (1959), "On successive settings of an arc on the circumference of a circle", Fundamenta Mathematicae, 46 (2): 187–189, doi:10.4064/fm-46-2-187-189, MR 0104651
  12. ^ Halton, John H. (1965), "The distribution of the sequence ", Proc. Cambridge Philos. Soc., 61 (3): 665–670, doi:10.1017/S0305004100039013, MR 0202668
  13. ^ Slater, Noel B. (1967), "Gaps and steps for the sequence ", Proc. Cambridge Philos. Soc., 63 (4): 1115–1123, doi:10.1017/S0305004100042195, MR 0217019
  14. ^ van Ravenstein, Tony (1988), "The three-gap theorem (Steinhaus conjecture)", Journal of the Australian Mathematical Society, Series A, 45 (3): 360–370, doi:10.1017/S1446788700031062, MR 0957201
  15. ^ Mayero, Micaela (2000), "The three gap theorem (Steinhaus conjecture)", Types for Proofs and Programs: International Workshop, TYPES'99, Lökeberg, Sweden, June 12–16, 1999, Selected Papers, Lecture Notes in Computer Science, 1956, Springer, pp. 162–173, arXiv:cs/0609124, doi:10.1007/3-540-44557-9_10, ISBN 978-3-540-41517-6
  16. ^ Marklof, Jens; Strömbergsson, Andreas (2017), "The three gap theorem and the space of lattices", American Mathematical Monthly, 124 (8): 741–745, arXiv:1612.04906, doi:10.4169/amer.math.monthly.124.8.741, hdl:1983/b5fd0feb-e42d-48e9-94d8-334b8dc24505, MR 3706822
  17. ^ a b Liang, Frank M. (1979), "A short proof of the distance theorem", Discrete Mathematics, 28 (3): 325–326, doi:10.1016/0012-365X(79)90140-7, MR 0548632
  18. ^ a b Shiu, Peter (2018), "A footnote to the three gaps theorem", American Mathematical Monthly, 125 (3): 264–266, doi:10.1080/00029890.2018.1412210, MR 3768035
  19. ^ Slater, N. B. (1950), "The distribution of the integers for which ", Proc. Cambridge Philos. Soc., 46 (4): 525–534, doi:10.1017/S0305004100026086, MR 0041891
  20. ^ Florek, K. (1951), "Une remarque sur la répartition des nombres ", Colloq. Math., 2: 323–324
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