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A Combinatorial Approach to the Inscribed Square Problem

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A Combinatorial Approach to the Inscribed Square Problem A Combinatorial Approach to the Inscribed Square Problem Elizabeth Kelley Francis Edward Su, Advisor Rob Thompson, Reader Department of Mathematics May, 2015 Copyright c 2015 Elizabeth Kelley. The author grants Harvey Mudd College and the Claremont Colleges Library the nonexclusive right to make this work available for noncommercial, educational purposes, provided that this copyright statement appears on the reproduced ma- terials and notice is given that the copying is by permission of the author. To dis- seminate otherwise or to republish requires written permission from the author. Abstract The inscribed square conjecture, also known as Toeplitz’ conjecture or the square peg problem, asserts that every Jordan curve in the Euclidean plane admits an inscribed square. Although there exists no proof of the general conjecture, there are affirmative proofs of the conjecture subject to addi- tional local smoothness conditions. The weakest of these local smooth- ness conditions include the special trapezoid criterion, due to Matschke, and local monotonicity, due to Stromquist. We develop several combinato- rial approaches to the inscribed square problem, explicitly locate inscribed squares for conic sections, and explore the existence of inscribed squares in the Koch snowflake, which we prove to be not locally monotone. Contents Abstract iii 1 Historical Development of the Inscribed Square Problem 1 1.1 Statement of the Problem . .1 1.2 Past Work . .2 1.3 Related Problems . .6 2 Conic Sections 9 3 Koch Snowflake 17 3.1 Inscribed Squares in the Koch Snowflake . 20 4 Discretizing the Parameter Space 35 4.1 Pivot vertex and side length . 36 4.2 Two pivot vertices . 38 4.3 Pivot vertex and angle . 43 5 Future Work 47 Bibliography 49 Chapter 1 Historical Development of the Inscribed Square Problem 1.1 Statement of the Problem The inscribed square problem is also commonly referred to as "Toeplitz’ conjecture" and "the square peg problem". Formulated by Otto Toeplitz in 1911, it can be simply stated as Conjecture 1.1.1 (Toeplitz). Every simple continuous closed curve in the plane admits an inscribed square. Simple continuous closed curves are known as Jordan curves and can be intuitively understood as injective maps from the unit circle into the Euclidean plane. It’s important to note that there is no requirement that the area of an inscribed square lie totally within the inscribing curve. Because the only requirement is that the vertices of the inscribed square lie on the inscribing curve, it’s possible for some portion of the inscribed square to live outside the curve. Perhaps because it is such an elegantly stated problem, there exists a large body of work on the inscribed square conjecture. In its most general form, as in Conjecture 1.1.1, the inscribed square problem remains open. By imposing additional smoothness conditions on the curve, many math- ematicians have been able to verify the conjecture in particular cases. In- terestingly, all of these proofs involve parity arguments, (Matschke (2014)) which naturally suggests that there may exist combinatorial proof strate- gies for at least these particular cases, if not for the conjecture in its full generality. 2 Historical Development of the Inscribed Square Problem Figure 1.1 An example of a square inscribed by a simple, closed curve. Note that a portion of the square lives outside the curve. Knowing that there are solutions to the Toeplitz conjecture for Jordan curves subject to various smoothness conditions, it’s natural to wonder if one - or many - of those partial solutions could be extended to a general solution using a simple convergence argument, as follows: Let the generic Jordan curve g be approximated as a sequence of sufficiently smooth curves g , each of which inscribes at least one square, Sq ; by the compactness f ng n of the plane, there exists a convergent subsequence Sq whose limit is f nk g a square, Sq, inscribed in the original curve g. Naively, this argument ap- pears to offer a simple proof of the general conjecture. Without some way to impose a non-vanishing lower bound on the area of Sqnk , however, the sequence of squares could converge to a degenerate point rather than to an inscribed square with non-zero area. Having dispensed with the hope that the solutions for sufficiently smooth curves might easily give rise to a general solution, we proceed to develop an understanding of the various smoothness conditions for which there ex- ists an affirmative proof for the conjecture. 1.2 Past Work Due to the extent of the body of work on this conjecture, we will confine the majority of our discussion to the weaker local smoothness conditions, the only known global condition, and conditions that admit proofs whose tech- niques seem plausibly extensible to further cases or perhaps the inscribed square problem in generality. Past Work 3 S1 γ s2 s3 p1 p2 s4 s1 p4 p3 Figure 1.2 An example of a special trapezoid of size e inscribed in curve g. Note that the size of the special trapezoid corresponds to the clockwise angle between points s1 and s4 on S1. Currently, the weakest local smoothness condition, proved by Matschke (Matschke (2011)), is: Theorem 1.2.1. The Jordan curve g : S1 , R2 inscribes a square if there exists ! some 0 < e < 2p such that g contains no special trapezoids of size e. In this context, a special trapezoid is defined as a set of four points s1, s2, s3, s4 1 1 f g 2 S (where the si are labeled clockwise on S ) for which the points pi = g(si) satisfy the following conditions: p p = p p = p p > p p , jj 1 − 2jj k 2 − 3k k 3 − 4k k 4 − 1k p p = p p . k 1 − 3k k 2 − 4k The size of a special trapezoid is the length of the clockwise arc from s1 to s4 in S1. A curve g is said to meet the special trapezoid criterion if there exists some 0 < e < 2p for which g does not inscribe a special trapezoid of size e. The set of curves meeting this criterion is open and dense in f : S1 , R2 f ! g (Matschke (2011)). The second weakest local smoothness condition with a known affirma- tive proof is local-monotonicity (Stromquist (1989)). A curve g : S1 , R2 is ! considered locally monotone if every x S1 has a corresponding neighbor- 2 hood U and non-zero vector h such that g contains no chords parallel jU to h. More formally, g is a locally monotone curve if for all x S1 there 2 exists a neighborhood U such that for any x , x , x , x S1 with x < x 1 2 3 4 2 1 2 4 Historical Development of the Inscribed Square Problem and x3 < x4, the points g(x1), g(x2), g(x3), and g(x4) meet the dot product condition g(x ) g(x ), g(x ) g(x ) > 0. h 2 − 1 4 − 3 i γ(x1) γ(x2) Figure 1.3 An example of a chord between points g(x1) and g(x2). Note that the chord is oriented from g(x1) to g(x2) - because g is clockwise oriented, the orientations of the chord and curve are considered to match. 1 Conventionally, g inherits the clockwise orientation of S . For all x1, x2 1 2 S , the corresponding chord in g is oriented from g(x1) to g(x2). Geomet- rically, then, a curve can be understood to be locally monotone when for every point on g there exists a corresponding neighborhood in which any pair of chords whose orientation match g form an angle strictly less than 90◦. η U(x) γ Figure 1.4 Reproduced from Sagols and Marín (2011), this is an example of a locally monotone curve. The region of the plane spanned by the chords in g U is shaded green. The existence of a non-zero vector h not contained in thisj region means that g is locally monotone at x. Because this is true for any point x S1, the curve g is locally monotone. If g were not locally monotone at x, the shaded2 region would extend to the entire plane. In 1989, Stromquist (Stromquist (1989)) proved: Theorem 1.2.2. If g : S1 , R2 is locally monotone, then g inscribes a square. ! Past Work 5 Although almost all of the affirmative proofs for the inscribed square problem rely on local smoothness conditions, Matschke (Matschke (2011)) offers one global criterion for the existence of inscribed squares: Theorem 1.2.3. Let A = x R2 1 x 1 + p2 . If g : S1 , A is a f 2 j ≤ k k ≤ g ! closed continuous curve in A that’s non-zero in p1(A) = Z, then g inscribes a square with side length at least p2. The curves that meet this global criterion form an open set in the class of continuous functions from S1 to R2 and are not necessarily injective Matschke (2011). Other classes of curves with existing affirmative proofs include piece- wise analytic curves with finitely many inflection points and other sin- gularities where the right and left hand tangents exist at the non-smooth points (Emch (1916)); a class of curves slightly larger than C2 (Schnirelman (1944)); quadrilaterals (Hebbert (1914-1915)); convex curves (Zindler (1921),Chris- tensen (1950)); analytic curves (Jerrard (1961)); curves with symmetry about a point or line (Nielsen and Wright (1995)); piecewise linear curves (Pak (2009)); curves with bounded total curvature and no cusps Cantarella et al. (2014); C1 curves (Cantarella et al. (2014)); and other special cases. Recently, Pettersson, Tverberg, and Óstergard◦ (Pettersson et al.
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