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 additional local smoothness conditions. The weakest of these local smoothness conditions include the special trapezoid criterion, due to Matschke, and local monotonicity, due to Stromquist. We develop several combinatorial 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 strategies 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 γ be approximated as a sequence of sufﬁciently smooth curves γ , each of which inscribes at least one square, Sq ; by the compactness { n} n of the plane, there exists a convergent subsequence Sq whose limit is { nk } a square, Sq, inscribed in the original curve γ. Naively, this argument appears 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 exists 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 γ. 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 γ : S1 , R2 inscribes a square if there exists → some 0 < e < 2π such that γ 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 { } ∈ S (where the si are labeled clockwise on S ) for which the points pi = γ(si) satisfy the following conditions:

p p = p p = p p > p p , || 1 − 2|| 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 γ is said to meet the special trapezoid criterion if there exists some 0 < e < 2π for which γ does not inscribe a special trapezoid of size e. The set of curves meeting this criterion is open and dense in f : S1 , R2 { → } (Matschke (2011)). The second weakest local smoothness condition with a known afﬁrma- tive proof is local-monotonicity (Stromquist (1989)). A curve γ : S1 , R2 is → considered locally monotone if every x S1 has a corresponding neighbor- ∈ hood U and non-zero vector η such that γ contains no chords parallel |U to η. More formally, γ is a locally monotone curve if for all x S1 there ∈ exists a neighborhood U such that for any x , x , x , x S1 with x < x 1 2 3 4 ∈ 1 2 4 Historical Development of the Inscribed Square Problem

and x3 < x4, the points γ(x1), γ(x2), γ(x3), and γ(x4) meet the dot product condition γ(x ) γ(x ), γ(x ) γ(x ) > 0. h 2 − 1 4 − 3 i

γ(x1)

γ(x2)

Figure 1.3 An example of a chord between points γ(x1) and γ(x2). Note that the chord is oriented from γ(x1) to γ(x2) - because γ is clockwise oriented, the orientations of the chord and curve are considered to match.

1 Conventionally, γ inherits the clockwise orientation of S . For all x1, x2 1 ∈ S , the corresponding chord in γ is oriented from γ(x1) to γ(x2). Geomet- rically, then, a curve can be understood to be locally monotone when for every point on γ there exists a corresponding neighborhood in which any pair of chords whose orientation match γ form an angle strictly less than 90◦.

U(x)

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 γ U is shaded green. The existence of a non-zero vector η not contained in this| region means that γ is locally monotone at x. Because this is true for any point x S1, the curve γ is locally monotone. If γ were not locally monotone at x, the shaded∈ region would extend to the entire plane.

In 1989, Stromquist (Stromquist (1989)) proved: Theorem 1.2.2. If γ : S1 , R2 is locally monotone, then γ 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 + √2 . If γ : S1 , A is a { ∈ | ≤ k k ≤ } → closed continuous curve in A that’s non-zero in π1(A) = Z, then γ inscribes a square with side length at least √2.

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 piecewise 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. (2014)) considered a version of the inscribed square problem on an n n grid. For × the class of finite Jordan curves that live on the n n grid, , they offer × J 0 the following conjecture:

Conjecture 1.2.4. Every Jordan curve J inscribes a square whose side ∈ J 0 length is at least i(jJ)/√2

Here, i(J) denotes the side length of the largest square contained in the interior of J. If true, then simple convergence arguments can be used to show that this conjecture implies Toeplitz’ conjecture (Pettersson et al. (2014)). Cur- rently, it has been explicitly computationally veriﬁed for up to the 13 13 × grid. Because the computational complexity of the problem grows quickly as n increases, it would be impractical to use the same methods to verify larger cases. The reduction of the inscribed square problem to a problem on a discrete grid suggests, however, that combinatorial tactics might render the problem tractable. 6 Historical Development of the Inscribed Square Problem

1.3 Related Problems

There has also been signiﬁcant investigation into variants on the inscribed problem. Among others, these variants include: Inscribed Triangles: It’s known that any smooth γ : S1 , R2 inscribes → a triangle (Matschke (2014)). If the condition is relaxed to continuous γ, Nielsen (Nielsen (1992)) proved the following:

Theorem 1.3.1. Suppose T is an arbitrary triangle and γ : S1 , R2 is contin- → uous. Then γ admits inﬁnitely many inscribed triangles similar to T. Fixing the vertex of smallest degree in T renders the set of corresponding vertices on γ dense in γ.

Inscribed Rectangles: One natural relaxation of the inscribed square conjecture is to ask if every Jordan curve admits an inscribed rectangle. If the aspect ratio of the rectangle isn’t fixed, then there exists an affirmative proof due to Vaughan Meyerson (1981). If we require that the rectangle have a fixed aspect ratio, however, then the problem remains open even when we impose strong conditions like smoothness or piecewise linearity on γ. For smooth curves, the open problem can be stated as

Conjecture 1.3.2. All γ : S1 , R2 C∞ admit an inscribed rectangle with → ∈ aspect ratio r > 0.

When r = 1, this conjecture reduces to the inscribed square problem for smooth curves, which we already know has an affirmative proof. There are presently no results for other values of r, however. If we relax the smoothness condition to instead consider piecewise linear γ, then there exist solutions for arbitrary r if γ is close to an ellipse (Makeev (2005)) and for r = √3 if γ is close to convex (Matschke (2011)). Investigations into the number of inscribed squares: There has also been significant investigation into the number of squares inscribed by various classes of curves. For any n 1, Popvassilev (Popvassilev (2008)) and ≥ Sagols-Marín (Sagols and Marín (2011)) have constructed, respectively, smooth convex curves and piecewise linear curves which admit exactly n inscribed squares. Matschke (Matschke (2011)) offered a result for the parity of the number of squares inscribed by generic smooth immersed curves. Van Hei- jst (van Heijst (2014)) proved that any real algebraic curve of degree n in R2 n4 5n2+4n inscribes either infinitely many squares or at most − 4 . Related Problems 7

Higher dimensions: A natural extension of the inscribed square problem is to higher dimensions, where it becomes the question: “do all (or any) (n 1)-spheres smoothly embedded in Rn inscribe an n-cube?" Al- − though the answer is negative for most smooth embeddings Sn 1 , Rn, − → (Matschke (2014)) there exists an afﬁrmative proof for centrally symmetric convex bodies in R3 (Hausel et al. (2002)). Inscribed crosspolytopes (Ma- keev (2003), Karasev (2009)) and inscribed simplices (Gromov (1969)) are addressed by the following theorems:

Theorem 1.3.3 (Makeev, Karasev). If n is an odd prime power, then all smooth embeddings Γ : Sn 1 , Rn admit an inscribed regular n-dimensional crosspoly- − → tope.

Theorem 1.3.4 (Gromov). Every compact set S Rn with a C1 boundary and ⊂ non-zero Euler characteristic admits an inscribed arbitrary simplex, up to similarity on δS.

Beyond these cases, the higher dimensional version of the inscribed square problem remains open.

Chapter 2

Conic Sections

In order to develop some intuition for the inscribed square problem, and for how we might identify and approach curves that don’t meet one of the conditions with an existing afﬁrmative proof, we will begin by exploring particular curves. Because the properties of convex curves will be convenient for our explorations of various parameter space discretizations in Section Three, it’s natural for us to begin our exploration of the inscribed square problem with convex curves. We will speciﬁcally consider conic sections, as having some intuition for how the conjecture interacts with conic sections will be convenient for us later. Generically, conic sections have the form

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0, where A, B, and C are not all simultaneously zero. Conic sections are clas- siﬁed as ellipses if B2 4AC < 0, as parabolae if B2 4AC = 0, and as − − hyperbolae if B2 4AC > 0. − 2.0.1 Ellipses Generically, an ellipse is a plane curve composed of the set of points that satisfy the equation

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0, 10 Conic Sections

where B2 4AC < 0. We deﬁne −

ABD BC A C 2B

∆ = BCE , δ1 = , δ2 = − − , and G = A + C AB 2BA C

DEF −

which we can then use to explicitly describe the location of an inscribed square for a generic ellipse:

Proposition 2.0.5. The generic ellipse

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0

admits an inscribed square with vertices (c, c), (c, c), ( c, c), and ( c, c) − − − − where 1 c = q 1 1 a2 + b2 and s s 2∆ 2∆ a = , b = δ G + √δ δ G √δ − 2 − − 2 Proof. Using appropriate coordinate transformations, any ellipse can be x2 y2 written in the canonical form a2 + b2 = 1. We begin by recalling that the center of a generic ellipse, (xc, yc) , is given by

CD BE xc = − , δ1 AE BD yc = − . δ1 The counter-clockwise angle between the x axis and the major axis of the − ellipse is given by:  0 b = 0 and a < c   π  2 b = 0 and a > c 1 A C θ = cot− ( − ) 2B b = 0 and a < c  2  1 A C 6  cot− ( − )  π + 2B b = 0 and a > c 2 2 6 11

One method of rewriting an ellipse in standard form is to rotate the ellipse so its major axis is aligned with the x-axis. This is done by setting

x0 = cos(θ)x sin(θ)y, − y0 = sin(θ)x + cos(θ)y. The semi-major and semi-minor axes of the transformed ellipse are then s 2∆ a = , δ G + √δ − 2 s 2∆ b = . δ G √δ − − 2 and the ellipse can be rewritten in canonical form as

x 2 y 2 0 + 0 = 1 a2 b2 Using the explicit derivation of Hardin, et al, (Hardin et al. (2009)) for the location of an inscribed square in an ellipse which is symmetric about the origin, we can explicitly write the vertices of an inscribed square as ( c, c) where ± ± 1 c = q . 1 + 1 ± a2 b2 Because c is real and ﬁnite for all possible values of a and b (excluding the degenerate a = b = 0 case, where the ellipse would actually be a single degenerate point), this explicit solution demonstrates the existence of an inscribed square in any ellipse.

Because circles are a special type of ellipse, this proposition has the following immediate corollary: Corollary 2.0.6. Any circle admits inﬁnitely many inscribed squares.

2.0.2 Parabolae Parabolae are generically written as

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 where B2 4AC = 0. Although any parabola can be rewritten in canonical − form, there’s no utility in detailing the explicit procedure, since: 12 Conic Sections

v3 4 y

3 v4

2 v2 1

x 2 1 1 2 v1 − − Figure 2.1 An illustration of the impossibility of a parabola admitting an inscribed square (left) and the ‘up-down-up-down’ pattern required for the vertices of any inscribed square (right).

Proposition 2.0.7. There are no parabolae that admit an inscribed square.

Proof. Let the vertices of a potential inscribed square be labeled v1, v2, v3, and v4, beginning counterclockwise from the lower left corner. Each vertex can be represented in terms of its coordinates as vi = (vi,1, vi,2). To- gether, this labeling convention and the geometry of a square require that v4,2, v3,2 > v1,2 and v3,2 > v2,2. This ‘up-down-up-down’ pattern cannot occur for four points on a single parabola, since it would require that the parabola have at least two points of inﬂection and parabolae are quadratic functions.

2.0.3 Hyperbolae Generically, a hyperbola is a plane curve that can be written as

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0, 13 where B2 4AC > 0. We deﬁne −

A B D B 2 2 A 2 D = and D = B E 1 2 2 C 2 B C 2 D E 2 2 F and subsequently deﬁne λ and λ as the roots of λ2 (A + C)λ + D = 0. 1 2 − 1 These can then be used to deﬁne s s D D a = 2 and b = 2 − λ1D1 − λ2D1 which allow us to state the following proposition:

Proposition 2.0.8. If b > a , then the generic hyperbola | | | | Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 admits an inscribed square with vertices (c, c), ( c, c), (c, c), and ( c, c) − − − − where 1 c = q 1 1 a2 − b2 Proof. With the appropriate coordinate transformations, all hyperbolae can 2 2 be written in the canonical form x y = 1. To observe this, we ﬁrst recall a2 − b2 that the location of the center of the hyperbola, (xc, yc), is given by

EB CD 4 2 xc = − B2 AC 4 − EA DB 2 4 yc = − B2 AC 4 − Letting ζ = x x and η = y y , we then use a change of coordinates to − c − c rewrite the generic equation as

D Aζ2 + Bζη + Cη2 = 2 − D1 14 Conic Sections

The angle θ between the principle axes of the hyperbola and the standard positive x-axis is given by 1 1 B θ = tan− 2 A C − In order to rewrite the hypberola in canonical form, its principle axes must be rotated to match the standard coordinate axes. By setting

x0 = cos(θ)ζ sin(θ)η, − y0 = sin(θ)ζ + cos(θ)η,

the hyperbola can be rewritten in canonical form as

x 2 y 2 0 0 = 1 a2 − b2 Because all hyperbolae can be transformed into canonical form, demon- strating a condition for the existence of an inscribed square in the generic 2 2 hyperbola x y = 1 is sufﬁcient to demonstrate a general condition for a2 − b2 the existence of inscribed squares in hyperbolae. From Figure 2.2, we observe that the drawn inscribed rectangle becomes a square if the condition x = y is met. Using this condition and the canonical equation for a hy- ∗ ∗ perbola, we can solve for x∗ as 1 x∗ = q . ± 1 1 a2 − b2 Because x is only real if 1 > 1 , this gives the condition b > a for ∗ a2 b2 | | | | the existence of an inscribed square in a hyperbola. 15

2 y

x* 1 y* x 2 1 1 2 − −

1 −

2 − Figure 2.2 A square inscribed in a hyperbola, with lengths x∗ and y∗ marked.

Chapter 3

Koch Snowﬂake

Having completed our investigation of conic sections, we will next explore and locate multiple inscribed squares in the Koch snowflake, which we will prove is not locally monotone. It is currently not known whether the Koch snowflake satisfies the special trapezoid criterion; since that criterion is the weakest criterion for which the conjecture is known to be true, this makes the snowflake an intriguing curve. Understanding the Koch snowflake, the applicability of the various affirmative conditions, and the existence of inscribed squares in the snowflake will allow us to develop some intuition for how the Toeplitz conjecture interacts with fractal curves. In particular, explicitly locating inscribed squares may offer some insight into the con- structibility of counterexamples.

Figure 3.1 The ﬁrst three generations of the Koch Snowﬂake fractal. The green lines indicate line segments that were present in the previous generation, but have been removed in the current generation.

The Koch snowﬂake, a well-studied fractal, is recursively constructed from an equilateral triangle by repeatedly applying the following procedure: divide each line segment into three equal parts; on each segment, 18 Koch Snowﬂake

draw an outward-pointing equilateral triangle with the middle third as its base; and then remove that middle third. Each iteration is referred to as a generation, with the original equilateral triangle constituting generation zero. Each generation is composed of line segments which meet at points called nodes. The uniform limit of these generations is the Koch snowflake itself. Crucially, we note that the nodes from every generation are points on the Koch snowflake. The Koch snowflake is the uniform limit of the curves in each generation. Although it is not currently known if the Koch snowflake meets the special trapezoid criterion (Matschke (2011)), we can demonstrate that it does not meet Stromquist’s criterion:

Theorem 3.0.9. The Koch snowﬂake is not locally monotone.

Proof. In this context, we will find it most intuitive to employ the geometric notion of local monotonicity. Recall that a curve must satisfy the conditions for local monotonicity at every point in order to be locally monotone. As such, showing that a single point fails to meet the criterion is sufficient to show that the entire curve is not locally monotone. We claim that not only does such a point exist for the Koch snowflake, but in fact the snowflake is not locally monotone at any point. This is because any neighborhood B(p, e) around a point p on the snowflake must contain the planted star structure shown in Figure 3.2.

Figure 3.2 An example of the planted star construction in an arbitrary generation of the Koch snowflake (left) and a depiction of the possible orientations of the chords in a neighborhood containing a planted star (right). Because the chords span the entire plane, proving that an arbitrary neighborhood around a particular point on the Koch snowflake must contain a planted star suffices to show that the snowflake is not locally monotone.

Suppose that the point p is introduced in generation n of the Koch snowflake. Then any neighborhood B(p, e) around p on the generation 19 n precursor to the snowflake must contain a line segment that will be sub- divided in subsequent generations. This is shown in Figure 3.4 for a point on the second generation snowflake. By the method of its construction, we know that line segments introduced in generation n of the snowflake have 1 length 3n and that it only takes two steps for a line segment to produce a planted star. A planted star that first appears in generation n therefore has 1 √3 width 3n and height 2 3n , as shown in Figure 3.3. ·

1 3n

√3 2 3n ·

Figure 3.3 The dimensions of a planted star construction that ﬁrst appears in generation n of the Koch snowﬂake. This macrostructure will persist in future generations

1 ∞ √3 ∞ Because both 3n n=1 and 2 3n n=1 converge to 0, for any e > 0 there { } { · } 1 √3 must exist some N N such that e > n and n for all n > N. As such, ∈ 3 2 3 any ball of radius e around point p will contain· the planted star structure after sufficient generations. From Figure 3.2, we observe that the planted star contains a pair of orthogonal chords. From the development of the geometric intuition for local monotonicity developed in Chapter 1, we recall that the existence of a pair of orthogonal chords in any arbitrary neighborhood of p means that the snowflake is not locally monotone at p. Although the planted star argument is directly applicable only to points appearing in one of the generations of the snowflake, it can be easily extended to cover points that appear only in the limit. Let q be a point on the snowflake that appears only in the limit and consider an arbitrary neighborhood B(q, δ) around q. Since line segments in generation n of the Koch 1 1 ∞ snowflake are of length n , and n converges to 0, we know that the 3 { 3 }n=1 neighborhood contains nodes of the Koch snowflake. Let p be some node in B(q, δ). Let e = δ d(p, q). Then we observe that B(p, e) B(q, δ) and − ⊂ therefore, by our previous argument, B(p, e) must contain a planted star 20 Koch Snowflake

structure. We conclude that there are no points at which the Koch snowﬂake is locally monotone.

Figure 3.4 An example of an arbitrary neighborhood around a point on the second generation Koch snowﬂake. After sufﬁciently many subsequent generations, this neighborhood will contain the planted star macrostructure.

Corollary 3.0.10. Any fractal that contains a structure with orthogonal chords that, due to the self-similarity of the fractal, appears in arbitrarily small neighbor- hoods is not locally monotone.

Proof. If such a structure exists, then the corollary can be proven using an argument analogous to the one used for Theorem 3.0.9.

3.1 Inscribed Squares in the Koch Snowﬂake

Inscribed squares can be experimentally located in early generations of the snowflake, but locating inscribed squares in further generations requires a more systematic approach. In order to develop such an approach, we can begin by establishing a sensible coordinate system to describe points on the snowflake. Inscribed Squares in the Koch Snowflake 21

Figure 3.5 Explicitly located inscribed squares for the ﬁrst three generations of the Koch snowﬂake. Note that none of these inscribed squares exist in more than one generation.

Since the structure of the Koch snowﬂake is based on equilateral triangles, it’s natural to employ a barycentric coordinate system. In barycentric coordinates, the main vertices of a triangle are assigned the labels (1, 0, 0), (0, 1, 0), and (0, 0, 1), as shown in Figure 3.6(a). Each point in the interior or on the edges of the triangle then has a unique barycentric coordinate representation (x1, x2, x3) such that x1 + x2 + x3 = 1. In this representation, the magnitude of xi indicates the proximity of the point to the i-th corner of the triangle. Because subsequent generations of the Koch snowﬂake are formed by replacing the interior third of line segments, we will frequently want to discuss the vertex points introduced by dividing a particular line segment 1 2 into thirds, which often have decimal barycentric coordinates like ( 3 , 3 , 0) 1 2 or (0, 3 , 3 ). This makes it convenient to use ternary representations for our coordinates. Ternary numbers are base-3 representations using only the digits 0, 1, and 2. Through this document, we will denote decimal numbers as usual and denote ternary numbers using a subscript. For example, the decimal number 25 can be written in ternary as 25 = 2 32 + 2 31 + 1 30 = 221 . 10 · · · 3 It’s also possible to represent fractions using negative powers of three, although they may have non-terminating ternary representations. For exam- 5 ple, the fraction 8 can be written in ternary as

5 1 2 3 4 = 3− + 2 3− + 3− + 2 3− + = (0.12) 8 · · ··· 3 Powers of three, however, are easily and compactly expressible in ternary. Because the method of construction of the Koch snowﬂake means that we’re 22 Koch Snowﬂake

often interested in points formed via trisection, this poses a tremendous advantage for us and is a compelling reason to adopt the use of ternary representations.

1 2 1 2 Decimal 3 3 9 9 0.7 0.8 0.4 Ternary 0.1 0.2 0.01 0.02 .21 0.22 0.11

Table 3.1 Common ternary representations needed to describe the coordinates of points on the Koch snowﬂake.

We will adopt the convention of assigning labels to the main vertices of the generation zero Koch snowflake by beginning with the lower left corner and proceeding counterclockwise, as shown in Figure 3.6(a). Other points on the curve and the vertices of future generations can then be assigned coordinates based on their distances from those main vertices. For example, consider the edge between (1, 0, 0) and (0, 1, 0). When that edge is trisected and the middle segment replaced with an equilateral triangle, as shown in Figure 3.6(b), two new vertices are introduced on the original line segment. If we consider walking along the original edge from (1, 0, 0) to (0, 1, 0), the first of the new vertices marks the point where we’ve traveled one-third of the distance and the second vertex marks the point where we’ve traveled two-thirds of the distance. As such, the vertices are respectively assigned 2 the coordinates (0.2, 0.1, 0) and (0.1, 0.2, 0) (recall that 0.23 = 3 and 0.1 = 1 3 ). Visual aids like Figure 3.7 can also be used to determine the coordinates of a particular vertex. The coordinates of all snowflake vertices obey the following rules:

1. All vertices have ternary barycentric coordinate representations (a1, a2, a3) such that ∑i ai = 1. 2. Vertices introduced in the ith generation of the snowflake have coordinates with exactly i decimal places. In fact, the first rule extends to all points on the snowflake. Non-vertex points do not obey the second rule. Non-vertex points introduced in the ith generation have one coordinate with a zero in every decimal place after the i-th decimal place, that remains constant along a given line segment. This coordinate is referred to as the fixed coordinate of the point. The remaining coordinates are fixed up to the (i 1)-th decimal place, but can assume any − value in the ith decimal place and beyond. An example of this behavior is shown in Figure 3.8, where some of the points introduced in the first Inscribed Squares in the Koch Snowflake 23

(0,0,1)

(0,0,1) (.0 ,0,.2 ) (0,.0 ,.2 )

(.1 ,-.0 ,.2) (-.0 ,.1 ,.2) (.2,-.1,.2) (-.1,.2,.2) (.1,0,.2) (0,.1,.2)

(.2,-.0 ,.1 ) (-.0 ,.2,.1 )

( ,0, ) (0, , ) (.2,0,.1) (0,.2,.1)

(.2 ,0,.0 ) (0,.2 ,.0 )

(.2,.1,0) (.1,.2,0) (1,0,0) (0,1,0) (.2 ,.0 ,0) (.0 ,.2 ,0)

(.2,.1 ,-.0 ) (.1 ,.2,-.0 ) (1,0,0) (0,1,0)

( , ,0) (.2,.2,-.1)

a. Generation zero b. Generation one

Figure 3.6 Vertex and side coordinate labels for generations zero and one of the Koch snowﬂake. Vertex labels are in blue and side labels are in black.

Figure 3.7 A visualization of lines of constant coordinates for the first (left), second (middle), and third (right) coordinates for the second generation of the Koch snowflake. Such visualizations are useful in determining the ternary barycentric coordinates of particular vertices or points on the snowflake. Because this is the second generation snowflake, each line represents an increment of 0.013. So, for example, a point on the furthest left green line would have the first coordinate 0.223, a point on the next line would have coordinate 0.0213, and so forth. generation snowflake are shown. To illustrate this rule more concretely, let us consider the set of points introduced in the first generation snowflake. We know that the vertices of the generation zero Koch snowflake are 24 Koch Snowflake

(1, 0, 0), (0, 1, 0), (0, 0, 1) and that the points along its edges can be de- { } scribed as (0._, 0, 0._), (0._, 0._, 0), (0, 0._, 0._) where the “_" symbol indi- { } cates that any sequence of ternary digits can follow as long as the three coordinates sum to one, as shown in Figure 3.6. We can note that the zero generation vertices all have exactly zero decimal places and that the non- vertex points in generation zero also have no fixed values after the decimal place, in accordance with our rule. In the first generation snowflake, the following vertices are introduced: (.2, .1, .2) , ( .1, .2, .2) , (.1, 0, .2) , (0, .1, .2) , (2, 0, 0.1) { − − (0, .2, .1) , ( 2., .1, 0) , (.1, .2, 0) , (.2, .2, .1) − − } We can observe that each of these vertices has coordinates with exactly one decimal place. Further, the non-vertex points introduced in generation one can be described as (.0_, 0, .2_) , (0, .0_, .2_) , ( .0_, .1_, .2) , ( .0_, .2, .1_) , (0, .2_, .0_) { − − (0, .2_, .0_) , (.0_, .2_, 0) , (.1_, .2, .0_) , (.2, .1_, .0_) , (.2_, .0_, 0) − − (.2_, 0, .0_) , (.2, .0_, .1_) , (.1_, .0_, .2) − − } From this explicit list, we observe that each set of non-vertex points has one coordinate that may assume any value in its first decimal place, but must have a zero in all subsequent decimal places, and two coordinates where one decimal place is fixed and the remaining decimal places may assume any value. Note that the fixed coordinate may be equal to zero - because our requirement is simply that the fixed coordinate have zeros in every decimal place beyond the i-th decimal place, zero is an acceptable value. Now that we have a basic characterization of the points on the Koch snowflake, and some understanding of how we might begin building the i-th generation of the Koch snowflake from the (i 1)-th generation, a nat- − ural next question is: can the coordinates of the set of vertices introduced in the i-th generation of the snowflake be systematically computed from the set of vertices present in the (i 1)-th generation? If so, can that description − be extended to the non-vertex points? While we have made some progress in answering this question, we do not yet have a complete method to describe the vertices introduced in the i-th generation. The patterns that we have found apply only for i 2 - be- ≥ cause the vertices in the zeroth and first generations can be easily explicitly described, however, this is not particularly problematic. For vertices introduced in the ith generation (where i 2), we make the ≥ following observations: Inscribed Squares in the Koch Snowflake 25

(0,0,1) (0,0,1)

(-.0 ,.1 ,.2) (0,.1,.2) (-.1,.2,.2)

(-.0 ,.2,.1 )

(0,.2,.1)

(1,0,0) (0,1,0) (1,0,0) (0,1,0)

Figure 3.8 An example of some of the points introduced in the first generation koch snowflake. Note that the vertices from generation zero have no decimal places, the vertices introduced in generation one have exactly one decimal place, and the non-vertex points introduced in generation one have fixed values only the first decimal place but may have any value in subsequent decimal places.

1. Some of the vertices introduced in the i-th generation are produced by modifying line segments introduced in the (i 1)-th generation. − This set of vertices can always be partitioned into pairs such that the vertices in a given pair have coordinates that follow one of three patterns:

(a1, a2, a3) and (a1, a3, a2)

(a1, a2, a3) and (a3, a2, a1)

(a1, a2, a3) and (a2, a1, a3) That is, it can be divided into pairs with one fixed coordinate and a pair of alternate coordinates. For example, Figure 3.9 shows vertices in the 2nd generation that are introduced by modifying a line segment in the 1st generation. This modification produces six new vertices, which can be paired as: ( .02, .11, .21) and ( .02, .21, .11), − − ( .01, .2, .11) and ( .01, .11, .2), − − ( .02, .12, .2) and ( .02, .2, .12). − − Here, the first pair of vertices has a fixed first coordinate (-.02) and has the pair of values .11 and .21 that act as either the second or third coordinate. 26 Koch Snowflake

(-.02,.11,.21)

(0,.1,.2) (-.01,.11,.2) (-.02,.12,.2) (-.1,.2,.2)

(-.02,.2,.12)

(-.01,.2,.11) (-.02,.21,.11)

(0,.2,.1)

Figure 3.9 An example of two sets of vertex points introduced in the second generation Koch snowflake via a "positive push" procedure. Here, the first generation vertex ( .1, .2, .2) gives rise to the second generation vertex pair ( .02, .11, .21), ( −.02, .21, .11) , marked in blue. Because .1 < .2 , the{ − first coordinate was− incremented} by 0.01 and then the other two| − coordinates| | | were either incremented by 0.01 or decremented by 0.02, as per our rule.

2. The same set of vertices can be divided into pairs produced by a positive push and pairs produced by a negative push. Each pair of these vertices can be uniquely associated with the nearest vertex introduced in the (i 1)-th generation, (a , a , a ). Arbitrarily, let a < a , a . − 1 2 3 | 1| | 2| | 3| Then vertex pairs produced by a positive push follow the pattern i i i i i i (a + 3− , a + 3− , a 2 3− ) and(a + 3− , a 2 3− , a + 3− ), 1 2 3 − · 1 3 − · 2 whereas vertex pairs produced by a negative push follow the pattern i i i i i i (a 3− , a 3− , a + 2 3− ) and (a 3− , a + 2 3− , a 3− ). 1 − 2 − 3 · 1 − 3 · 2 − An example of a pair of vertices produced via a positive push is shown in Figure 3.9. 3. Vertices introduced in the i-th generation by subdividing line segments that appeared previous to the (i 1)-th generation can also − be partitioned into pairs that share a single constant coordinate. Al- though there are some apparent patterns in the coordinates for these vertices, we do not currently have a global rule for how such coordinate pairs can be computed from the coordinates of vertices in previous generations. Inscribed Squares in the Koch Snowﬂake 27

Figure 3.10 Ternary barycentric coordinates for the vertices of the fourth generation Koch snowflake. Vertex coordinates are shown in black for vertices introduced in the zeroth, first, second, and third generations. The coordinates of vertices introduced in the fourth generation are shown in green, black, and red depending on their geometric location on the snowflake.

Although we have not yet obtained a complete characterization of the points on the Koch snowﬂake, we are optimistic that these observations, when completed, will yield such a description. 28 Koch Snowﬂake

3.1.1 Centrally Symmetric Inscribed Squares In order to justify the utility of devoting time to developing a characterization of points (or at least vertices) on the Koch snowflake, we will briefly explain how such a characterization might be used in an affirmative proof of the existence of an inscribed square. For early generations of the Koch snowflake, it’s possible to empirically locate a set of four line segments whose vertices inscribe a pair of rectangles with complementary longest sides, as shown in Figure 3.11. Any rectangle whose vertices lie on those line segments may be labeled either "T" (if it’s taller than it is wide) or "W" (if it’s wider than it is tall). The fact that the two extremal inscribed rectangles (i.e., the two rectangles with the largest and smallest possible width) have opposite labels allows a simple continuity argument to be made for the existence of an inscribed square with vertices somewhere in the interior of the line segments. This allows us to bound the location of the inscribed square in a particular generation.

Figure 3.11 For both the second (left) and third (right) generations of the Koch snowﬂake, this diagram shows an inscribed rectangle with greater width than height (in purple) and with greater height than width (in green). Because these rectangles occur on the vertices of line segments (highlighted in orange), the continuity of those line segments implies the existence of an inscribed square with vertices located somewhere along those segments.

Having a characterization of the vertices in the i-th generation of the Koch snowflake would allow us to bound the location of the inscribed square in that generation. Let us denote the i-th generation of the swnoflake as Si and the length of its line segments as si. Because we know that lim si 0, being able to bound the location of the vertices of a square i ∞ → → inscribed in Si would allow us to explicitly locate the square inscribed in Inscribed Squares in the Koch Snowflake 29 the limit of S , assuming the limit is not degenerate (e.g., a point). { i} 3.1.2 Other Inscribed Squares This centrally symmetric square is not the only square inscribed by the Koch snowflake, however. Consider the line segments connecting the vertices (1, 0, 0) and (0, 1, 0) and ( .2, .1, .2) and ( .1, .2, .2), which exist in − − − the second generation of the snowflake. In the next generation, the middle third of the points are removed. In each subsequent generation, the open middle third of each remaining line segment is removed. In the limit, the set of remaining points is homeomorphic to the Cantor set in the unit interval, as illustrated in Figure 3.12. Perhaps unsurprisingly, these parallel Cantor sets contain the vertices of at least one inscribed square. In order to justify this claim, we will begin by exploring the relevant properties of the Cantor set.

Figure 3.12 An illustration of how the Cantor set is contained in the Koch snowﬂake. Because subsequent generations of the Koch snowﬂake are formed by replacing the interior third of each line segment, in the limit the only points remaining on this line will be a set of points homeomorphic to the Cantor set in the unit interval.

We begin by recalling the construction of the Cantor set, which is it- eratively created via a process similar to the one used to construct the Koch snowflake. Beginning with the unit interval [0, 1], the Cantor set is formed by repeatedly removing the open middle third of each line seg- 1 2 ment. In the first step, the interval 3 , 3 is removed from [0, 1], leaving 0, 1 2 , 1. In the next step, the intervals 1 , 2 and 7 , 8 are removed, 3 ∪ 3 9 9 9 9 leaving 0, 1 2 , 1 2 , 7 8 , 1. If C denotes the set of points re- 9 ∪ 9 3 ∪ 3 9 ∪ 9 n maining after this process has been repeated n times, the set Cn can be 30 Koch Snowflake

written recursively as C0 = [0, 1] and Cn 1 2 Cn 1 C = − + − . n 3 ∪ 3 3 T∞ The Cantor set itself, C = n=0 Cn, can be explicitly written as

n 1 ∞ 3 − 1 [ [− 3k + 1 3k + 2 C = [0, 1] , \ 3n 3n n=1 k=0 The Cantor set is a well-studied mathematical object and there exists a signiﬁcant body of work on distances in the Cantor set. We will ﬁnd the following theorem, which describes the set of possible distances between points in the Cantor set, useful in asserting the existence of an inscribed square whose vertices lie in parallel Cantor sets. Although there are several existing proofs of this theorem (Randolph (1940) Steinhaus (1917)) Utz (1951), we will present a particularly elegant geometric proof due to Utz (Utz (1951)). Theorem 3.1.1 (Steinhaus (1917), Randolph (1940), Utz (1951)). Given any c [0, 1], there exist numbers x and y, which represent points of the Cantor set, ∈ such that y x = c | − |

Proof. Consider the unit square S = [0, 1] [0, 1] on the typical Cartesian × plane. We will use M to denote the set of points (x, y) [0, 1] [0, 1] such ∈ × that x and y are points in the respective Cantor sets on the x and y axes. Proving the theorem is then equivalent to showing that if c [0, 1], then ∈ the line y = x + c intersects M at least once. The square S can be divided into three equal vertical strips by dividing the corresponding interval on the x-axis into three equal parts. We can then remove the strip corresponding to the open middle interval, leaving two closed rectangular strips. The same procedure can be repeated on the y-axis, removing the middle horizontal strip. This produces four closed S squares, each of which has area equal to 9 . As illustrated in Figure 3.13, any line y = x + c, with c [0, 1], that ∈ intersects S must intersect one of the remaining four squares. Note that such a line can only be tangent to a single of the four squares if the tangency occurs at the upper left or lower right corner of S, in which case the point of tangency is itself a point in M. Because y = mx + c translates continuously Inscribed Squares in the Koch Snowﬂake 31

(0,1)

(0,0) (1,0)

Figure 3.13 An example of the described procedure for reducing S to four S squares with area 9 that illustrates the impossibility of a line y = x + c, with c [0, 1], having a single point of tangency anywhere except the upper left or lower∈ right corners of S. as c varies, this means that any line intersecting S must intersect at least one of the four smaller squares. That is, it’s not possible for such a line to only pass through the strips that were removed from S. Note that these four squares contain the points in C C , which matches our intuition since 1 × 1 the squares were produced by a division procedure analogous to the ﬁrst step used to produce the Cantor set. We can denote the set of points in the intersection of A and y = mx + b as I1. Let A denote one of the four squares that has a non-empty intersection with y = x + c. The same division procedure can be repeated on A to produce four smaller squares which consist of the points in C C . By 2 × 2 the same argument, the line must share a point with at least one of these squares. Again, we denote the points in the intersection as I2. This argument can be applied inductively to create the monotonically decreasing sequence of sets I ∞ . Since each I is a closed subset of [0, 1] [0, 1], we { n}n=1 n × know that the In are compact sets. Since it’s known that the intersection T∞ of a nested collection of compact sets is non-empty, we know that n=0 In is non-empty and therefore y = mx + b and M must have a non-empty intersection.

The coordinates (x, y) of the points in that intersection give pairs of 32 Koch Snowﬂake

points in the Cantor set that are distance c apart. Note that there may be multiple points in the intersection, so there may be multiple pairs of points in the Cantor set that are a given distance apart. With this fact established, we are now ready to prove the following theorem: Theorem 3.1.2. The Koch snowﬂake admits an inscribed square whose vertices lie in the parallel Cantor sets shown in Figure 3.14

a b (.2,-.1,.2) (-.1,.2,.2)

√.2

(1,0,0) (0,1,0) c d

Figure 3.14 An example of an inscribed square whose vertices lie in the parallel Cantor sets. Note that this square is not centrally symmetric; in fact, there are no centrally symmetric inscribed squares whose vertices lie in these Cantor sets.

Proof. Using the previously developed barycentric ternary coordinate system for the snowﬂake and scaling by the length of the Cantor sets, the vertical distance between the parallel Cantor sets can be explicitly computed in ternary as: 1 q d = (.2 1)2 + ( .1 0)2 + (.2 0)2 √2 − − − − 1 = √.01 + .01 + .11 √2 √.2 = √2 Inscribed Squares in the Koch Snowﬂake 33 which is equivalent to √1/3 in the more familiar decimal system. Because d [0, 1], the result of Theorem 3.1.1 guarantees the existence of points in ∈ the Cantor set which are distance d apart. In fact, there exists an explicit algorithm for computing x and y Randolph (1940). Because √.2/√2 is a non-terminating, non-repeating ternary number, however, this algorithm is similarly non-terminating and can only be used to compute approximate locations. It should also be noted that x and y are not necessarily unique, so there may be multiple inscribed squares whose vertices lie in the Cantor set. If we assume that x and y are the exact locations of points in the Can- tor set, on the interval [0, 1], that are distance d apart, however, we can use basic trigonometry to explicitly locate the vertices of the corresponding inscribed square. Because the ternary representations of these vertices are non-repeating and non-terminating, it’s perhaps more useful to write their barycentric coordinates using the decimal representation:

2 2 2 a = √2x, √2(1 x), 3 − 3 − − 3 2 2 2 b = √2y, √2(1 y), 3 − 3 − − 3 c = 1 √2x, 1 √2(1 x), 0 − − − d = 1 √2y, 1 √2(1 y), 0 − − −

Note that because x and y are not necessarily unique, these coordinates may represent a family of inscribed squares rather than a single square.

Chapter 4

Discretizing the Parameter Space

Thus far, we have devoted our attention to particular cases of the conjecture. In this chapter, we will look more broadly at discrete approaches to the general conjecture. In our exploration of the existing body of work on the particular cases of inscribed square problem, we noted that the existing affirmative proofs rely on parity arguments. Because combinatorial proofs so often rely on parity arguments, this observation perhaps suggests that the judicious application of combinatorial methods might render the conjecture tractable, or at least permit its resolution for additional classes of curves. In order to apply combinatorial arguments to continuous curves, however, we will need to first determine some method of discretizing the problem. Rather than considering methods of discretizing a generic Jordan curve, we will explore various discretizations of the solution space of possible inscribed squares. Although there has been some investigation into discretizations of Jordan curves (Sagols and Marín (2009), Pettersson et al. (2014)), the results are still currently confined to curves in the digital plane. The literature does not currently include any investigation into discretizations of the solution space. In this chapter, we will explore several potential discretizations and the associated parameter spaces. For each construction, we will explore various labeling schemes and properties of the appropriate parameter space, offering commentary on the potential utility of each discretization scheme. 36 Discretizing the Parameter Space

0 v1

v2 v4

Figure 4.1 An example of one of the squares determined by a possible set of values for v1 and d. Note that v2 > v1 and that the labels v1, v2, v3 and v4 are assigned counter-clockwise, proceeding from the pivot vertex, v1.

4.1 Pivot vertex and side length

Let γ : S1 R2 be a convex Jordan curve in the plane. A square with at → least two vertices on γ can be parametrized by the length of its sides and the location of one of its vertices, referred to here as the pivot vertex. Label the vertices of the square as v1, v2, v3, and v4, where v1 is the pivot vertex and, without loss of generality, v2 is constrained to lie on γ. Let d denote the side length of the square. Because γ can be understood as a mapping from the unit circle into the plane, the location of v1 can be naturally represented by values in the interval [0, 2π). By convention, we orient γ counter-clockwise so v2 > v1. The assignment of the vertex labels v3 and v4 follows the same convention, proceeding clockwise from v2. Let X be the parameter space X = (v , d) : v [0, 2π), d [0, ∞) . { 1 1 ∈ ∈ } A square becomes an inscribed square when v3 and v4 both lie on γ. Each pair (v1, d) in X can be assigned a label based on whether v3 and v4 lie inside or outside of γ. We use the label "I" to denote points inside the curve and the label "O" to denote points outside the curve. Each pair (v1, d) in X is assigned two labels, based on the locations of v3 and v4. Pairs for which both v3 and v4 lie inside the curve are labeled II, pairs for which both lie outside the curve are labeled OO, and mixed pairs are labeled either IO or OI. Points that fall exactly on γ are associated with the label "G". Pivot vertex and side length 37

Within X, nearby pairs with different labels in a particular coordinate correspond to a square where either v3 or v4 lies on γ. For example, locating nearby points labeled IO and OO would suggest that there must be a nearby pair labeled GO, where v3 lies on γ. As such, ﬁnding two nearby pairs whose labels switch in both coordinates suggests the existence of a nearby pair labeled GG, with both v3 and v4 on γ. Since such a pair corresponds to the desired inscribed square, locating that pair would allow us to both demonstrate the existence of and locate a square inscribed in γ.

v3 v 4 v3 v4 v4 v1 v2 v3 v 2 v v 1 v2 1 II IO OO

Figure 4.2 Examples of II, IO, and OO squares.

v1 0 2π

Figure 4.3 The parameter space for the (v1, d) parameterization can be visu- alized as an annulus. The shaded edge of this representation of the parameter space corresponds to degenerate squares with zero area, since d = 0, which are excluded from the set of possible inscribed squares. 38 Discretizing the Parameter Space

4.2 Two pivot vertices

Let γ : [0, 1] R2 be a curve in R2. Any square with at least two vertices → on γ can be alternately parametrized by the location of two adjacent vertices. The vertices of the square are assigned labels v1, v2, v3, and v4 and we retain the convention of assigning vertex labels in the counter-clockwise direction. Arbitrarily, we can choose v1 and v2 to serve as our “pivot vertices". If t [0, 1], then γ(t) is a point on the curve γ. Choose t , t , t , t so ∈ 1 2 3 4 that γ(ti) = √i. Here, let X be the parameter space

X = (v , v ) : t , t [0, 1] . { 1 2 1 2 ∈ }

As in the previous parameterization, each (t1, t2) pair is assigned a label from the set II, OO, IO, OI where the ﬁrst and second labels are deter- { } mined by the locations of v3 and v4, respectively. So the label II corresponds to a square with both v3 and v4 inside γ, the label IO corresponds to a square with v3 inside γ and v4 outside γ, etc. Based on this parameterization, we can create an alternate visualization of the parameter space.

2π

0 t1 2π

Figure 4.4 The parameter space for the (v1, v2) parameterization. Note that this parameter space wraps both horizontally and vertically. Here, the shaded diagonal lines indicates the positions where v1 = v2 and so the corresponding square is simply a degenerate point.

Now that we understand the basic construction of the (t1, t2) parameter space, it’s appropriate to begin exploring its properties. For a ﬁxed mesh size, we can envision the parameter space as a triangulated discrete grid where each vertex of the triangulation represents a (v1, v2) pair. Each vertex Two pivot vertices 39

v3 v2 v1 v4

v4 v1 v2 v3

Figure 4.5 An illustration of the "II" and "OO" labeling resulting from squares with sufﬁciently small side length. Note that this behavior relies on the convexity of γ. Interchanging the locations of v1 and v2 is equivalent to reﬂecting the square over the line γ(t1)γ(t2).

is assigned a label based on the locations of v3, v4, as previously explained. In this labeling scheme, an edge whose vertices are labeled "I*" and "O*" (where the * could be either I or O) corresponds to a place where v3 passes from inside γ to outside γ - it must, therefore, lie on γ at some intermediate point. An edge with vertices labeled "*I" and "*O" has the same behavior for v4. In order to locate an inscribed square - for which v1, v2, v3 and v4 must all lie on γ - we are therefore looking for an edge that represents a transition for both vertex labels. Because the parameter space of an arbitrary Jordan curve is difficult to parametrize, we will begin by confining our discussion to the class of convex curves. Although there is an existing affirmative topological proof of the conjecture for convex curves, a combinatorial solution would both be constructive and potentially extensible to further cases. If γ is a smooth, convex curve, then the parameter space obeys the following rules:

1. Vertices sufﬁciently close to the diagonal of degenerate squares (i.e., (v1, v2) pairs in which v1 and v2 are sufﬁciently close together) assume "II" labels above the diagonal and "OO" labels below the diagonal.

2. Reflecting a given point about the degenerate diagonal flips its labels. So, for example, the reflection of a vertex labeled "IO" would be labeled "OI". 40 Discretizing the Parameter Space

I I I I

Figure 4.6 The first (left) and second (right) cases. The curve, γ, is shown in blue. The labeled points represent one vertex of the square before and after being reflected about v1v2. The orange ball represents the neighborhood Ne(g) for some g γ that serves as v1 for the original square and v2 for its reflection. The shaded∈ and unshaded portions of the ball represent points outside and inside of the curve, respectively.

Although the justification for the first rule is self-evident, the second rule is non-obvious and requires additional explanation. We can first note that in this parameterization, reflecting a point about the degenerate diagonal is equivalent to swapping the locations of v1 and v2. Because of the counter-clockwise convention for labeling the square vertices, this has the effect of reflecting the square over the line γ(t1)γ(t2). We can then state and prove this labeling rule as: Proposition 4.2.1. Reflecting a point about the degenerate diagonal flips its labels. That is, if the point (v1, v2) is labeled "AB" then the corresponding point (v2, v1) will be labeled "BA." Proof. Since the conjecture requires that γ be a Jordan curve, we know from the Jordan curve theorem that γ divides the plane into an interior and ex- terior region. Let g be the point on γ that represents the location of v1 for some square, Sq, with vertices γ(t ), γ(t ), v , v of which only γ(t ) and { 1 2 3 4} 1 γ(t2) are on γ. The reflection of Sq about γ(t1)γ(t2) is denoted Sq0 and has vertices γ(t ), γ(t ), v , v where γ(t ) = γ(t ) = g and γ(t ) = γ(t ). { 10 20 30 40 } 20 1 10 2 By way of contradiction, let us assume that both v4 and v40 are inside γ and are therefore labeled ‘I’. Consider a neighborhood Ne(g), with e > 0, suf-

ﬁciently small such that γ appears locally linear. Let η = v4v40 (shown in green in Figure 4.6). Within Ne(g), γ could either be transverse, as in Fig- ure 4.6(a), or parallel, as in Figure 4.6, to η. If γ is transverse to η, then the Jordan curve theorem requires that points on one side of γ be outside the

curve. In order for points between g and v40 to be outside γ, however, γ Two pivot vertices 41 would have to be concave at some point after g, contradicting the assump- tion that γ is convex. Next, we must consider the case where γ is parallel to η. If γ remained parallel to η for the entirety of the line segment η, then both v4 and v40 would lie on the line. Because we are speciﬁcally considering a square with exactly two vertices on γ, we know therefore that there must exist some point, g0, where γ curves away from and ceases to be parallel to η. Considering Ne(g0) allows us to conclude, by the same argument as in the transverse case, that v4 and v40 cannot both be inside γ and must, therefore, have opposite labels. The same argument can be made about the labels of v3 and v30 . If γ is further constrained to actually be a circle, then the parameter space obeys the additional rule:

3. Vertices along a given diagonal all assume the same label.

In this parameter space, moving along a diagonal is equivalent to rotating 1 the locations of v1 and v2 along S while ﬁxing the side length between them. The fact that a circle is centrally symmetric requires that the label associated with a given (v1, v2) pair be the same if the side length remains ﬁxed. If γ isn’t a circle, but instead some other convex curve, then its parameter space obeys the following, somewhat weaker labeling rules:

1. Let d denote the length of the longest chord of γ. If γ(v ) max | 1 − γ(v ) > d , then (v , v ) must be labeled ‘OO’. 2 | max 1 2 2. Let d denote the length of the shortest cord of γ. If γ(v ) γ(v ) < min | 1 − 2 | dmin, then the corresponding diagonal in the parameter space has both ‘OO’ and ‘II’ labels.

3. All points below the degenerate diagonal, where γ(v1) = γ(v2), are labeled ‘OO’.

Although these rules are not sufficient for making an argument for the existence of an inscribed square in an arbitrary convex curve, they are sufficient for making such an argument when γ is a circle. Although circles are addressed by several of the existing affirmative proofs, we offer a novel proof:

Theorem 4.2.2. Any circle γ inscribes a square with non-zero area. 42 Discretizing the Parameter Space

OO II OO II OO GO II OO GO II OO GO IO II OO IO II OO IO II OO II OO II OO II OO II OO OI II OO OI II OO OI II OO GO II OO GO II OO GO II OO II OO II OO II

Figure 4.7 An example of the (v1, v2) parameter space for circular γ. Here, we observe that a diagonal of ‘GO’ labels reﬂects over the degenerate diagonal into a diagonal of ‘GI’ labels. The label ﬂipping rule, however, would require that the ‘GO’ label become an ‘OG’ label rather than a ‘GI’ label. This contradiction indicates the presence of an inscribed square.

Proof. An illustration of the parameter space for circular γ is shown in Fig- ure 4.7. The bands of "II" and "OO" labels along the degenerate diagonal require the existence of bands of "OO" and "II" labels near the top left and bottom right corners. As per the labeling rules for circular γ, the labels along a given diagonal must be uniform. In order to transition from the band "II" labels to the band "OO" labels, there must therefore be intermediate diagonals with "IO" and "OI" labels. In order for a vertex label to change from "I" to "O", however, there must be an intermediate location where that vertex lies exactly on γ and is assigned the labeled "G". This gives rise to diagonals with "GO" and "IG" labels. The reflection rule requires that a diagonal of "GO" labels above the degenerate diagonal reflect into a diagonal of "OG" labels. From Figure 4.7, however, we observe that it actually reflects into a diagonal labeled "GI". This contradiction indicates the presence of an inscribed square, since it forces the "I" and "O" labels to actually be "G" labels and a "GG" point in the parameter space corresponds to a square with all of its vertices on γ. Pivot vertex and angle 43

Figure 4.8 An example of a rectangle constructed as explained above from the pair (v1, φ). Because v1v2 > v4v1 and v3 is inside γ, the pair (v1, φ) would be assigned the label ‘RI’.| | | |

4.3 Pivot vertex and angle

In this parameterization, we switch our focus to rectangles with at least three vertices on γ. We retain the counter-clockwise convention for both the orientation of γ and the labeling of square vertices. If γ is smooth or C1, any such rectangle can be uniquely parameterized by the location of a pivot vertex, v1, and the angle φ between v1v2 and γ.We refer to φ as the pivot angle. Once again, v1 may assume any value in the interval [0, 2π) and φ may assume any value in the interval (0, π). Let X be the parameter space

X = (v , φ) : v [0, 2π), φ (0, π) . { 1 1 ∈ ∈ } Given (v , φ), the location of v is fixed such that v v , γ = φ and, in 1 2 h 1 2 i order to ensure the construction is rectangular, the location of v4 is fixed such that v v , v , v = π. The final vertex, v , is then placed to complete h 4 1 1 2i 3 the rectangle. This construction requires that v1, v2, and v4 be on γ, but imposes no such condition on v3, which can be inside, on, or outside γ. Each (v , φ) pair is assigned a label from the set LI, LO, RI, RO where 1 { } 44 Discretizing the Parameter Space

2π

0 φ π

Figure 4.9 The parameter space for the (v1, φ) parameterization is isomorphic to an annulus. The shaded edges, where φ = 0, π, represent values of φ for which no valid construction exists. The dotted lines represent the various values of φ and π φ for each value of v . Currently, the relationship between max − max 1 φmax and v1 is unknown.

the label "L" stands for "left" and the label "R" stands for "right". The first label, which is either "L" or "R", is determined by the relative lengths of v1v2 and v v . If v v > v v , then (v , φ) is assigned the label "R". Otherwise, 4 1 | 1 2| | 4 1| 1 it’s assigned the label "L". More intuitively, the pair (v1, φ) is assigned an "L" or "R" label based on whether the left or right side, as seen by an ob- server standing on v1 and facing the interior of γ, of the corresponding inscribed rectangle is longer. The second label, which is either "I" or "O", is determined by whether v3 lies inside or outside γ. If v3 falls exactly on γ, then we assign it the special label "G". For a particular γ and pivot point v , there exists a unique point g γ 1 ∈ that maximizes v g (i.e., for which the cord from v to g attains its max- | 1 | 1 imum length). We define φ = v g, γ . When φ = φ , the side v v max h 1 i max 1 2 lies along the chord of maximum length, and so must be longer than v4v1 in the resulting construction and (v1, φ) must have a ‘R’ label. Similarly, if φ = π φ , then v v > v v and (v , φ) must have an ‘L’ label. − max | 4 1| | 1 2| 1 The transition from ‘R’ to ‘L’ labels must, therefore, take place somewhere between φ and π φ . max − max For a fixed mesh size, the parameter space can again be understood as a discrete grid where each point represents a distinct (v1, φ) pair. Locating an inscribed square corresponds to located an edge with all four labels present Pivot vertex and angle 45 on its vertices (e.g., ‘LI’ or ‘RO’ labels or ‘RI’ or ‘LO’ labels). Although there are many potential strategies for locating such an edge, our immediate goal is to begin by narrowing our bounds for the transition between ‘R’ and ‘L’ labels. In order to locate an edge where both labels transition, we’re then interested in answering the following questions:

1. Is it possible to establish similar bounds for the transition between ‘I’ and ‘O’ labels?

2. Does the set of ‘L’-‘R’ transition edges form a continuous line in the parameter space?

3. If so, is it possible for that line to have only ‘I’ or only ‘O’ labels along it? If not, then locating that ‘I’-‘O’ transition would yield the desired type of edge.

In order to explore these questions, we created an interactive model of this construction in Geogebra, shown in Figure 4.10, which allows the user to control v1 and φ and then observe what label the resulting construction ought to be assigned. This model allows for qualitative exploration of the behavior of this construction and will hopefully prove useful in answering the previous questions. 46 Discretizing the Parameter Space

Figure 4.10 An example of a rectangle constructed via this technique for an arbitrary conic section. In this model, φ can be controlled via the slider shown at the top right and v1 can be controlled by manually dragging the pivot vertex, highlighted in green, along γ. Chapter 5

Future Work

As the somewhat disparate nature of the preceding chapters perhaps indicates, this thesis has been a largely exploratory project. We began with an investigation of the existing literature about the inscribed square problem, particularly focusing on the weaker conditions for which the conjecture has been verified: the special trapezoid criterion (Matschke (2011)) and local monotonicity (Stromquist (1989)). Subsequently, we shifted to an exploration of particular families of curves. We identified explicit conditions for the existence of inscribed squares in conic sections and located two distinct types of inscribed squares in the Koch snowflake. We demonstrated that the Koch snowflake is not locally monotone. The question of whether or not it meets the special trapezoid criterion remains open, however. Answering that question would determine whether our work constitutes a novel result or an alternate proof for a subclass of curves meeting the special trapezoid criterion. Additionally, we defined and explored three discretization schemes for the parameter space of Jordan curves in the plane. The two pivot vertex discretization scheme developed in section 4.2 yields a result for circular γ and appears tractable for convex curves in general. Future work on these discretization schemes might begin with that discretization scheme, given that it has already yielded some small results. It might also include some work on the other discretization schemes.

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