Squares and other polygons inscribed in curves Elizabeth Denne [email protected] Wellesley College April 10, 2012 Introduction Some Topology Rectangles Squares Other Polygons Thanks A proof from “the book” Theorem (Proof: H. Vaughan 1977) Let γ be a continuous closed curve in the plane without self-intersections, then there are four points on γ that are the vertices of a rectangle. The proof is purely topological. Principle • Two objects are the “same" if there is a homeomorphism between them. • Homeomorphism: 1-to-1, onto, continuous map with a continuous inverse. • Topological properties are preserved by homeomorphisms (ex. connectedness). Introduction Some Topology Rectangles Squares Other Polygons Thanks Setting up the proof — a topological detour • Let γ be a continuous simple closed curve in the plane. • Let M(γ) = ffu; vg j u; v 2 γg (order doesn’t matter). • Since γ is homeomorphic to the standard unit circle S1, we understand M(γ) by understanding M(S1). • Observe that S1 × S1 is represented by (u; v) • S1 (v; u) • (0; x) • • (u; u) • S1 (x; 0) Introduction Some Topology Rectangles Squares Other Polygons Thanks Visualizing a torus S1 × S1 Gluing a Torus - YouTube http://www.youtube.com/watch?v=0H5_h-RB0T8 Introduction Some Topology Rectangles Squares Other Polygons Thanks Structure of M(γ) = ffu; vg j u; v 2 γg • S1 × S1 has both (u; v) and (v; u). • Keep one half of S1 × S1. Cut this in half then reglue. a a a a m m m •x m x• x• m • a x a a a a Fact M(γ) is homeomorphic to a Möbius band. Also, M(γ)’s boundary corresponds to pairs of points fu; ug. (This can be thought of as a copy of γ.) Introduction Some Topology Rectangles Squares Other Polygons Thanks Möbius bands are non-orientable surfaces M.C. Escher Introduction Some Topology Rectangles Squares Other Polygons Thanks Real Projective Plane Definition Real projective plane RP2 = flines in R3 through the origing • Each line meets the unit sphere S2 in two antipodal points. • Identify a line ` with ` \ S2 = fx; −xg. • Take half of S2 and glue together y and −y on the equator. • RP2 is a surface without boundary. •x y • • 0 • • • 0 −y −x• •x Introduction Some Topology Rectangles Squares Other Polygons Thanks Visualizing RP2 Learn more about surfaces: Beginning Topology, Sue Goodman. Introduction Some Topology Rectangles Squares Other Polygons Thanks Boy’s surface How can you visualize topological objects? A Topological Picture Book by George Francis Introduction Some Topology Rectangles Squares Other Polygons Thanks Real Projective Space and Möbius Bands Fact RP2 − disc = Mb with boundary of disc as edge, or RP2 = Mb [ disc joined along their boundaries. disc •b •c e • a• •b c • •a • d a a b• • • •c Möbius e• •d band • • • • c a a b Introduction Some Topology Rectangles Squares Other Polygons Thanks A different view of the Möbius band... ... the Sudanese Möbius band. Introduction Some Topology Rectangles Squares Other Polygons Thanks Back to proof Theorem (Proof: H. Vaughan 1977) Let γ be a continuous closed curve in the plane without self-intersections, then there are four points on γ that are the vertices of a rectangle. • γ simple closed curve in the plane. • γ bounds a disc D (Schoenflies Theorem). • M(γ) = ffu; vg j u; v 2 γg homeomorphic to Möbius band. • Boundary of M(γ) is a copy of γ. • We know that gluing together a disc and Möbius band along their boundary gives RP2. 2 • RP = M(γ) [γ D Introduction Some Topology Rectangles Squares Other Polygons Thanks Back to proof Theorem (Proof: H. Vaughan 1977) Let γ be a continuous closed curve in the plane without self-intersections, then there are four points on γ that are the vertices of a rectangle. • γ simple closed curve in the plane. • γ bounds a disc D (Schoenflies Theorem). • M(γ) = ffu; vg j u; v 2 γg homeomorphic to Möbius band. • Boundary of M(γ) is a copy of γ. • We know that gluing together a disc and Möbius band along their boundary gives RP2. 2 • RP = M(γ) [γ D Introduction Some Topology Rectangles Squares Other Polygons Thanks Back to proof Theorem (Proof: H. Vaughan 1977) Let γ be a continuous closed curve in the plane without self-intersections, then there are four points on γ that are the vertices of a rectangle. • γ simple closed curve in the plane. • γ bounds a disc D (Schoenflies Theorem). • M(γ) = ffu; vg j u; v 2 γg homeomorphic to Möbius band. • Boundary of M(γ) is a copy of γ. • We know that gluing together a disc and Möbius band along their boundary gives RP2. 2 • RP = M(γ) [γ D Introduction Some Topology Rectangles Squares Other Polygons Thanks Proof continued. Define F : RP2 ! R3 by 2 3 Fjdisc is an inclusion of R ,! R , and FjM(γ) is u+v 2 F(fu; vg) = ( 2 ; jju − vjj) (take midpoint in R , length uv). • F agrees on γ and is a continuous function. • Fact: F cannot be an embedding (since RP2 is a closed non-orientable surface). • Fjdisc is an embedding (by construction). • Problem points are on FjM(γ). • u+v a+b There are pairs fu; vg, fa; bg with 2 = 2 and jju − vjj = jja − bjj. • Thus uv and ab are the diagonals of a rectangle. Introduction Some Topology Rectangles Squares Other Polygons Thanks Proof continued. Define F : RP2 ! R3 by 2 3 Fjdisc is an inclusion of R ,! R , and FjM(γ) is u+v 2 F(fu; vg) = ( 2 ; jju − vjj) (take midpoint in R , length uv). • F agrees on γ and is a continuous function. • Fact: F cannot be an embedding (since RP2 is a closed non-orientable surface). • Fjdisc is an embedding (by construction). • Problem points are on FjM(γ). • u+v a+b There are pairs fu; vg, fa; bg with 2 = 2 and jju − vjj = jja − bjj. • Thus uv and ab are the diagonals of a rectangle. Introduction Some Topology Rectangles Squares Other Polygons Thanks Proof continued. Define F : RP2 ! R3 by 2 3 Fjdisc is an inclusion of R ,! R , and FjM(γ) is u+v 2 F(fu; vg) = ( 2 ; jju − vjj) (take midpoint in R , length uv). • F agrees on γ and is a continuous function. • Fact: F cannot be an embedding (since RP2 is a closed non-orientable surface). • Fjdisc is an embedding (by construction). • Problem points are on FjM(γ). • u+v a+b There are pairs fu; vg, fa; bg with 2 = 2 and jju − vjj = jja − bjj. • Thus uv and ab are the diagonals of a rectangle. Introduction Some Topology Rectangles Squares Other Polygons Thanks Proof continued. Define F : RP2 ! R3 by 2 3 Fjdisc is an inclusion of R ,! R , and FjM(γ) is u+v 2 F(fu; vg) = ( 2 ; jju − vjj) (take midpoint in R , length uv). • F agrees on γ and is a continuous function. • Fact: F cannot be an embedding (since RP2 is a closed non-orientable surface). • Fjdisc is an embedding (by construction). • Problem points are on FjM(γ). • u+v a+b There are pairs fu; vg, fa; bg with 2 = 2 and jju − vjj = jja − bjj. • Thus uv and ab are the diagonals of a rectangle. Introduction Some Topology Rectangles Squares Other Polygons Thanks Proof continued. Define F : RP2 ! R3 by 2 3 Fjdisc is an inclusion of R ,! R , and FjM(γ) is u+v 2 F(fu; vg) = ( 2 ; jju − vjj) (take midpoint in R , length uv). • F agrees on γ and is a continuous function. • Fact: F cannot be an embedding (since RP2 is a closed non-orientable surface). • Fjdisc is an embedding (by construction). • Problem points are on FjM(γ). • u+v a+b There are pairs fu; vg, fa; bg with 2 = 2 and jju − vjj = jja − bjj. • Thus uv and ab are the diagonals of a rectangle. Introduction Some Topology Rectangles Squares Other Polygons Thanks The Square Peg problem Question (Toeplitz, 1911) Let γ be a closed curve in the plane with no self-intersections. Are there four points on γ that are the vertices of a square? Still an open question! Known results require γ to be “smooth enough”. Introduction Some Topology Rectangles Squares Other Polygons Thanks History of the Square Peg Problem 1. Emch, 1917, proved for piecewise analytic arcs 2. Schnirelmann, 1929, proved for bounded curvature∗ 3. Christiansen, 1950, for convex curves 4. Ogilvy, 1950, for “nice enough” curves 5. Jerrard, 1959, for analytic curves 6. Guggenheim, 1965, fixes an error in Schnirelmann∗ 7. Stromquist, 1989, for a class somewhat larger than C1 8. (Griffiths, 1991, contains serious errors) 9. Nielsen, Wright, 2002, centrally symmetric C0 curves 10. Zivaljevik, 2008, preprint 11. Cantarella, D, McCleary, joint work-in-progress. Introduction Some Topology Rectangles Squares Other Polygons Thanks Our results: Squares Theorem (with Cantarella, McCleary) Let γ be a simple closed continuously differentiable curve in R2. Then there are an odd (or infinite) number of inscribed squares abcd (in order) on γ. What about space curves, or curves in Rk ? Introduction Some Topology Rectangles Squares Other Polygons Thanks Our results: Squares Theorem (with Cantarella, McCleary) Let γ be a simple closed continuously differentiable curve in Rk . Then there are an odd (or infinite) number of square-like quadrilaterals abcd inscribed in order on γ with equal sides jabj = jbcj = jcdj = jdaj and equal diagonals jacj = jbdj. Corollary For a simple closed C1 plane curve, there are an odd (or infinite) number of inscribed squares. Theorem (corners don’t matter) For a curve of finite total curvature without cusps, there is at least one inscribed square.