Dissecting a Square into an Odd Number of Triangles of Almost Equal Area

Size: px

Start display at page:

Download "Dissecting a Square into an Odd Number of Triangles of Almost Equal Area"

Rhoda Christal Long
5 years ago
Views:

1 Dissecting a Square into an Odd Number of Triangles of Almost Equal Area Jean-Philippe Labbé, Günter Rote, Günter M. Ziegler Freie Universität Berlin

2 Dissecting a Square into an Odd Number of Triangles of Almost Equal Area Jean-Philippe Labbé, Günter Rote, Günter M. Ziegler Freie Universität Berlin n = 9 triangles root-mean-square (RMS) error: The optimum among dissections with at most 8 nodes.

3 Dissecting a Square into an Odd Number of Triangles of Almost Equal Area Jean-Philippe Labbé, Günter Rote, Günter M. Ziegler Freie Universität Berlin THEOREM (P. Monsky, 1970) If n is odd, there is no dissection of the square into n triangles of equal area

4 Dissecting a Square into an Odd Number of Triangles of Almost Equal Area Jean-Philippe Labbé, Günter Rote, Günter M. Ziegler Freie Universität Berlin THEOREM (P. Monsky, 1970) If n is odd, there is no dissection of the square into n triangles of equal area dissection triangulation

5 Measuring Area Deviation areas a 1,..., a n, target area = 1 n Root-mean-square error (RMS, standard deviation): RMS := 1 n n (a i 1 n )2 i=1 Range: range := max 1 i n a i min 1 i n a i range 2 n RMS range

6 Lower and Upper Bounds range 1 2 2O(n) (doubly-exponential) Proof: Gap theorems from real algebraic geometry A family of dissections for every n with range 1 n log 2 n 5 = 1 2 (superpolynomial) Ω(log2 n) Previous results: Numerical experiments, exhaustive enumeration for small n (Katja Mansow, 2003) A family of triangulations with range 1/n 3 (Bernd Schulze, 2011)

7 Overview Introduction: Problem statement and results Review of Monsky s proof (2-adic valuation, Sperner s lemma) Modeling the problem Lower bound via a gap theorem Numerical experiments Systematic construction (Thue-Morse sequence) More numerical experiments Speculations

8 Monsky: 3-coloring of the plane 2-adic valuation of R coloring of R R with three colors A, B, C Crucial property: A rainbow triangle cannot have area 0 or 1 n for odd n.

9 Monsky: 3-coloring of the plane 2-adic valuation of R coloring of R R with three colors A, B, C Crucial property: A rainbow triangle cannot have area 0 or 1 n for odd n. Parity argument like for Sperner s lemma: If the boundary of a polygon has an odd number of AB-colored edges, then every dissection has an odd number of rainbow triangles. B A C A

10 Lower Bound: Modeling Collinearity Lock at all maximal line segments

11 Lower Bound: Modeling Collinearity Lock at all maximal line segments

12 Lower Bound: Modeling Collinearity Lock at all maximal line segments Open them up Triangulate them arbitrarily. combinatorial triangulation of a 4-gon, with additional zero-area triangles Z

13 Lower Bound: Modeling Collinearity Lock at all maximal line segments Open them up Triangulate them arbitrarily. combinatorial triangulation of a 4-gon, with additional zero-area triangles Z

14 Lower Bound: Modeling Collinearity Lock at all maximal line segments Open them up Triangulate them arbitrarily. combinatorial triangulation of a 4-gon, with additional zero-area triangles Z Area 0 does not enforce collinearity!

15 Area Deviation Polynomial n triangles, areas a 1,..., a n v unknown vertex positions (apart from the 4 fixed corners of the square)

16 Area Deviation Polynomial n triangles, areas a 1,..., a n v unknown vertex positions (apart from the 4 fixed corners of the square)

17 Area Deviation Polynomial n triangles, areas a 1,..., a n v unknown vertex positions (apart from the 4 fixed corners of the square) n T ( x) = (a i ( x) 1 n )2, x R 2v i=1... a degree-4 polynomial, RMS = T ( x)/n

18 Area Deviation Polynomial n triangles, areas a 1,..., a n v unknown vertex positions (apart from the 4 fixed corners of the square) n T ( x) = (a i ( x) 1 n )2, x R 2v i=1... a degree-4 polynomial, RMS = T ( x)/n z zero-area triangles, areas b 1,..., b z z Z( x) = (b j ( x) 0) 2 j=1

19 Area Deviation Polynomial n triangles, areas a 1,..., a n v unknown vertex positions (apart from the 4 fixed corners of the square) n T ( x) = (a i ( x) 1 n )2, x R 2v i=1... a degree-4 polynomial, RMS = T ( x)/n z zero-area triangles, areas b 1,..., b z z Z( x) = (b j ( x) 0) 2 j=1 T ( x) + Z( x) min!, x R 2v

20 Lower-Bound Argument min{ RMS 2 n dissection } = min{ T ( x) x R 2v, x is a dissection } min{ T ( x) x R 2v, Z( x) = 0 } min{ T ( x) + Z( x) x R 2v, Z( x) = 0 } min{ T ( x) + Z( x) x R 2v } > 0 (à la Sperner and Monsky)

21 Lower-Bound Argument min{ RMS 2 n dissection } = min{ T ( x) x R 2v, x is a dissection } min{ T ( x) x R 2v, Z( x) = 0 } min{ T ( x) + Z( x) x R 2v, Z( x) = 0 } min{ T ( x) + Z( x) x R 2v } > 0 (à la Sperner and Monsky)

22 Lower-Bound Argument min{ RMS 2 n dissection } = min{ T ( x) x R 2v, x is a dissection } min{ T ( x) x R 2v, Z( x) = 0 } min{ T ( x) + Z( x) x R 2v, Z( x) = 0 } min{ T ( x) + Z( x) x R 2v } > 0 (à la Sperner and Monsky)

23 Gap Theorems An algebraic number α 0 cannot be arbitarily close to 0. (depending on the degree and the size of the coefficients)

24 Gap Theorems An algebraic number α 0 cannot be arbitarily close to 0. (depending on the degree and the size of the coefficients) The DMM bound ( Davenport Mahler Mignotte ) [ Emiris Mourrain Tsigaridas,2010 ] polynomial f( x) of degree d in k variables integer coefficients with τ bits f(x) > 0 on the unit simplex in R k = min{ f(x) x unit simplex } m DMM 1 = 2 d(d 1) (k 1) ((d log 2 k+τ+1)(k+1)+(k 2 +3k+1) log 2 d+d+2k+1) 2 m DMM

25 Gap Theorems An algebraic number α 0 cannot be arbitarily close to 0. (depending on the degree and the size of the coefficients) The DMM bound ( Davenport Mahler Mignotte ) [ Emiris Mourrain Tsigaridas,2010 ] polynomial f( x) of degree d in k variables integer coefficients with τ bits f(x) > 0 on the unit simplex in R k = min{ f(x) x unit simplex } m DMM 1 = 2 d(d 1) (k 1) ((d log 2 k+τ+1)(k+1)+(k 2 +3k+1) log 2 d+d+2k+1) 2 m DMM THEOREM: If the unit square is dissected into an odd number n of triangles, the range of areas is at least 1/2 2Ω(n).

26 Computer Experiments 1. Generate all combinatorial types of triangulations/dissections [ plantri by Brinkmann and McKay ] 2a. [ Katja Mansow 2003 ] for triangulations: Minimize the range numerically [ minmax command of Matlab ] 2b. For dissections: Minimize the squared error (RMS): Find critical points of T ( x) + λz( x). system of polynomial equations [ Bertini of Bates, Hauenstein, Sommese, Wampler ]

27 Computer Experiments n = 3, RMS = , range =

28 Computer Experiments n = 5, RMS = the best triangulation found by Mansow: range =

29 Computer Experiments n = 7, RMS = the RMS-optimal solutions with at most 8 vertices: the best triangulation found by Mansow: range =

30 Computer Experiments n = 9, RMS = the RMS-optimal solution with at most 8 vertices: best triangulation found by Mansow: 9 vertices, range = !

31 A Systematic Construction (0, 1) (1, 1) ( ) 1, 1 2 n... range = O(1/n 3 ) (0, 0) (1, 0)

32 A Systematic Construction (0, 1) (1, 1) ( ) 1, 1 2 n... n 1 (mod 4) range = O(1/n 5 ) (0, 0) (1, 0)

33 A Systematic Construction n 2 log 2 n 1 filler triangles THEOREM: range 8n2 n log 2 n (1 + O( log n n )) 2 log 2 n triangles use the Thue-Morse sequence s 1 s 2 s 3... =

34 Estimating the error a 1 a2 1/n 2/n a 3... a i a i+1... a n 1 O

35 Estimating the error a 1 a2 1/n 2/n a 3... a i a i+1... a n 1 O

36 Estimating the error a 1 a2 1/n 2/n a 3... a i a i+1... a n 1 O n 1 i=1 ( 1 iu 1 (i 1)U ) si! 1, U := 4 n 2

37 Estimating the error a 1 a2 1/n 2/n a 3... a i a i+1... a n 1 O W := n 1 i=1 n 1 i=1 ( 1 iu 1 (i 1)U s i ln 1 iu 1 (i 1)U ) si! 1, U := 4 n 2! 0

38 Estimating the error W := n 1 i=1 n 1 i=1 s i ln 1 iu 1 (i 1)U s i ln 1 iu ± ε 1 (i 1)U ± ε! 0! = 0 W small ε small. = Concentrate on small W! n 1 s i ln(1 iu) = i=1 n 1 i=1 s i ( iu i2 2 U 2 i3 3 U 3 ) Try to cancel the first powers of U

39 Annihilate Powers = = = = = 1536 Theorem (E. Prouhet 1851) If f is a polynomial of degree < k then 2 k i=1 s i f(i) = 0, for the Thue-Morse sequence s 1, s 2, s 3,....

40 Upper bound THEOREM: For every n, there is a dissection with range 8n2 n log 2 n (1 + O( log n n ))

41 Systematic is not Always Best n optimal sign sequence s ε = ±range/

42 Systematic is not Always Best Example: n = 33 Best sequence: ε = Thue-Morse: ε = Guarantee from theorem: ε

43 Heuristic Explanation W := n 1 i=1 s i ln 1 iu 1 (i 1)U! 0 RANDOM s i = ±1 W : approximately Gaussian with µ = 0 and σ U n n 3/2. Take N = 2 n 1 random samples from this distribution. f(x) 0 x

44 Heuristic Explanation W := n 1 i=1 s i ln 1 iu 1 (i 1)U! 0 RANDOM s i = ±1 W : approximately Gaussian with µ = 0 and σ U n n 3/2. Take N = 2 n 1 random samples from this distribution. f(x) Near x = 0, these samples are like a Poisson distribution with density λ = N f(0) 2 n /n 3/2 0 x 0 Smallest absolute value = 1/2λ n 3/2 /2 n

45 Triangulations? So far: Ideas for systematic computer experiments. No general analysis.

46 The Tarry-Escott Problem 2 k i=1 s i i d = 0, for d = 0, 1,..., k 1 Can you annihilate the first k powers with a SHORTER sign sequence? The Tarry-Escott Problem: Find two distinct sets of integers α 1,..., α n and β 1,..., β n such that α d α d n = β d β d n, for all d = 0, 1, 2,..., k 1 Try to make n as small as possible.

arxiv: v3 [math.mg] 6 Jun 2018

arxiv: v3 [math.mg] 6 Jun 2018 AREA DIFFERENCE BOUNDS FOR DISSECTIONS OF A SQUARE INTO AN ODD NUMBER OF TRIANGLES JEAN-PHILIPPE LABBÉ, GÜNTER ROTE, AND GÜNTER M. ZIEGLER arxiv:1708.02891v3 [math.mg] 6 Jun 2018 Abstract. Monsky s theorem