the bellows theorem introduction
play

The Bellows Theorem (Introduction) Giovanni Viglietta JAIST June - PowerPoint PPT Presentation

Dec 06, 2023 •787 likes •1.47k views

The Bellows Theorem (Introduction) Giovanni Viglietta JAIST June 26, 2018 The Bellows Theorem (Introduction) Real-life bellows The Bellows Theorem (Introduction) Real-life bellows Observation: All of them have elasticity or curved creases.

  1. The Bellows Theorem (Introduction) Giovanni Viglietta JAIST – June 26, 2018 The Bellows Theorem (Introduction)

  2. Real-life bellows The Bellows Theorem (Introduction)

  3. Real-life bellows Observation: All of them have elasticity or curved creases. Why? The Bellows Theorem (Introduction)

  4. Definition of bellows Wiktionary: “A bellows is a container which is deformable in such a way as to alter its volume, which has an outlet where one wishes to blow air.” The Bellows Theorem (Introduction)

  5. Definition of bellows Wiktionary: “A bellows is a container which is deformable in such a way as to alter its volume, which has an outlet where one wishes to blow air.” Problem: Is it possible to construct a “geometric bellows” in some mathematical sense? The Bellows Theorem (Introduction)

  6. Bellows in Flatland /A triangular linkage (rigid bars and joints) cannot be a bellows./ The Bellows Theorem (Introduction)

  7. Bellows in Flatland a b c 2 2 2 2 2 2 4 4 4 2 a b +2 b c + c a a b c 2 A = − − − 16 /Heron’s formula gives its area as a function of the edge lengths./ The Bellows Theorem (Introduction)

  8. Bellows in Flatland /However, all other closed polygonal linkages are flexible./ The Bellows Theorem (Introduction)

  9. Bellows in Flatland /However, all other closed polygonal linkages are flexible./ The Bellows Theorem (Introduction)

  10. Bellows in Flatland /However, all other closed polygonal linkages are flexible./ The Bellows Theorem (Introduction)

  11. Bellows in Flatland /However, all other closed polygonal linkages are flexible./ The Bellows Theorem (Introduction)

  12. Bellows in Flatland /However, all other closed polygonal linkages are flexible./ The Bellows Theorem (Introduction)

  13. Bellows in Flatland /However, all other closed polygonal linkages are flexible./ The Bellows Theorem (Introduction)

  14. Bellows in Flatland /However, all other closed polygonal linkages are flexible./ The Bellows Theorem (Introduction)

  15. Bellows in Flatland /However, all other closed polygonal linkages are flexible./ The Bellows Theorem (Introduction)

  16. Bellows in Flatland /However, all other closed polygonal linkages are flexible./ The Bellows Theorem (Introduction)

  17. Bellows in Flatland /If the linkage has a small hole, it can “breathe” air as it flexes./ The Bellows Theorem (Introduction)

  18. Bellows in Flatland /If the linkage has a small hole, it can “breathe” air as it flexes./ The Bellows Theorem (Introduction)

  19. Bellows in Flatland /If the linkage has a small hole, it can “breathe” air as it flexes./ The Bellows Theorem (Introduction)

  20. Bellows in Flatland /If the linkage has a small hole, it can “breathe” air as it flexes./ The Bellows Theorem (Introduction)

  21. Bellows in Flatland /If the linkage has a small hole, it can “breathe” air as it flexes./ The Bellows Theorem (Introduction)

  22. Bellows in Flatland /If the linkage has a small hole, it can “breathe” air as it flexes./ The Bellows Theorem (Introduction)

  23. Bellows in Flatland /If the linkage has a small hole, it can “breathe” air as it flexes./ The Bellows Theorem (Introduction)

  24. Bellows in Flatland /If the linkage has a small hole, it can “breathe” air as it flexes./ The Bellows Theorem (Introduction)

  25. Bellows in Flatland /If the linkage has a small hole, it can “breathe” air as it flexes./ The Bellows Theorem (Introduction)

  26. Polyhedral model To define a 3D model, let us generalize 2D linkages. Instead of rigid bars, we have rigid polygons. Instead of joints at vertices, we have hinges at edges. The Bellows Theorem (Introduction)

  27. Polyhedral model To define a 3D model, let us generalize 2D linkages. Instead of rigid bars, we have rigid polygons. Instead of joints at vertices, we have hinges at edges. Problem: Can such a polyhedron be a bellows? Can it even flex? The Bellows Theorem (Introduction)

  28. Euclid’s “definition” of equal polyhedra Euclid’s Elements, Book XI, Definition 10: Literal translation: Equal and similar solid figures are those contained by similar planes equal in multitude and in magnitude. Modern interpretation: Two polyhedra are equal if they have the same combinatorial structure and equal corresponding faces. The Bellows Theorem (Introduction)

  29. Euclid’s “definition” of equal polyhedra Euclid’s Elements, Book XI, Definition 10: Literal translation: Equal and similar solid figures are those contained by similar planes equal in multitude and in magnitude. Modern interpretation: Two polyhedra are equal if they have the same combinatorial structure and equal corresponding faces. = ⇒ Euclid seems to disallow the existence of flexible polyhedra! The Bellows Theorem (Introduction)

  30. Simson’s critique to Euclid Simson, 1756: Euclid’s statement cannot be a definition, but a theorem that ought to be proved. Also, the statement is not universally true: The Bellows Theorem (Introduction)

  31. Simson’s critique to Euclid Simson, 1756: Euclid’s statement cannot be a definition, but a theorem that ought to be proved. Also, the statement is not universally true: Heath, 1908: To be fair, Euclid only applies his definition to prove equality of convex polyhedra with trihedral vertices. For these polyhedra, Euclid’s statement is obviously true, because a trihedral vertex is rigid. The Bellows Theorem (Introduction)

  32. Convex polyhedra are rigid Cauchy’s arm lemma: α α > α > α 3 2 3 2 > α 4 α α > α 1 4 1 d > d Theorem (Legendre-Cauchy, 1813) Two convex polyhedra are equal if they have the same combinatorial structure and equal corresponding faces. The Bellows Theorem (Introduction)

  33. Convex polyhedra are rigid Cauchy’s arm lemma: α α > α > α 3 2 3 2 > α 4 α α > α 1 4 1 d > d Theorem (Legendre-Cauchy, 1813) Two convex polyhedra are equal if they have the same combinatorial structure and equal corresponding faces. Corollary Convex polyhedra are rigid. = ⇒ Polyhedral bellows must be non-convex. The Bellows Theorem (Introduction)

  34. Existence of flexible polyhedra Theorem (Bricard, 1897) There exist self-intersecting flexible octahedra. The Bellows Theorem (Introduction)

  35. Existence of flexible polyhedra Theorem (Bricard, 1897) There exist self-intersecting flexible octahedra. The Bellows Theorem (Introduction)

  36. Existence of flexible polyhedra Theorem (Bricard, 1897) There exist self-intersecting flexible octahedra. The Bellows Theorem (Introduction)

  37. Existence of flexible polyhedra Theorem (Bricard, 1897) There exist self-intersecting flexible octahedra. The Bellows Theorem (Introduction)

  38. Existence of flexible polyhedra Theorem (Bricard, 1897) There exist self-intersecting flexible octahedra. The Bellows Theorem (Introduction)

  39. Existence of flexible polyhedra Theorem (Bricard, 1897) There exist self-intersecting flexible octahedra. Theorem (Gluck, 1975) Almost all polyhedra of genus 0 are rigid. The Bellows Theorem (Introduction)

  40. Existence of flexible polyhedra Theorem (Bricard, 1897) There exist self-intersecting flexible octahedra. Theorem (Gluck, 1975) Almost all polyhedra of genus 0 are rigid. Theorem (Connelly, 1977) There exist (non-self-intersecting) flexible polyhedra. The Bellows Theorem (Introduction)

  41. Steffen’s flexible polyhedron Theorem (Steffen, 1979) There is a flexible polyhedron with 9 vertices (smallest possible). The Bellows Theorem (Introduction)

  42. Steffen’s flexible polyhedron Theorem (Steffen, 1979) There is a flexible polyhedron with 9 vertices (smallest possible). The Bellows Theorem (Introduction)

  43. Steffen’s flexible polyhedron Theorem (Steffen, 1979) There is a flexible polyhedron with 9 vertices (smallest possible). The Bellows Theorem (Introduction)

  44. Steffen’s flexible polyhedron Theorem (Steffen, 1979) There is a flexible polyhedron with 9 vertices (smallest possible). The Bellows Theorem (Introduction)

  45. Steffen’s flexible polyhedron Theorem (Steffen, 1979) There is a flexible polyhedron with 9 vertices (smallest possible). The Bellows Theorem (Introduction)

  46. Steffen’s flexible polyhedron Theorem (Steffen, 1979) There is a flexible polyhedron with 9 vertices (smallest possible). The Bellows Theorem (Introduction)

  47. Steffen’s flexible polyhedron Theorem (Steffen, 1979) There is a flexible polyhedron with 9 vertices (smallest possible). The Bellows Theorem (Introduction)

  48. Steffen’s flexible polyhedron Theorem (Steffen, 1979) There is a flexible polyhedron with 9 vertices (smallest possible). The Bellows Theorem (Introduction)

  49. Still no polyhedral bellows! Observation The (generalized) volume of all these flexible polyhedra remains constant throughout the flexing! In other words, althought these polyhedra are not rigid, none of them can blow air. The Bellows Theorem (Introduction)

Recommend

The Bellows Theorem (Proof) Giovanni Viglietta JAIST July 5, 2018 The Bellows Theorem

The Bellows Theorem (Proof) Giovanni Viglietta JAIST July 5, 2018 The Bellows Theorem

The Bellows Theorem (Proof) Giovanni Viglietta JAIST July 5, 2018 The Bellows Theorem (Proof) Bellows theorem: statement Theorem (Sabitov, 1996) The volume V of a polyhedron (of any genus) with edge lengths 1 , , e

1.22k views • 93 slides
On Theorem Proving for Program Checking Historical perspective and recent developments Maria

On Theorem Proving for Program Checking Historical perspective and recent developments Maria

Outline Introduction Where is theorem proving in program checking Inside theorem proving Big and little engines together: a new theorem proving style Decision procedures with speculative inferences Current and future challenges On Theorem

944 views • 69 slides
Rockingham (Bellows Falls) BO 1442(41) Alternatives Presentation Meeting TH-448 (Depot Street)

Rockingham (Bellows Falls) BO 1442(41) Alternatives Presentation Meeting TH-448 (Depot Street)

Rockingham (Bellows Falls) BO 1442(41) Alternatives Presentation Meeting TH-448 (Depot Street) Bridge #53 over the Power Dam Canal November 27, 2018 Introductions Laura Stone, P.E. VTrans Scoping Engineer Todd Sumner, P.E. VTrans Project

944 views • 41 slides
A Mechanical Model to Simulate Interactively a Bending Actuator Composed of three Parallel Bellows

A Mechanical Model to Simulate Interactively a Bending Actuator Composed of three Parallel Bellows

A Mechanical Model to Simulate Interactively a Bending Actuator Composed of three Parallel Bellows P. Joli 1 *, N. Seguy 1 , Z.Q. Feng 2 1 Laboratoire Systmes Complexes , Universit d'Evry , 40 rue du Pelvoux , 91020 Evry, France e-mail:

370 views • 10 slides
We sustain Fall, 2018 Cohort. Internship Ana Bellows College 2 2018 Fall cohort Interns

We sustain Fall, 2018 Cohort. Internship Ana Bellows College 2 2018 Fall cohort Interns

We sustain Fall, 2018 Cohort. Internship Ana Bellows College 2 2018 Fall cohort Interns by majors Duration: 5 th Sep- 7 Th Dec Intern number: 9 5 Domestic & 4 International students Natural science Engineering Nursing

432 views • 12 slides
Automated Theorem Proving 1/4: Introduction and Propositional Theorem Proving A.L. Lamprecht

Automated Theorem Proving 1/4: Introduction and Propositional Theorem Proving A.L. Lamprecht

Automated Theorem Proving 1/4: Introduction and Propositional Theorem Proving A.L. Lamprecht Course Program Semantics and Verfication 2020, Utrecht University September 21, 2020 Lecture Notes Automated Reasoning by Gerard A.W. Vreeswijk.

609 views • 40 slides
31. Stokes Theorem Stokes theorem is to Greens theorem, for the work done, as the

31. Stokes Theorem Stokes theorem is to Greens theorem, for the work done, as the

31. Stokes Theorem Stokes theorem is to Greens theorem, for the work done, as the divergence theorem is to Greens theorem, for the flux. Both are 3D generalisations of 2D theorems. Theorem 31.1 (Stokes Theorem) . Let C be any closed

406 views • 3 slides
29. The divergence theorem Theorem 29.1 (Divergence Theorem; Gauss, Ostrogradsky) . Let S be a

29. The divergence theorem Theorem 29.1 (Divergence Theorem; Gauss, Ostrogradsky) . Let S be a

29. The divergence theorem Theorem 29.1 (Divergence Theorem; Gauss, Ostrogradsky) . Let S be a closed surface bounding a solid D , oriented outwards. Let F be a vector field with continuous partial derivatives. Then F d

471 views • 3 slides
Kleenes Theorem 4-0 Kleenes Theorem Theorem For every language L (over

Kleenes Theorem 4-0 Kleenes Theorem Theorem For every language L (over

Griffith University 3130CIT Theory of Computation (Based on slides by Harald Sndergaard of The University of Melbourne) Kleenes Theorem 4-0 Kleenes Theorem Theorem For every language L (over a finite

422 views • 17 slides
PCP Theorem [PCP Theorem is] the most important result in complexity theory since Cooks

PCP Theorem [PCP Theorem is] the most important result in complexity theory since Cooks

PCP Theorem [PCP Theorem is] the most important result in complexity theory since Cooks Theorem. Ingo Wegener, 2005 Computational Complexity, by Fu Yuxi PCP Theorem 1 / 119 S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy. Proof

1.53k views • 120 slides
The Replacement Theorem Theorem (Theorem 1.10) Let V be a vector space and suppose G and L are

The Replacement Theorem Theorem (Theorem 1.10) Let V be a vector space and suppose G and L are

The Replacement Theorem Theorem (Theorem 1.10) Let V be a vector space and suppose G and L are finite subsets of V such that V = Span ( G ) , |G| = n , L is linearly independent, and |L| = m . Then m n and there is a set H G , such that

468 views • 36 slides
A Quantitative Assessment of Flight Training Effectiveness in Mixed Reality Peter Bellows 1 , Amy

A Quantitative Assessment of Flight Training Effectiveness in Mixed Reality Peter Bellows 1 , Amy

IT 2 EC 2020 A Quantitative Assessment of Flight Training Effectiveness in Mixed Reality Presentation/Panel A Quantitative Assessment of Flight Training Effectiveness in Mixed Reality Peter Bellows 1 , Amy Dideriksen 1 , Joe Williams 1 , Tom

234 views • 4 slides
1 What is a Theorem Prover? What is a Theorem? Theorem: A formalizable statement which is

1 What is a Theorem Prover? What is a Theorem? Theorem: A formalizable statement which is

Prerequisites Course 2D1453, 2006-07 Undergraduate logic and discrete maths CS literacy Advanced Formal Methods Some functional programming experience useful The theorem prover Isabelle is programming in SML Some SML

644 views • 4 slides
22. Greens Theorem Theorem 22.1 (Greens Theorem) . If C is a positively oriented closed curve

22. Greens Theorem Theorem 22.1 (Greens Theorem) . If C is a positively oriented closed curve

22. Greens Theorem Theorem 22.1 (Greens Theorem) . If C is a positively oriented closed curve enclosing a region R then curl F d r = F d A. C R The circle in the centre of the integral sign is simply to emphasize

355 views • 4 slides
A Parallelized Theorem Prover for Interactive Theorem Proving David L. Rager, Warren A. Hunt Jr.,

A Parallelized Theorem Prover for Interactive Theorem Proving David L. Rager, Warren A. Hunt Jr.,

Introduction Parallelism Primitives Parallelizing ACL2 Evaluate Approach Conclusion A Parallelized Theorem Prover for Interactive Theorem Proving David L. Rager, Warren A. Hunt Jr., and Matt Kaufmann ragerdl@defthm.com, hunt@cs.utexas.edu,

835 views • 59 slides
Introduction to Complexity and Computability NTIN090 Petr Kuera 2019/20 1/167 Introduction

Introduction to Complexity and Computability NTIN090 Petr Kuera 2019/20 1/167 Introduction

Introduction to Complexity and Computability NTIN090 Petr Kuera 2019/20 1/167 Introduction Syllabus 6 algorithms and schemes Deterministic space and time hierarchy theorems 9 Savitchs theorem 8 1 7 Rices theorem 3/167 2 5

2k views • 167 slides
Generalized Intermediate Value Theorem Intermediate Value Theorem Theorem Intermediate Value

Generalized Intermediate Value Theorem Intermediate Value Theorem Theorem Intermediate Value

Generalized Intermediate Value Theorem Intermediate Value Theorem Theorem Intermediate Value Theorem Suppose f is continuous on [ a , b ] and let N be any number between f ( a ) and f ( b ) , where f ( a ) = f ( b ) . Then there exists a

59 views • 4 slides
From Shang Gao Magic 15 If a 2 + b 2 = c 2 , then ( a , b , c ) is called to be a Background

From Shang Gao Magic 15 If a 2 + b 2 = c 2 , then ( a , b , c ) is called to be a Background

Magic 15 Magic 15 Background Two Square Theorem Galois Finite Fields Magic 15: A STORY OVER 380 YEARS Proofs of 2-Square Theorem Three Square Theorem Four Square Theorem Universal Polynomials Magic 15 Proof of Fifteen Theorem 15

1.31k views • 97 slides
SMTtoTPTP - A Converter for Theorem Proving Formats Peter Baumgartner Introduction TPTP

SMTtoTPTP - A Converter for Theorem Proving Formats Peter Baumgartner Introduction TPTP

SMTtoTPTP - A Converter for Theorem Proving Formats Peter Baumgartner Introduction TPTP (Thousands of Problems for Theorem Proving) Languages : clause logic, [typed]FOL[+arithmetics], HOL Problem library : > 20k problems Infrastructure :

391 views • 38 slides
The BorsukUlam Theorem Anthony Carbery University of Edinburgh & Maxwell Institute for

The BorsukUlam Theorem Anthony Carbery University of Edinburgh & Maxwell Institute for

The BorsukUlam Theorem Anthony Carbery University of Edinburgh & Maxwell Institute for Mathematical Sciences May 2010 () 1 / 43 Outline Outline Brouwer fixed point theorem 1 BorsukUlam theorem 2 Introduction Case n = 2

794 views • 35 slides
NOTES ON THE VAPNIK-CHERVONENKIS THEOREM: BACKGROUND AND PROOF ROLAND WALKER 1. Introduction

NOTES ON THE VAPNIK-CHERVONENKIS THEOREM: BACKGROUND AND PROOF ROLAND WALKER 1. Introduction

NOTES ON THE VAPNIK-CHERVONENKIS THEOREM: BACKGROUND AND PROOF ROLAND WALKER 1. Introduction Vladimir Vapnik and Alexey Chervonenkis proved their eponymous theorem in 1968. The original Russian proof was published in 1971 and then translated to

278 views • 13 slides
Lecture 18: PCP Theorem and Hardness of Approximation I Arijit Bishnu 26.04.2010 Introduction

Lecture 18: PCP Theorem and Hardness of Approximation I Arijit Bishnu 26.04.2010 Introduction

Introduction to Approximation Algorithm PCP Theorem Lecture 18: PCP Theorem and Hardness of Approximation I Arijit Bishnu 26.04.2010 Introduction to Approximation Algorithm PCP Theorem Outline 1 Introduction to Approximation Algorithm 2 PCP

779 views • 54 slides
Ch04. Maximum Theorem, Implicit Function Theorem and Envelope Theorem Ping Yu Faculty of

Ch04. Maximum Theorem, Implicit Function Theorem and Envelope Theorem Ping Yu Faculty of

Ch04. Maximum Theorem, Implicit Function Theorem and Envelope Theorem Ping Yu Faculty of Business and Economics The University of Hong Kong Ping Yu (HKU) MIFE 1 / 27 The Maximum Theorem 1 The Implicit Function Theorem 2 The Envelope

599 views • 28 slides
Understanding the Prime Number Theorem Misunderstood Monster or Beautiful Theorem? Liam Fowl

Understanding the Prime Number Theorem Misunderstood Monster or Beautiful Theorem? Liam Fowl

Introduction Complex Plane Complex functions and Analytic Continuation Gamma Function Laplace Transform Zeta Function Understanding the Prime Number Theorem Misunderstood Monster or Beautiful Theorem? Liam Fowl September 5, 2014 Liam Fowl

618 views • 20 slides

More recommend