The Shadows of a Cycle Cannot All Be Paths CCCG 2015 Prosenjit - - PowerPoint PPT Presentation

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The Shadows of a Cycle Cannot All Be Paths CCCG 2015 Prosenjit - - PowerPoint PPT Presentation

Dec 01, 2022 962 likes •2.06k views

The Shadows of a Cycle Cannot All Be Paths CCCG 2015 Prosenjit Bose, Jean-Lou De Carufel, Michael G. Dobbins, Heuna Kim, Giovanni Viglietta Kingston August 10, 2015 The Shadows of a Cycle Cannot All Be Paths Oskars maze A 3D maze

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SLIDE 1

The Shadows of a Cycle Cannot All Be Paths

CCCG 2015 Prosenjit Bose, Jean-Lou De Carufel, Michael G. Dobbins, Heuna Kim, Giovanni Viglietta Kingston – August 10, 2015

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 2

Oskar’s maze

A 3D maze designed in the 1980s by Oskar van Deventer. To move the rods around, one has to solve three 2D mazes.

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 3

Oskar’s maze

/Observation: the maze on each face must be a tree./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 4

Oskar’s maze

/If there were a cycle, part of the structure would fall off./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 5

Oskar’s maze

/If there were a cycle, part of the structure would fall off./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 6

Oskar’s maze

/But can there be cycles in the “internal” 3D maze?/

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 7

Rickard’s curve

A 3D cycle whose shadows are all trees.

(Illustration by Afra Zomorodian.)

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 8

Goucher’s “treefoil”

A trefoil knot whose shadows are all trees.

(Illustration courtesy of Adam P. Goucher.)

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 9

In this talk...

Can the shadows of a 3D cycle be all paths? (Open problem from CCCG 2007)

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 10

In this talk...

Can the shadows of a 3D cycle be all paths? NO. (Open problem from CCCG 2007)

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 11

In this talk...

Can the shadows of a 3D cycle be all paths? NO. (Open problem from CCCG 2007) Can the shadows of a 3D path be all cycles?

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 12

In this talk...

Can the shadows of a 3D cycle be all paths? NO. (Open problem from CCCG 2007) Can the shadows of a 3D path be all cycles? YES.

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 13

In this talk...

Can the shadows of a 3D cycle be all paths? NO. (Open problem from CCCG 2007) Can the shadows of a 3D path be all cycles? YES. Can the shadows of a 3D path be all convex cycles?

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 14

In this talk...

Can the shadows of a 3D cycle be all paths? NO. (Open problem from CCCG 2007) Can the shadows of a 3D path be all cycles? YES. Can the shadows of a 3D path be all convex cycles? NO.

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 15

In this talk...

Can the shadows of a 3D cycle be all paths? NO. (Open problem from CCCG 2007) Can the shadows of a 3D path be all cycles? YES. Can the shadows of a 3D path be all convex cycles? NO. What about higher dimensions?

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 16

In this talk...

Can the shadows of a 3D cycle be all paths? NO. (Open problem from CCCG 2007) Can the shadows of a 3D path be all cycles? YES. Can the shadows of a 3D path be all convex cycles? NO. What about higher dimensions? Rickard’s curve generalizes to any dimension.

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 17

The shadows of a 3D cycle cannot all be paths

/Suppose that the shadows of a 3D cycle are all paths./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 18

The shadows of a 3D cycle cannot all be paths

/Suppose that the shadows of a 3D cycle are all paths./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 19

The shadows of a 3D cycle cannot all be paths

Definition: a strand is a minimal sub-curve connecting the top and bottom faces of the bonding box.

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 20

The shadows of a 3D cycle cannot all be paths

/Claim: a lateral shadow has two (internally disjoint) strands./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 21

The shadows of a 3D cycle cannot all be paths

τ

/Suppose it has a unique strand τ./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 22

The shadows of a 3D cycle cannot all be paths

σ τ

a b

/Then, the shadow of any strand σ of the 3D cycle is τ./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 23

The shadows of a 3D cycle cannot all be paths

σ

a b

τ

′

b

′

a ′

τ

/Let τ ′ be the other lateral shadow of σ./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 24

The shadows of a 3D cycle cannot all be paths

σ

a b

τ

′

b

′

a ′

τ

/Let τ ′ be the other lateral shadow of σ./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 25

The shadows of a 3D cycle cannot all be paths

σ

a b

τ

′

b

′

a ′

τ

/The shadow of a point moving from a to b moves from a′ to b′./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 26

The shadows of a 3D cycle cannot all be paths

σ

a b

τ

′

b

′

a ′

τ

/As it keeps moving, its shadow traverses τ ′ again, from b′ to a′./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 27

The shadows of a 3D cycle cannot all be paths

σ

a b

τ

′

b

′

a ′

τ

′

σ

/Thus a second strand σ′ is found, whose shadow is again τ ′./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 28

The shadows of a 3D cycle cannot all be paths

σ τ

′

τ

′

σ

/But the shadows of σ and σ′ cannot both be τ. Contradiction!/

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 29

The shadows of a 3D cycle cannot all be paths

/Hence a shadow has two (internally disjoint) vertical strands./

The Shadows of a Cycle Cannot All Be Paths

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The shadows of a 3D cycle cannot all be paths

/By a symmetric argument, it also has two horizontal strands./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 31

The shadows of a 3D cycle cannot all be paths

/Claim: no 2D path has two vertical and two horizontal strands./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 32

The shadows of a 3D cycle cannot all be paths

/Start with a horizontal strand./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 33

The shadows of a 3D cycle cannot all be paths

/Extend it to a vertical strand./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 34

The shadows of a 3D cycle cannot all be paths

/Extend it with a second horizontal strand./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 35

The shadows of a 3D cycle cannot all be paths

/If a second vertical strand is drawn, the curve self-intersects./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 36

The shadows of a 3D cycle cannot all be paths

/The other cases are similar.../

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 37

The shadows of a 3D path can be all cycles

/An orthogonal chain whose shadows are polygons./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 38

The shadows of a 3D path can be all cycles

/A 5-segment chain whose shadows are polygons./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 39

The shadows of a 3D path can be all cycles

/Claim: no such chain has fewer than five segments./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 40

The shadows of a 3D path can be all cycles

/Suppose it has four segments (the smaller cases are trivial)./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 41

The shadows of a 3D path can be all cycles

/In each shadow, either the two endpoints coincide.../

The Shadows of a Cycle Cannot All Be Paths

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The shadows of a 3D path can be all cycles

/...Or the first and last segments overlap./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 43

The shadows of a 3D path can be all cycles

/But the two endpoints can coincide in at most one shadow./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 44

The shadows of a 3D path can be all cycles

/But the two endpoints can coincide in at most one shadow./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 45

The shadows of a 3D path can be all cycles

/In the other two shadows, the first and last segments overlap./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 46

The shadows of a 3D path can be all cycles

/Both segments lie on a plane orthogonal to the left shadow.../

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 47

The shadows of a 3D path can be all cycles

/...And to a plane orthogonal to the right shadow./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 48

The shadows of a 3D path can be all cycles

/If the two planes are not coincident, they determine a line./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 49

The shadows of a 3D path can be all cycles

/But the two segments cannot be collinear, or they overlap./

The Shadows of a Cycle Cannot All Be Paths

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The shadows of a 3D path can be all cycles

/Hence the two planes must be coincident./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 51

The shadows of a 3D path can be all cycles

/So the first and last segments are coplanar and disjoint./

The Shadows of a Cycle Cannot All Be Paths

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The shadows of a 3D path can be all cycles

/But then their top shadows are disjoint, as well. Contradiction!/

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 53

The shadows of a 3D path cannot all be convex cycles

/Suppose that the shadows of a 3D path are all convex cycles./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 54

The shadows of a 3D path cannot all be convex cycles

/Extrude one shadow to a cylindrical surface./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 55

The shadows of a 3D path cannot all be convex cycles

/Intersect it with the cylinder of another shadow./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 56

The shadows of a 3D path cannot all be convex cycles

/The 3D path must lie in the intersection./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 57

The shadows of a 3D path cannot all be convex cycles

/In general, the intersection consists of two parallel rectangles,/ /whose vertices are connected by four paths./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 58

The shadows of a 3D path cannot all be convex cycles

/The third shadow’s cylinder removes the rectangles’ interiors./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 59

The shadows of a 3D path cannot all be convex cycles

/What is left is the graph of a cube, or one of its minors./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 60

The shadows of a 3D path cannot all be convex cycles

/No embedding of a path in this graph has cycles as shadows.../

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 61

The shadows of a 3D path cannot all be convex cycles

/No embedding of a path in this graph has cycles as shadows.../

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 62

The shadows of a 3D path cannot all be convex cycles

/No embedding of a path in this graph has cycles as shadows.../

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 63

The shadows of a 3D path cannot all be convex cycles

/...Not even if the graph is a minor of the graph of a cube./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 64

The shadows of a 3D path cannot all be convex cycles

/...Not even if the graph is a minor of the graph of a cube./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 65

The shadows of a 3D path cannot all be convex cycles

/...Not even if the graph is a minor of the graph of a cube./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 66

The shadows of a 3D path cannot all be convex cycles

/...Not even if the graph is a minor of the graph of a cube./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 67

The shadows of a 3D path cannot all be convex cycles

/...Not even if the graph is a minor of the graph of a cube./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 68

The shadows of a 3D path cannot all be convex cycles

/...Not even if the graph is a minor of the graph of a cube./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 69

Generalizing Rickard’s curve to higher dimensions

/What does it mean to generalize Rickard’s curve?/

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 70

Generalizing Rickard’s curve to higher dimensions

/Note that the shadows of Rickard’s curve are contractible/ /(i.e., they deformation-retract to a point).../

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 71

Generalizing Rickard’s curve to higher dimensions

/Note that the shadows of Rickard’s curve are contractible/ /(i.e., they deformation-retract to a point).../

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 72

Generalizing Rickard’s curve to higher dimensions

/Note that the shadows of Rickard’s curve are contractible/ /(i.e., they deformation-retract to a point).../

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 73

Generalizing Rickard’s curve to higher dimensions

/Note that the shadows of Rickard’s curve are contractible/ /(i.e., they deformation-retract to a point).../

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 74

Generalizing Rickard’s curve to higher dimensions

/Note that the shadows of Rickard’s curve are contractible/ /(i.e., they deformation-retract to a point).../

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 75

Generalizing Rickard’s curve to higher dimensions

/Note that the shadows of Rickard’s curve are contractible/ /(i.e., they deformation-retract to a point).../

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 76

Generalizing Rickard’s curve to higher dimensions

/Note that the shadows of Rickard’s curve are contractible/ /(i.e., they deformation-retract to a point).../

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 77

Generalizing Rickard’s curve to higher dimensions

/...While Rickard’s curve, being a 1-sphere, is not contractible./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 78

Generalizing Rickard’s curve to higher dimensions

/Claim: there is a 2-sphere in R4 whose shadows are contractible./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 79

Generalizing Rickard’s curve to higher dimensions

t

/Think of the 4-dimensional space as a function of time./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 80

Generalizing Rickard’s curve to higher dimensions

t

/In each 3D frame, put a scaled copy of Rickard’s curve.../

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 81

Generalizing Rickard’s curve to higher dimensions

t t

= ∼ = ∼

t

/...So that the union of all frames is homeomorphic to a 2-sphere./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 82

Generalizing Rickard’s curve to higher dimensions

/The t-orthogonal shadow is the superimposition of all frames.../

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 83

Generalizing Rickard’s curve to higher dimensions

/...Which is contractible./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 84

Generalizing Rickard’s curve to higher dimensions

/...Which is contractible./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 85

Generalizing Rickard’s curve to higher dimensions

/...Which is contractible./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 86

Generalizing Rickard’s curve to higher dimensions

/...Which is contractible./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 87

Generalizing Rickard’s curve to higher dimensions

t

/In the other three shadows, each t-orthogonal slice is/ /a scaled copy of a shadow of Rickard’s curve./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 88

Generalizing Rickard’s curve to higher dimensions

t

/To contract it, first contract all slices simultaneously.../

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 89

Generalizing Rickard’s curve to higher dimensions

t

/To contract it, first contract all slices simultaneously.../

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 90

Generalizing Rickard’s curve to higher dimensions

t

/To contract it, first contract all slices simultaneously.../

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 91

Generalizing Rickard’s curve to higher dimensions

t

/To contract it, first contract all slices simultaneously.../

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 92

Generalizing Rickard’s curve to higher dimensions

t

/To contract it, first contract all slices simultaneously.../

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 93

Generalizing Rickard’s curve to higher dimensions

t

/...Then contract the resulting segment./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 94

Generalizing Rickard’s curve to higher dimensions

t

/...Then contract the resulting segment./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 95

Generalizing Rickard’s curve to higher dimensions

t

/...Then contract the resulting segment./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 96

Generalizing Rickard’s curve to higher dimensions

t

/By induction, for every d > 0, we can construct/ /a d-sphere in Rd+2 with contractible shadows./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 97

Open problems

/Note that each shadow of Rickard’s curve has four “leaves”./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 98

Open problems

/Note that each shadow of Rickard’s curve has four “leaves”./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 99

Open problems

/Problem: what is the minimum number of leaves/ /that each shadow of a Rickard-like curve must have?/

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 100

Open problems

/Our impossibility proof for the (2,2,2)-leaf case extends/ /to the (3,2,2)-leaf and the (3,3,2)-leaf cases.../

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 101

Open problems

/Our impossibility proof for the (2,2,2)-leaf case extends/ /to the (3,2,2)-leaf and the (3,3,2)-leaf cases.../

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 102

Open problems

/Our impossibility proof for the (2,2,2)-leaf case extends/ /to the (3,2,2)-leaf and the (3,3,2)-leaf cases.../

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 103

Open problems

/...But it does not extend to the (4,2,2)-leaf case and above./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 104

Open problems

/...But it does not extend to the (4,2,2)-leaf case and above./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 105

Open problems

/In this example, the right shadow is a path with a unique strand./

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 106

Open problems

/Problem: what if the curves cast four shadows instead of three? Can the four shadows of a cycle be trees? Can the four shadows of a path be cycles?/

The Shadows of a Cycle Cannot All Be Paths

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SLIDE 107

Summary

The shadows of a cycle in R3:

can be all trees cannot all be paths

The shadows of a path in R3:

can be all cycles cannot all be convex cycles

The shadows of a d-sphere in Rd+2:

can be all contractible

The Shadows of a Cycle Cannot All Be Paths