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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Generalizing Rickard’s curve to higher dimensions / Claim: there is a 2-sphere in R 4 whose shadows are contractible./ The Shadows of a Cycle Cannot All Be Paths
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
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
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
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
Generalizing Rickard’s curve to higher dimensions /...Which is contractible./ The Shadows of a Cycle Cannot All Be Paths
Generalizing Rickard’s curve to higher dimensions /...Which is contractible./ The Shadows of a Cycle Cannot All Be Paths
Generalizing Rickard’s curve to higher dimensions /...Which is contractible./ The Shadows of a Cycle Cannot All Be Paths
Generalizing Rickard’s curve to higher dimensions /...Which is contractible./ The Shadows of a Cycle Cannot All Be Paths
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
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
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
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
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
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
Generalizing Rickard’s curve to higher dimensions t /...Then contract the resulting segment./ The Shadows of a Cycle Cannot All Be Paths
Generalizing Rickard’s curve to higher dimensions t /...Then contract the resulting segment./ The Shadows of a Cycle Cannot All Be Paths
Generalizing Rickard’s curve to higher dimensions t /...Then contract the resulting segment./ The Shadows of a Cycle Cannot All Be Paths
Generalizing Rickard’s curve to higher dimensions t /By induction, for every d > 0, we can construct/ /a d -sphere in R d +2 with contractible shadows./ The Shadows of a Cycle Cannot All Be Paths
Open problems /Note that each shadow of Rickard’s curve has four “leaves”./ The Shadows of a Cycle Cannot All Be Paths
Open problems /Note that each shadow of Rickard’s curve has four “leaves”./ The Shadows of a Cycle Cannot All Be Paths
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
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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