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Pach's animal problem within the bounding box

Martin Tancer

Abstract

A collection of unit cubes with integer coordinates in $\mathbb R^3$ is an animal if its union is homeomorphic to the 3-ball. Pach's animal problem asks whether any animal can be transformed to a single cube by adding or removing cubes one by one in such a way that any intermediate step is an animal as well. Here we provide an example of an animal that cannot be transformed to a single cube this way within its bounding box.

Pach's animal problem within the bounding box

Abstract

A collection of unit cubes with integer coordinates in is an animal if its union is homeomorphic to the 3-ball. Pach's animal problem asks whether any animal can be transformed to a single cube by adding or removing cubes one by one in such a way that any intermediate step is an animal as well. Here we provide an example of an animal that cannot be transformed to a single cube this way within its bounding box.
Paper Structure (17 sections, 6 theorems, 1 equation, 17 figures)

This paper contains 17 sections, 6 theorems, 1 equation, 17 figures.

Table of Contents

  1. Introduction
  2. Pach's animal problem.
  3. Pach's animal problem within the bounding box.
  4. Sketch of the construction (mostly in dimension 2)
  5. Furch's construction.
  6. A 2-dimensional example.
  7. Full construction
  8. Box filling curves.
  9. The first expansion.
  10. The second expansion.
  11. Bends and straight segments after the second expansion.
  12. Doubling.
  13. Correctness of the construction.
  14. $A$ is an animal
  15. Additions and removals
  16. ...and 2 more sections

Key Result

Theorem 1

There is an animal $A$ such that it cannot be transformed to a single cube by additions or removals of cubes which are inside the bounding box of $A$. In fact, if we remove a cube from $A$ or add a cube to $A$ contained inside the bounding box, we never obtain an animal.

Figures (17)

  • Figure 1: Furch's knotted ball. All displayed cubes are removed from the box except the dark one. The picture we provide here is very similar to a picture in ziegler98.
  • Figure 2: The first expansion of a 2-dimensional example.
  • Figure 3: The second expansion of a 2-dimensional example. The squares on both sides of the picture should be understood as unit squares. The dimensions of the right right picture are $17 \times 27$ but it is shrunk due to space constraints.
  • Figure 4: Left: Joining the construction from Figure \ref{['f:second_expansion']} with its mirror copy. Now the dimensions are $17 \times 55$. Right: After adding or removing the squares in green we still have a 2-dimensional animal.
  • Figure 5: U-turn.
  • ...and 12 more figures

Theorems & Definitions (10)

  • Theorem 1
  • Lemma 2
  • Lemma 3
  • proof
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • proof : Proof of Lemma \ref{['l:red']}.
  • proof : Proof of Lemma \ref{['l:white']}.
  • proof : Proof of Lemma \ref{['l:black']}