Cube Tiling Research Articles

The Resolution of Keller’s Conjecture

We consider three graphs, $$G_{7,3}$$ , $$G_{7,4}$$ , and $$G_{7,6}$$ , related to Keller’s conjecture in dimension 7. The conjecture is false for this dimension if and only if at least one of the graphs contains a clique of size $$2^7 = 128$$ . We present an automated method to solve this conjecture by encoding the existence of such a clique as a propositional formula. We apply satisfiability solving combined with symmetry-breaking techniques to determine that no such clique exists. This result implies that every unit cube tiling of $$\mathbb {R}^7$$ contains a facesharing pair of cubes. Since a faceshare-free unit cube tiling of $$\mathbb {R}^8$$ exists (which we also verify), this completely resolves Keller’s conjecture.

Gluing and Cutting Cube Tiling Codes in Dimension Six

Let S be a set of arbitrary objects, and let $$s\mapsto s'$$ be a permutation of S such that $$s''=(s')'=s$$ and $$s'\ne s$$ . Let $$S^d=\{v_1\dots v_d:v_i\in S\}$$ . Two words $$v,w\in S^d$$ are dichotomous if $$v_i=w'_i$$ for some $$i\in [d]=\{1,\dots ,d\}$$ , and they form a twin pair if $$v_i'=w_i$$ and $$v_j=w_j$$ for every $$j\in [d]\setminus \{i\}$$ . A polybox code is a set $$V\subset S^d$$ in which every two distinct words are dichotomous. A polybox code V is a cube tiling code if $$|V|=2^d$$ . A 2-periodic cube tiling of $$\mathbb {R}^d$$ and a cube tiling of flat torus $$\mathbb {T}^d$$ can be encoded in a form of a cube tiling code. A twin pair v, w in which $$v_i=w_i'$$ is glue (at the ith position) if the pair v, w is replaced by one word u such that $$u_j=v_j=w_j$$ for every $$j\in [d]\setminus \{i\}$$ and $$u_i=*$$ , where $$*\notin S$$ is some extra fixed symbol. A word u with $$u_i=*$$ is cut (at the ith position) if u is replaced by a twin pair q, t such that $$q_i=t_i'\ne *$$ and $$u_j=q_j=t_j$$ for every $$j\in [d]\setminus \{i\}$$ . If $$V,W\subset S^d$$ are two cube tiling codes and there is a sequence of twin pairs which can be interchangeably gluing and cutting in a way which allows us to pass from V to W, then we say that W is obtained from V by gluing and cutting. In the paper it is shown that for every two cube tiling codes in dimension six one can be obtained from the other by gluing and cutting.

Towards resolving Keller’s cube tiling conjecture in dimension seven

Abstract A cube tiling of ℝ d is a family of pairwise disjoint cubes [0, 1) d + T = {[0, 1) d + t: t ∈ T} such that ∪ t∈T ([0, 1) d + t) = ℝ d . Two cubes [0, 1) d + t, [0, 1) d + s are called a twin pair if |t j −s j | = 1 for some j ∈ [d] = {1, ⋅, d} and t i = s i for every i ∈ [d]∖{j}. In 1930, Keller conjectured that in every cube tiling of ℝ d there is a twin pair. For x ∈ ℝ d and i ∈ [d], let L(T, x, i) be the set of all ith coordinates t i of vectors t ∈ T such that ([0, 1) d + t)∩([0, 1] d + x)≠∅ and t i ≤ x i . Let r − ( T ) = min x ∈ R d max 1 ≤ i ≤ d | L ( T , x , i ) | $r^-(T)=\min_{x\in \mathbb{R}^d} \max_{1\leq i\leq d}|L(T,x,i)|$ and r + ( T ) = max x ∈ R d max 1 ≤ i ≤ d | L ( T , x , i ) | $r^ + (T)=\max_{x\in \mathbb{R}^d} \max_{1\leq i\leq d}|L(T,x,i)|$ . Before 2019 it was known that Keller’s conjecture is true for dimensions d ≤ 6 and false for all dimensions d = 8. Moreover, in dimension 7 it was known to be true if r −(T) ≤ 2 or r +(T) = 5. The present paper resolves the case r +(T) = 4. At the end of 2019, when the paper was still under review, Brakensiek et al. resolved the cases r +(T) ∈ {3, 4, 6}, proving thereby Keller’s conjecture in dimension 7.

Hyperball packings related to truncated cube and octahedron tilings in hyperbolic space

In this paper, we study congruent and noncongruent hyperball (hypersphere) packings to the truncated regular cube and octahedron tilings. These are derived from the Coxeter truncated orthoscheme tilings $\{4,3,p\}$ $(6< p \in \mathbb{N})$ and $\{3,4,p\}$ $(4< p \in \mathbb{N})$, respectively, by their Coxeter reflection groups in hyperbolic space $\mathbb{H}^{3}$. We determine the densest hyperball packing arrangement and its density with congruent and noncongruent hyperballs. We prove that the locally densest (noncongruent half) hyperball configuration belongs to the truncated cube with a density of approximately $0.86145$ if we allow $6< p \in \mathbb{R}$ for the dihedral angle $2\pi/p$. This local density is larger than the B\or\oczky--Florian density upper bound for balls and horoballs. But our locally optimal noncongruent hyperball packing configuration cannot be extended to the entire hyperbolic space $\mathbb{H}^3$. We determine the extendable densest noncongruent hyperball packing arrangement to the truncated cube tiling $\{4,3,p=7\}$ with a density of approximately $0.84931$.

Hyperball packings related to truncated cube and octahedron tilings in hyperbolic space

In this paper, we study congruent and noncongruent hyperball (hypersphere) packings to the truncated regular cube and octahedron tilings. These are derived from the Coxeter truncated orthoscheme tilings $\{4,3,p\}$ $(6&lt; p \in \mathbb{N})$ and $\{3,4,p\}$ $(4&lt; p \in \mathbb{N})$, respectively, by their Coxeter reflection groups in hyperbolic space $\mathbb{H}^{3}$. We determine the densest hyperball packing arrangement and its density with congruent and noncongruent hyperballs. We prove that the locally densest (noncongruent half) hyperball configuration belongs to the truncated cube with a density of approximately $0.86145$ if we allow $6&lt; p \in \mathbb{R}$ for the dihedral angle $2\pi/p$. This local density is larger than the B\"or\"oczky--Florian density upper bound for balls and horoballs. But our locally optimal noncongruent hyperball packing configuration cannot be extended to the entire hyperbolic space $\mathbb{H}^3$. We determine the extendable densest noncongruent hyperball packing arrangement to the truncated cube tiling $\{4,3,p=7\}$ with a density of approximately $0.84931$.

On the structure of cube tiling codes

Let S be a set of arbitrary objects, and let Sd={v1...vd:vi∈S}. A polybox code is a set V⊂Sd with the property that for every two words v,w∈V there is i∈[d] with vi′=wi, where a permutation s↦s′ of S is such that s′′=(s′)′=s and s′≠s. If |V|=2d, then V is called a cube tiling code. Cube tiling codes determine 2-periodic cube tilings of Rd or, equivalently, tilings of the flat torus Td={(x1,…,xd)(mod2):(x1,…,xd)∈Rd} by translates of the unit cube as well as r-perfect codes in Z4r+2d in the maximum metric. By a structural result, cube tiling codes for d=4 are enumerated. It is computed that there are 27,385 non-isomorphic cube tiling codes in dimension four, and the total number of such codes is equal to 17,794,836,080,455,680. Moreover, some procedure of passing from a cube tiling code to a cube tiling code in dimensions d≤5 is given.

Rigid polyboxes and keller's conjecture

Abstract A cube tiling of ℝ d is a family of axis-parallel pairwise disjoint cubes [0,1) d + T = {[0,1) d +t : t ∈ T} that cover ℝ d . Two cubes [0,1) d + t, [0,1) d + s are called a twin pair if their closures have a complete facet in common. In 1930, Keller conjectured that in every cube tiling of ℝ d there is a twin pair. Keller's conjecture is true for dimensions d ≤ 6 and false for all dimensions d ≥ 8. For d = 7 the conjecture is still open. Let x ∈ ℝ d , i ∈ [d], and let L(T, x, i) be the set of all ith coordinates ti of vectors t ∈ T such that ([0,1) d +t) ∩ ([0,1] d +x) ≠ ø and ti ≤ xi . Let r −(T) = min x∈ℝ d max1≤i≤d |L(T,x,i)| and r +(T) = max x∈ℝ d max1≤i≤d |L(T,x,i)|. It is known that Keller's conjecture is true in dimension seven for cube tilings [0,1)7 + T for which r −(T) ≤ 2. In the present paper we show that it is also true for d = 7 if r +(T) ≥ 6. Thus, if [0,1) d + T is a counterexample to Keller's conjecture in dimension seven, then r −(T), r +(T) ∈ {3, 4, 5}.

On Keller’s Conjecture in Dimension Seven

A cube tiling of $\mathbb{R}^d$ is a family of pairwise disjoint cubes $[0,1)^d+T=\{[0,1)^d+t:t\in T\}$ such that $\bigcup_{t\in T}([0,1)^d+t=\mathbb{R}^d$. Two cubes $[0,1)^d+t$, $[0,1)^d+s$ are called a twin pair if $|t_j-s_j|=1$ for some $j\in [d]=\{1,\ldots, d\}$ and $t_i=s_i$ for every $i\in [d]\setminus \{j\}$. In $1930$, Keller conjectured that in every cube tiling of $\mathbb{R}^d$ there is a twin pair. Keller's conjecture is true for dimensions $d\leq 6$ and false for all dimensions $d\geq 8$. For $d=7$ the conjecture is still open. Let $x\in \mathbb{R}^d$, $i\in [d]$, and let $L(T,x,i)$ be the set of all $i$th coordinates $t_i$ of vectors $t\in T$ such that $([0,1)^d+t)\cap ([0,1]^d+x)\neq \emptyset$ and $t_i\leq x_i$. Let $r^-(T)=\min_{x\in \mathbb{R}^d}\; \max_{1\leq i\leq d}|L(T,x,i)|$ and $r^+(T)=\max_{x\in \mathbb{R}^d}\; \max_{1\leq i\leq d}|L(T,x,i)|$. It is known that if $r^-(T)\leq 2$ or $r^+(T)\geq 6$, then Keller's conjecture is true for $d=7$. In the present paper we show that it is also true for $d=7$ if $r^+(T)=5$. Thus, if $[0,1)^d+T$ is a counterexample to Keller's conjecture in dimension seven, then $r^-(T),r^+(T)\in \{3,4\}$.

Partitions and balanced matchings of an [formula omitted]-dimensional cube

A non-empty set A⊆X=X1×⋯×Xn is a box if A=A1×⋯×An and Ai⊆Xi for each i∈[n]. Two boxes A,B⊂X are dichotomous if Ai=Xi∖Bi for some i∈[n]. Using a cube tiling code of Rn designed by Lagarias and Shor, a certain class of partitions of an n-dimensional cube into 2n pairwise dichotomous boxes is constructed. Additionally, for every prime number n≥3 perfect matchings of the graph of the unit cube [0,1]n with faulty vertices (0,…,0) and (1,…,1) in which the number of edges in every direction is the same are presented.

Enumerating Cube Tilings

Cube tilings formed by $$n$$ -dimensional $$4\mathbb Z ^n$$ -periodic hypercubes with side $$2$$ and integer coordinates are considered here. By representing the problem of finding such cube tilings within the framework of exact cover and using canonical augmentation, pairwise nonisomorphic 5-dimensional cube tilings are exhaustively enumerated in a constructive manner. There are 899,710,227 isomorphism classes of such tilings, and the total number of tilings is 638,560,878,292,512. It is further shown that starting from a 5-dimensional cube tiling and using a sequence of switching operations, it is possible to generate any other cube tiling.

The Structure of Cube Tilings Under Symmetry Conditions

Let m 1,…,m d be positive integers, and let G be a subgroup of ℤ d such that m 1ℤ×⋯×m d ℤ⊆G. It is easily seen that if a unit cube tiling [0,1) d +t,t∈T, of ℝ d is invariant under the action of G, then for every t∈T, the number |T∩(t+ℤ d )∩[0,m 1)×⋯×[0,m d )| is divisible by |G|. We give sufficient conditions under which this number is divisible by a multiple of |G|. Moreover, a relation between this result and the Minkowski–Hajos theorem on lattice cube tilings is discussed.

On the structure of cube tilings of [formula omitted] and [formula omitted

A family of translates of the unit cube [0,1)d+T={[0,1)d+t:t∈T}, T⊂Rd, is called a cube tiling of Rd if cubes from this family are pairwise disjoint and ⋃t∈T[0,1)d+t=Rd. A non-empty set B=B1×⋯×Bd⊆Rd is a block if there is a family of pairwise disjoint unit cubes [0,1)d+S, S⊂Rd, such that B=⋃t∈S[0,1)d+t and for every t,t′∈S there is i∈{1,…,d} such that ti−ti′∈Z∖{0}. A cube tiling of Rd is blockable if there is a finite family of disjoint blocks B, |B|>1, with the property that every cube from the tiling is contained in exactly one block of the family B. We construct a cube tiling T of R4 which, in contrast to cube tilings of R3, is not blockable. We give a new proof of the theorem saying that every cube tiling of R3 is blockable.

The Coin Exchange Problem and the Structure of Cube Tilings

It is shown that if $[0,1)^d+t$, $t\in T$, is a unit cube tiling of $\mathbb{R}^d$, then for every $x\in T$, $y\in \mathbb{R}^d$, and every positive integer $m$ the number $|T\cap (x+\mathbb{Z}^d)\cap([0,m)^d+ y)|$ is divisible by $m$. Furthermore, by a result of Coppersmith and Steinberger on cyclotomic arrays it is proven that for every finite discrete box $D=D_1\times\cdots\times D_d \subseteq x+\mathbb{Z}^d$ of size $m_1\times \cdots\times m_d$ the number $|D\cap T|$ is a linear combination of $m_1,\ldots, m_d$ with non-negative integer coefficients. Several consequences are collected. A generalization is presented.

On the structure of cube tilings and unextendible systems of cubes in low dimensions

We give a structural description of cube tilings and unextendible cube packings of R3. We also prove that up to dimension 4 each cylinder of a cube tiling contains a column and demonstrate by an example that the latter result does not hold in dimension 5.

A Cube Tiling of Dimension Eight with No Facesharing

A cube tiling of eight-dimensional space in which no pair of cubes share a complete common seven-dimensional face is constructed. Together with a result of Perron, this shows that the first dimension in which such a tiling can exist is seven or eight.

Cube tiling and covering a complete graph

A collection of translates of an n-dimensional cube forms a tiling if the n-space is covered by its elements and no two cubes have a common interior point. If two n-dimensional cubes share a whole common (n−1)-dimensional face then they are called a twin pair. In 1930 Keller conjectured that in any n-dimensional cube tiling there is a twin pair. In 1982 Lawrence, using geometrical argumentation, gave a graph theoretical form for Keller's conjecture. We derive a slightly different graph test from its group theoretical version

A reduction of Keller's conjecture

A family of translates of a closedn-dimensional cube is called a cube tiling if the union of the cubes is the wholen-space and their interiors are disjoint. According to a famous unsolved conjecture of O. H. Keller, two of the cubes in ann-dimensional cube tiling must share a complete (n − 1)-dimensional face. In this paper we shall prove that to solve Keller's conjecture it is sufficient to examine certain factorizations of direct sum of finitely many cyclic group of order four.