Cube Packing Research Articles

Three Cube Packing for All Dimensions

Let Vn(d) denote the least number, such that every collection of n d-cubes with total volume 1 in d-dimensional (Euclidean) space can be packed parallelly into some d-box of volume Vn(d). We show that V3(d)=r1−dd if d≥11 and V3(d)=1r+1rd+1r−rd+1 if 2≤d≤10, where r is the only solution of the equation 2(d−1)kd+dkd−1=1 on 22,1 and (k+1)d(1−k)d−1dk2+d+k−1=kddkd+1+dkd+kd+1 on 22,1, respectively. The maximum volume is achieved by hypercubes with edges x, y, z, such that x=2rd+1−1/d, y=z=rx if d≥11, and x=rd+(1r−r)d+1−1/d, y=rx, z=(1r−r)x if 2≤d≤10. We also proved that only for dimensions less than 11 are there two different maximum packings, and for all dimensions greater than 10, the maximum packing has the same two smallest cubes.

Packing of non-blocking four-dimensional cubes into the unit cube

Packing of non-blocking four-dimensional cubes into the unit cube

Packing of non-blocking cubes into the unit cube

Any collection of non-blocking cubes, whose total volume does not exceed 1/3, can be packed into the unit cube.

Online Bin Packing of Squares and Cubes

In the d-dimensional online bin packing problem, d-dimensional cubes of positive sizes no larger than 1 are presented one by one to be assigned to positions in d-dimensional unit cube bins. In this work, we provide improved upper bounds on the asymptotic competitive ratio for square and cube bin packing problems, where our bounds do not exceed 2.0885 and 2.5735 for square and cube packing, respectively. To achieve these results, we adapt and improve a previously designed harmonic-type algorithm, and apply a different method for defining weight functions. We detect deficiencies in the state-of-the-art results by providing counter-examples to the current best algorithms and their analysis, where the claimed bounds were 2.1187 for square packing and 2.6161 for cube packing.

Математичні моделі та алгоритми оптимальної упаковки куль та кубів у сферичний та кубічний контейнери

Розглянуто математичні моделі та алгоритми оптимальної збалансованоїрозрідженої упаковки куль та кубів у сферичний та кубічний контейнери.Збалансованою розрідженою (задаються допустимі відстані між обʼєктами)упаковкою об’єктів у зовнішній контейнер є така їх упаковка, коли центрваги сімейства об’єктів збігається з центром зовнішнього контейнера, авідстані між об’єктами та відстані від них до зовнішнього контейнера булине менші за наперед задані величини. Наведено математичні моделі, послідовні та паралельні алгоритми розв’язання задач знаходження збалансованої розрідженої упаковки куль різних радіусів у сферичний та кубічнийконтейнери. Наведено математичну модель задачі знаходження збалансованої розрідженої упаковки кубів у куб мінімального обʼєму за умови, щосторони всіх кубів паралельні осям координат, та опис негладкої штрафноїфункції для пошуку локальних мінімумів задачі. Досліджувані задачівідносяться до класу NP-важких задач. Математичні моделі представленібагатоекстремальними задачами нелінійного програмування. Для пошукунайкращого допустимого розвʼязку застосовується метод мультистарту усполученні з r-алгоритмом Шора. Для цього задача зводиться до задачібезумовної оптимізації за допомогою штрафних функцій у вигляді функціймаксимуму, а для пошуку локальних мінімумів із набору стартових точокзастосовуються методи мінімізації негладких функцій, що базуються навикористанні програмних реалізацій r-алгоритму. Математичні моделі тапослідовні і паралельні алгоритми, що розглядаються, можна використатидля розробки програмних засобів розв’язування задач знаходження збалансованої розрідженої упаковки сферичних та кубічних обʼєктів у сферичніта кубічні контейнери. Матеріал представлено в трьох розділах. У розд. 1 наведено математичну модель та алгоритми розв’язання задачі знаходження збалансованої розрідженої упаковки куль різних радіусів у сферичнийконтейнер. Описано послідовний та паралельний алгоритми знаходженнянайкращого допустимого розв’язку задач. У розд. 2 наведено математичну модель та алгоритми розв’язання задачі знаходження збалансованої розрідженоїупаковки куль різних радіусів у кубічний контейнер. Описано послідовний тапаралельний алгоритми знаходження найкращого допустимого розв’язкузадач. У розд. 3 наведено математичну модель задачі знаходження збалансованої розрідженої упаковки кубів у кубічний контейнер. Наведено описнегладкої штрафної функції для пошуку локальних мінімумів задачі.

Vibration-driven fabrication of dense architectured panels

<h2>Summary</h2> Self-assembly at the macro-scale is a promising pathway for fabrication, but the assembly process and mechanisms are still poorly understood. We examine the vibration-induced assembly of hard cubic grains as a potential route for the rapid fabrication of architectured materials and structures. We performed assembly experiments with various combinations of vibration amplitudes and frequencies to map the different states of the system. The results show that the acceleration normalized by gravity cannot fully capture the phase transitions or the mechanisms governing cubes packing and that amplitude and frequency must be considered independently. We used discrete element modeling to duplicate experiments and then single-grain models to find the effective mechanisms involved in the packing and phase transition of cubes. Both cube rotation and bouncing govern packing, while bouncing has an additional role in the phase transition. These findings provide guidelines for the assembly of complex materials, for example, topologically interlocked materials.

Crystal structure of the new silicide LaNi11.8–11.4Si1.2–1.6

Abstract The intermetallic compound LaNi11.8–11.4Si1.2–1.6 was synthesized by arc-melting and its crystal structure was determined using powder and single-crystal X-ray diffraction data. The compound adopts the cubic CaCu6.5Al6.5-type structure (space group Fm 3 ‾ $\bar{3}$ c, Pearson code cF112, Z = 8), which is a partially ordered ternary derivative of the NaZn13 type: a = 11.256(4) Å, V = 1426.1(15) Å3, R = 0.0133, wR = 0.0285 for 93 reflections with I &gt; 2 σ(I) for LaNi11.4Si1.6; a = 11.25486(8) Å, V = 1425.68(2) Å3, R p = 4.17%, R wp = 5.85%, R B = 3.44% for LaNi11.8Si1.2. One of its crystallographic positions (96i) is occupied by a mixture of Ni and Si atoms. The structure of this new silicide can be represented as a packing of Ni-centered icosahedra and La-centered snub cubes, which are packed in a CsCl-related manner.

A note on perfect packing of squares and cubes

We show that squares of sides of length $$ 1$$ , $$2^{-t}$$ , $$3^{-t}$$ , $$4^{-t}$$ , $$\ldots $$ can be packed perfectly into a square, provided $$ 1/2 < t \le 2/3$$ . Moreover, we prove that cubes of edges of length $$ 1$$ , $$2^{-t}$$ , $$3^{-t}$$ , $$4^{-t}$$ , $$ \ldots $$ can be packed perfectly into a box, provided $$ 2/3 < t \le 4/11$$ .

CUBE PACKINGS IN EUCLIDEAN SPACES

In this paper, we study some cube packing problems. In particular, we are interested in compact subsets of R n , n ⩾ 2 , which contain boundaries of cubes with all side lengths in (0,1). We show here that such sets must have lower box dimension at least n − 0.5 , and we will also provide sharp examples. We also show here that such sets must be large in general in a precise sense which is also introduced in this paper.

On Parallel Packing and Covering of Squares and Cubes

Suppose that T is a right triangle with leg lengths 1 and $$\sqrt{2}$$, $$T_{r}$$ is a trirectangular tetrahedron with three right-angle edges lengths 1, 1 and $$\sqrt{2}$$. And let {$$S_{n}$$} be a sequence of the homothetic copies of a square S with a side parallel to some leg of T, {$$C_{n}$$} be a sequence of the homothetic copies of a cube C with a face parallel to the base face of $$T_{r}$$. We first determine the bound of sums of areas of squares from the sequence {$$S_{n}$$} that permits a parallel packing of T. Then we show the bound of sums of volumes of cubes from the sequence {$$C_{n}$$} that permits a parallel covering of $$T_{r}$$.

Efficient online packing of 4-dimensional cubes into the unit cube

Any sequence of 4-dimensional cubes of total volume not greater than 1/8 can be online packed into the unit cube.

Perfect packing of cubes

It is known that $${\sum_{i =1}^\infty {1/ i^2}={\pi^2/6}}$$ . We can ask what is the smallest $${\epsilon}$$ such that all the squares of sides of length $${1, 1/2, 1/3, \ldots}$$ can be packed into a rectangle of area $${{\pi^2/6}+\epsilon}$$ . A packing into a rectangle of the right area is called perfect packing. Chalcraft [4] packed the squares of sides of length $${1, 2^{-t}, 3^{-t}, \ldots}$$ and he found perfect packings for $${1/2 < t \le 3/5}$$ . We generalize this problem and pack the 3-dimensional cubes of sides of length $${1, 2^{-t}, 3^{-t}, \ldots}$$ into a right rectangular prism of the right volume. Moreover we show that there is a perfect packing for all t in the range $${0.36273 \le t \le 4/11}$$ .

Octanuclear Ni(ii) cubes based on halogen-substituted pyrazolates: synthesis, structure, electrochemistry and magnetism

The employment of halogen-substituted pyrazole ligands in nickel(II) clusters afforded three anionic Ni(II) cubes, (HNEt3)2[Ni8(Xpz)12(OH)6] (X = Cl, 1·2CH3CN; X = Br, 2·2CH3CN; X = I, 3·12CH3CN; pz = pyrazolate). Clusters 1–3 are isolated as dianionic compounds which have similar cube geometry with each vertex occupied by a Ni(II) atom and eight μ4-OH− capped on eight square faces. The Xpz ligands adopt a bidentate mode to bind paired Ni(II) atoms on the edge of the cube. Interestingly, the three Ni8 cubes have similar symmetry, size and shape but pack together to form different lattice symmetry, which should be dictated by the halogen-related supramolecular interactions (C–X⋯H, C–X⋯π and X⋯X halogen bonding) in different crystals. The electrochemistry of 1–3 showed a Ni(II)/Ni(I) redox couple with E1/2 at ca. 900 mV and the oxidization potential order is 1 > 3 > 2 depending on the halogen substituents. 1–3 exhibit excellent electrocatalytic performances toward the oxidation of nitrite. The different packing of Ni8 cubes in 1–3 also has an important influence on their magnetic behaviors. Complex 2 showed antiferromagnetic couplings between the Ni(II) ions within the cubes, whereas 1 and 3 exclusively exhibited mixed ferromagnetic and antiferromagnetic properties, leading to frustration and typical spin glass behavior.

A Simple Expression for the Tortuosity of Gas Transport Paths in Solid Oxide Fuel Cells’ Porous Electrodes

Based on the three-dimensional (3D) cube packing model, a simple expression for the tortuosity of gas transport paths in solid oxide fuel cells’ (SOFC) porous electrodes is developed. The proposed tortuosity expression reveals the dependence of the tortuosity on porosity, which is capable of providing results that are very consistent with the experimental data in the practical porosity range of SOFC. Furthermore, for the high porosity (&gt;0.6), the proposed tortuosity expression is also accurate. This might be helpful for understanding the physical mechanism for the tortuosity of gas transport paths in electrodes and the optimization electrode microstructure for reducing the concentration polarization.

Formation and liquid permeability of dense colloidal cube packings.

The liquid permeability of dense random packings of cubic colloids with rounded corners is studied for solid hematite cubes and hollow microporous silica cubes. The permeabilities of these two types of packings are similar, confirming that the micropores in the silica shell of the hollow cubes do not contribute to the permeability. From the Brinkman screening length √k of ∼16 nm, we infer that the relevant pores are indeed intercube pores. Furthermore, we relate the permeability to the volume fraction and specific solid volume of the cubes using the Kozeny-Carman relation. The Kozeny-Carman relation contains a constant that accounts for the topology and size distribution of the pores in the medium. The constant obtained from our study with aspherical particles is of the same order of magnitude as those from studies with spherical and ellipsoidal particles, which supports the notion that the Kozeny-Carman relation is applicable for any dense particle packing with (statistically) isotropic microstructures, irrespective of the particle shape.

New results on torus cube packings and tilings

We consider the sequential random packing of integral translates of cubes [0, N]n into the torus ℤn/2Nℤn. Two particular cases are of special interest: (1) N = 2, which corresponds to a discrete case of tilings, and (2) N = ∞, which corresponds to a case of continuous tilings. Both cases correspond to some special combinatorial structure, and we describe here new developments.

New Results on Torus Cube Packings and Tilings

We consider sequential random packing of integral translate of cubes $[0,N]^n$ into the torus $Z^n / 2NZ^n$. Two special cases are of special interest: (i) The case $N=2$ which corresponds to a discrete case of tilings (considered in \cite{cubetiling,book}) (ii) The case $N=\infty$ corresponds to a case of continuous tilings (considered in \cite{combincubepack,book}) Both cases correspond to some special combinatorial structure and we describe here new developments.

Robust Approximation Schemes for Cube Packing

Square bin packing, where square items are to be packed into a minimum number of unit size squares, and its multidimensional generalization, cube packing, where $d$-dimensional cubic items are to be packed into a minimum number of unit size cubes of the same dimension, are well-studied generalizations of the classical one-dimensional bin packing problem. These problems are known to have asymptotic polynomial time approximation schemes (APTAS). In this paper we design robust approximation schemes for these problems. At each step, a robust algorithm receives a single input item to be added to the packing. The new item must be packed and the packing can be modified, but the total volume of items which may migrate between bins, or change their positions inside bins, must be bounded by a constant factor times the volume of the new item. That is, the solution is created by slightly adjusting the solution upon the arrival of each new item. Previous approximation schemes partition items into three types according...

Rigidity and the chessboard theorem for cube packings

We are concerned with subsets of Rd that can be tiled with translates of the half-open unit cube in a unique way. We call them rigid sets. We show that the set tiled with [0,1)d+s, s∈S, is rigid if for any pair of distinct vectors t, t′∈S the number |{i:|ti−ti′|=1}| is even whenever t−t′∈{−1,0,1}d. As a consequence, we obtain the chessboard theorem which reads that for each packing [0,1)d+s, s∈S, of Rd, there is an explicitly defined partition {S0,S1} of S such that the sets tiled with the systems [0,1)d+s, s∈Si, where i=0,1, are rigid. The technique developed in the paper is also applied to demonstrate certain structural results concerning cube tilings of Rd.

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.