Related Research Articles
In geometry, a cube is a three-dimensional solid object bounded by six square faces. It has twelve edges and eight vertices. It can be represented as a rectangular cuboid with six square faces, or a parallelepiped with equal edges. It is an example of many type of solids: Platonic solid, regular polyhedron, parallelohedron, zonohedron, and plesiohedron. The dual polyhedron of a cube is the regular octahedron.
In geometry, rhombicuboctahedron is an Archimedean solid with 26 faces, consisting of 8 equilateral triangles and 18 squares. It is named by Johannes Kepler in his 1618 Harmonices Mundi, being short for truncated cuboctahedral rhombus, with cuboctahedral rhombus being his name for a rhombic dodecahedron.
An impossible object is a type of optical illusion that consists of a two-dimensional figure which is instantly and naturally understood as representing a projection of a three-dimensional object but cannot exist as a solid object. Impossible objects are of interest to psychologists, mathematicians and artists without falling entirely into any one discipline.
Packing problems are a class of optimization problems in mathematics that involve attempting to pack objects together into containers. The goal is to either pack a single container as densely as possible or pack all objects using as few containers as possible. Many of these problems can be related to real-life packaging, storage and transportation issues. Each packing problem has a dual covering problem, which asks how many of the same objects are required to completely cover every region of the container, where objects are allowed to overlap.
Ronald Lewis Graham was an American mathematician credited by the American Mathematical Society as "one of the principal architects of the rapid development worldwide of discrete mathematics in recent years". He was president of both the American Mathematical Society and the Mathematical Association of America, and his honors included the Leroy P. Steele Prize for lifetime achievement and election to the National Academy of Sciences.
A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of geometries.
The Kepler conjecture, named after the 17th-century mathematician and astronomer Johannes Kepler, is a mathematical theorem about sphere packing in three-dimensional Euclidean space. It states that no arrangement of equally sized spheres filling space has a greater average density than that of the cubic close packing and hexagonal close packing arrangements. The density of these arrangements is around 74.05%.
In mathematics, a perfect magic cube is a magic cube in which not only the columns, rows, pillars, and main space diagonals, but also the cross section diagonals sum up to the cube's magic constant.
In mathematics, a P-multimagic square is a magic square that remains magic even if all its numbers are replaced by their kth powers for 1 ≤ k ≤ P. 2-multimagic squares are called bimagic, 3-multimagic squares are called trimagic, 4-multimagic squares tetramagic, and 5-multimagic squares pentamagic.
A polycube is a solid figure formed by joining one or more equal cubes face to face. Polycubes are the three-dimensional analogues of the planar polyominoes. The Soma cube, the Bedlam cube, the Diabolical cube, the Slothouber–Graatsma puzzle, and the Conway puzzle are examples of packing problems based on polycubes.
Klaus Friedrich Roth was a German-born British mathematician who won the Fields Medal for proving Roth's theorem on the Diophantine approximation of algebraic numbers. He was also a winner of the De Morgan Medal and the Sylvester Medal, and a Fellow of the Royal Society.
David Breyer Singmaster was an American-British mathematician who was emeritus professor of mathematics at London South Bank University, England. He had a huge personal collection of mechanical puzzles and books of brain teasers. He was most famous for being an early adopter and enthusiastic promoter of the Rubik's Cube. His Notes on Rubik's "Magic Cube" which he began compiling in 1979 provided the first mathematical analysis of the Cube as well as providing one of the first published solutions. The book contained his cube notation which allowed the recording of Rubik's Cube moves, and which quickly became the standard.
Four-dimensional space (4D) is the mathematical extension of the concept of three-dimensional space (3D). Three-dimensional space is the simplest possible abstraction of the observation that one needs only three numbers, called dimensions, to describe the sizes or locations of objects in the everyday world. For example, the volume of a rectangular box is found by measuring and multiplying its length, width, and height. This concept of ordinary space is called Euclidean space because it corresponds to Euclid's geometry, which was originally abstracted from the spatial experiences of everyday life.
In mathematics, a magic cube of order is an grid of natural numbers satisying the property that the numbers in the same row, the same column, the same pillar or the same length- diagonal add up to the same number. It is a -dimensional generalisation of the magic square. A magic cube can be assigned to one of six magic cube classes, based on the cube characteristics. A benefit of this classification is that it is consistent for all orders and all dimensions of magic hypercubes.
Square packing is a packing problem where the objective is to determine how many congruent squares can be packed into some larger shape, often a square or circle.
In geometry, Keller's conjecture is the conjecture that in any tiling of n-dimensional Euclidean space by identical hypercubes, there are two hypercubes that share an entire (n − 1)-dimensional face with each other. For instance, in any tiling of the plane by identical squares, some two squares must share an entire edge, as they do in the illustration.
Water retention on random surfaces is the simulation of catching of water in ponds on a surface of cells of various heights on a regular array such as a square lattice, where water is rained down on every cell in the system. The boundaries of the system are open and allow water to flow out. Water will be trapped in ponds, and eventually all ponds will fill to their maximum height, with any additional water flowing over spillways and out the boundaries of the system. The problem is to find the amount of water trapped or retained for a given surface. This has been studied extensively for random surfaces.
In geometry, sphere packing in a cube is a three-dimensional sphere packing problem with the objective of packing spheres inside a cube. It is the three-dimensional equivalent of the circle packing in a square problem in two dimensions. The problem consists of determining the optimal packing of a given number of spheres inside the cube.
Random sequential adsorption (RSA) refers to a process where particles are randomly introduced in a system, and if they do not overlap any previously adsorbed particle, they adsorb and remain fixed for the rest of the process. RSA can be carried out in computer simulation, in a mathematical analysis, or in experiments. It was first studied by one-dimensional models: the attachment of pendant groups in a polymer chain by Paul Flory, and the car-parking problem by Alfréd Rényi. Other early works include those of Benjamin Widom. In two and higher dimensions many systems have been studied by computer simulation, including in 2d, disks, randomly oriented squares and rectangles, aligned squares and rectangles, various other shapes, etc.
In geometry, it is possible to fill 3/4 of the volume of three-dimensional Euclidean space by three sets of infinitely-long square prisms aligned with the three coordinate axes, leaving cubical voids; John Horton Conway, Heidi Burgiel and Chaim Goodman-Strauss have named this structure tetrastix.
References
- ↑ Andreas Stiller: Der Würfel ist gefallen. Kleinster perfekter magischer Würfel gefunden. heise.de, 1 December 2003 (German)
- ↑ Konhauser, Joseph D. E.; Daniel J. Velleman; S. Wagon (1996). Which way did the bicycle go?: and other intriguing mathematical mysteries. Cambridge University Press. p. 105. ISBN 978-0-88385-325-2.
- ↑ Niederman, Derrick (2009). Number freak: from 1 to 200— the hidden language of numbers revealed. Penguin. p. 63. ISBN 978-0-399-53459-1.
- ↑ Knecht, Craig; Walter Trump; Daniel ben-Avraham; Robert M. Ziff (2012). "Retention capacity of random surfaces". Physical Review Letters. 108 (4): 045703. arXiv: 1110.6166 . Bibcode:2012PhRvL.108d5703K. doi:10.1103/PhysRevLett.108.045703. PMID 22400865.
- ↑ Notes on Magic Squares and Cubes by Walter Trump
