query: a cube with a side length of blank unit is called a unit cube

pos: Unit cube unit hypercube of "n" dimensions is formula_1, the square root of "n" and the (Euclidean) length of the vector (1,1,1...1,1) in "n"-dimensional space. Unit cube A unit cube, more formally a cube of side 1, is a cube whose sides are 1 unit long. The volume of a 3-dimensional unit cube is 1 cubic unit, and its total surface area is 6 square units. The term "unit cube" or unit hypercube is also used for hypercubes, or "cubes" in "n"-dimensional spaces, for values of "n" other than 3 and edge length 1. Sometimes the term "unit cube" refers in specific

pos: Unit cube unit hypercube of "n" dimensions is formula_1, the square root of "n" and the (Euclidean) length of the vector (1,1,1...1,1) in "n"-dimensional space. Unit cube A unit cube, more formally a cube of side 1, is a cube whose sides are 1 unit long. The volume of a 3-dimensional unit cube is 1 cubic unit, and its total surface area is 6 square units. The term "unit cube" or unit hypercube is also used for hypercubes, or "cubes" in "n"-dimensional spaces, for values of "n" other than 3 and edge length 1. Sometimes the term "unit cube" refers in specific

pos: Unit cube Unit cube A unit cube, more formally a cube of side 1, is a cube whose sides are 1 unit long. The volume of a 3-dimensional unit cube is 1 cubic unit, and its total surface area is 6 square units. The term "unit cube" or unit hypercube is also used for hypercubes, or "cubes" in "n"-dimensional spaces, for values of "n" other than 3 and edge length 1. Sometimes the term "unit cube" refers in specific to the set [0, 1] of all "n"-tuples of numbers in the interval [0, 1]. The length of the longest diagonal of a

pos: Prince Rupert's cube Prince Rupert's cube In geometry, Prince Rupert's cube (named after Prince Rupert of the Rhine) is the largest cube that can pass through a hole cut through a unit cube, i.e. through a cube whose sides have length 1, without splitting the cube into two pieces. Its side length is approximately 6% larger than that of the unit cube through which it passes. The problem of finding the largest square that lies entirely within a unit cube is closely related, and has the same solution. The original proposition posed by Prince Rupert of the Rhine was that a cube could

pos: Doubling the cube at solving what they saw as an obstinate but soluble problem. However, the nonexistence of a solution was finally proven by Pierre Wantzel in 1837. In algebraic terms, doubling a unit cube requires the construction of a line segment of length , where ; in other words, . This is because a cube of side length 1 has a volume of , and a cube of twice that volume (a volume of 2) has a side length of the cube root of 2. The impossibility of doubling the cube is therefore equivalent to the statement that is not a constructible

pos: Volume the other and in such cases the combined volume is not additive. In "differential geometry", volume is expressed by means of the volume form, and is an important global Riemannian invariant. In "thermodynamics", volume is a fundamental parameter, and is a conjugate variable to pressure. Any unit of length gives a corresponding unit of volume: the volume of a cube whose sides have the given length. For example, a cubic centimetre (cm) is the volume of a cube whose sides are one centimetre (1 cm) in length. In the International System of Units (SI), the standard unit of volume is

pos: Snub cube equally from the octahedron as the cube, called it snub cuboctahedron, with a vertical extended Schläfli symbol formula_1, and representing an alternation of a truncated cuboctahedron, which has Schläfli symbol formula_2. For a snub cube with edge length 1, its surface area and volume are: where "t" is the tribonacci constant If the original snub cube has edge length 1, its dual pentagonal icositetrahedron has side lengths Cartesian coordinates for the vertices of a snub cube are all the even permutations of with an even number of plus signs, along with all the odd permutations with an odd number of

pos: Hauyne exists between synthetic nosean and haüyne at 600 °C, but only limited solid solution occurs in the sodalite-nosean and sodalite-haüyne systems. Haüyne belongs to the hexatetrahedral class of the isometric system, 3m, space group P3n. It has one formula unit per unit cell (Z = 1), which is a cube with side length of 9 Å. More accurate measurements are as follows: All silicates have a basic structural unit that is a tetrahedron with an oxygen ion O at each apex, and a silicon ion Si in the middle, forming (SiO). In tectosilicates (framework silicates) each oxygen ion is shared

pos: Prince Rupert's cube cube after the hole is cut through it. For the maximally sized-inner cube with length 1.06... relative to the length 1 outer cube, constructing a model has been called "mathematically possible but practically impossible". For the example using two cubes of the same size, as originally proposed by Prince Rupert, model construction is possible. In a 1950 survey of the problem, D. J. E. Schrek published photographs of a model of a cube passing through a hole in another cube. Martin Raynsford has designed a template for constructing paper models of a cube with another cube passing through it; however,

pos: Volume integral on conventions)) has the form Integrating the function formula_7 over a unit cube yields the following result: formula_8 So the volume of the unit cube is 1 as expected. This is rather trivial however, and a volume integral is far more powerful. For instance if we have a scalar function formula_9 describing the density of the cube at a given point formula_10 by formula_11 then performing the volume integral will give the total mass of the cube: formula_12 Volume integral In mathematics—in particular, in multivariable calculus—a volume integral refers to an integral over a 3-dimensional domain, that is, it is

pos: Surface-area-to-volume ratio very large surface area for a given volume. The surface-area-to-volume ratio has physical dimension L (inverse length) and is therefore expressed in units of inverse distance. As an example, a cube with sides of length 1 cm will have a surface area of 6 cm and a volume of 1 cm. The surface to volume ratio for this cube is thus For a given shape, SA:V is inversely proportional to size. A cube 2 cm on a side has a ratio of 3 cm, half that of a cube 1 cm on a side. Conversely, preserving SA:V as size increases

pos: Sphere that can be evaluated to give the result Alternatively, this formula is found using spherical coordinates, with volume element so For most practical purposes, the volume inside a sphere inscribed in a cube can be approximated as 52.4% of the volume of the cube, since , where is the diameter of the sphere and also the length of a side of the cube and ≈ 0.5236. For example, a sphere with diameter 1 meter has 52.4% the volume of a cube with edge length 1 meter, or about 0.524 m. The surface area of a sphere of radius is: Archimedes

pos: Fat object square and a stick in the shape of a "b"×(1/"b") rectangle (with "b">1>(1/"b")). As "b" increases, the area of the enclosing cube (≈"b") increases, but the area of the enclosed cube remains constant (=1) and the total area of the shape also remains constant (=2). Thus the enclosing-cube-slimness can grow arbitrarily while the enclosed-cube-slimness remains constant (=√2). See this GeoGebra page for a demonstration. On the other hand, consider a rectilinear 'snake' with width "1/b" and length "b", that is entirely folded within a square of side length 1. As "b" increases, the area of the enclosed cube(≈1/"b") decreases, but

pos: 9-cube polytopes, called hypercubes. The dual of a 9-cube can be called a 9-orthoplex, and is a part of the infinite family of cross-polytopes. Cartesian coordinates for the vertices of a 9-cube centered at the origin and edge length 2 are while the interior of the same consists of all points ("x", "x", "x", "x", "x", "x", "x", "x", "x") with −1 < "x" < 1. Applying an "alternation" operation, deleting alternating vertices of the "9-cube", creates another uniform polytope, called a "9-demicube", (part of an infinite family called demihypercubes), which has 18 8-demicube and 256 8-simplex facets. 9-cube In geometry,

pos: Cubic yard Cubic yard A cubic yard (symbol yd) is an Imperial / U.S. customary (non-SI non-metric) unit of volume, used in the United States, Canada, and the UK. It is defined as the volume of a cube with sides of 1 yard (3 feet, 36 inches, 0.9144 meters) in length. The IEEE symbol for the cubic yard is yd. The following abbreviations are used: One cubic yard per second (1 yd/s) is a unit of volume flow rate. It corresponds to one cubic yard passing through a specified area every second. One cubic yard per minute (1 yd/min) is a unit

pos: Cube of the vertices are while the interior consists of all points ("x", "x", "x") with −1 < "x" < 1 for all "i". In analytic geometry, a cube's surface with center ("x", "y", "z") and edge length of "2a" is the locus of all points ("x", "y", "z") such that For a cube of edge length formula_2: As the volume of a cube is the third power of its sides formula_3, third powers are called "cubes", by analogy with squares and second powers. A cube has the largest volume among cuboids (rectangular boxes) with a given surface area. Also, a

pos: Ehrhart polynomial all positive integers . The Ehrhart polynomial of the interior of a closed convex polytope can be computed as: where is the dimension of . This result is known as Ehrhart–Macdonald reciprocity. Let be a -dimensional unit hypercube whose vertices are the integer lattice points all of whose coordinates are 0 or 1. In terms of inequalities, Then the -fold dilation of is a cube with side length , containing integer points. That is, the Ehrhart polynomial of the hypercube is . Additionally, if we evaluate at negative integers, then as we would expect from Ehrhart–Macdonald reciprocity. Many other figurate

pos: Flexural rigidity Poisson's ratio and cube of the plate's elastic thickness, it is a governing factor in both (1) and (2). Flexural Rigidity formula_6 formula_2 = Young's Modulus formula_8 = elastic thickness (~10–15 cm) formula_9 = Poisson's Ratio Flexural rigidity of a plate has units of Pa·m, i.e. one dimension of length less from the one for the rod, as it refers to the moment per unit length per unit of curvature, and not the total moment. I is termed as moment of inertia.J is denoted as 2nd moment of inertia/polar moment of inertia. Flexural rigidity Flexural rigidity is defined as the

pos: Square–cube law from mechanical engineering to biomechanics. It helps explain phenomena including why large mammals like elephants have a harder time cooling themselves than small ones like mice, and why building taller and taller skyscrapers is increasingly difficult. The square–cube law can be stated as follows: Represented mathematically: where formula_2 is the original surface area and formula_3 is the new surface area. where formula_5 is the original volume, formula_6 is the new volume, formula_7 is the original length and formula_8 is the new length. For example, a cube with a side length of 1 meter has a surface area of 6 m

pos: Pyramid (geometry) calculus for pyramids with rectangular bases. Consider a unit cube. Draw lines from the center of the cube to each of the 8 vertices. This partitions the cube into 6 equal square pyramids of base area 1 and height 1/2. Each pyramid clearly has volume of 1/6. From this we deduce that pyramid volume = height × base area / 3. Next, expand the cube uniformly in three directions by unequal amounts so that the resulting rectangular solid edges are "a", "b" and "c", with solid volume "abc". Each of the 6 pyramids within are likewise expanded. And each pyramid

pos: Cubic yard of volume flow rate. It corresponds to one cubic yard passing through a specified area every minute. Cubic yard A cubic yard (symbol yd) is an Imperial / U.S. customary (non-SI non-metric) unit of volume, used in the United States, Canada, and the UK. It is defined as the volume of a cube with sides of 1 yard (3 feet, 36 inches, 0.9144 meters) in length. The IEEE symbol for the cubic yard is yd. The following abbreviations are used: One cubic yard per second (1 yd/s) is a unit of volume flow rate. It corresponds to one cubic yard

pos: Affine geometry from a simple case such as a unit isosceles right angled triangle to give formula_5 i.e. 0.019860... or less than 2%, for all triangles. Familiar formulas such as half the base times the height for the area of a triangle, or a third the base times the height for the volume of a pyramid, are likewise affine invariants. While the latter is less obvious than the former for the general case, it is easily seen for the one-sixth of the unit cube formed by a face (area 1) and the midpoint of the cube (height 1/2). Hence it holds for

pos: Chamfer (geometry) all 120° angles. Because all its faces have an even number of sides with 180° rotation symmetry, it is a zonohedron. It is also the Goldberg polyhedron GP(2,0) or {4+,3}, containing square and hexagonal faces. The "chamfered cube" is the Minkowski sum of a rhombic dodecahedron and a cube of side length 1 when eight vertices of the rhombic dodecahedron are at formula_2 and its six vertices are at the permutations of formula_3. In geometry, the chamfered octahedron is a convex polyhedron constructed from the rhombic dodecahedron by truncating the 8 (order 3) vertices. It can also be called a

pos: Square root of 3 1. On the complex plane, this distance is expressed as mentioned below. It is the length of the space diagonal of a unit cube. The vesica piscis has a major axis to minor axis ratio equal to 1:, this can be shown by constructing two equilateral triangles within it. Multiplication of by the imaginary unit gives a square root of -3, an imaginary number. More exactly, (see square root of negative numbers). It is an Eisenstein integer. Namely, it is expressed as the difference between two non-real cubic roots of 1 (which are Eisenstein integers). In power engineering, the voltage

pos: Cube a "measure polytope". There are analogues of the cube in lower dimensions too: a point in dimension 0, a segment in one dimension and a square in two dimensions. The quotient of the cube by the antipodal map yields a projective polyhedron, the hemicube. If the original cube has edge length 1, its dual polyhedron (an octahedron) has edge length formula_5. The cube is a special case in various classes of general polyhedra: The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron; more generally this is referred to as a demicube.

pos: Casus irreducibilis by the axioms defining an ordered field, ω, ω, and 1 are all positive. But if ω>ω, then cubing both sides gives 1>1, a contradiction; similarly if ω>ω. The equation can be depressed to a monic trinomial by dividing by formula_4 and substituting (the Tschirnhaus transformation), giving the equation where Then regardless of the number of real roots, by Cardano's solution the three roots are given by where formula_8 ("k"=1, 2, 3) is a cube root of 1 (formula_9, formula_10, and formula_11, where is the imaginary unit). Here if the radicands under the cube roots are non-real, the cube roots

neg: Minimum bounding box algorithms bounding box of this coreset. A Matlab implementation of the algorithm is available. The minimal enclosing box of the regular tetrahedron is a cube, with side length 1/ that of the tetrahedron; for instance, a regular tetrahedron with side length fits into a unit cube, with the tetrahedron's vertices lying at the vertices (0,0,0), (0,1,1), (1,0,1) and (1,1,0) of the unit cube. Minimum bounding box algorithms In computational geometry, the smallest enclosing box problem is that of finding the oriented minimum bounding box enclosing a set of points. It is a type of bounding volume. "Smallest" may refer to volume,

neg: Prince Rupert's cube this question in "Scientific American", Kay R. Pechenick DeVicci and several other readers showed that the answer for the (3,4) case is the square root of the smaller of two real roots of the polynomial formula_7, which works out to approximately 1.007435. For formula_8, the optimal side length of the largest square in an formula_5-dimensional hypercube is either formula_10 or formula_11, depending on whether formula_5 is even or odd respectively. Prince Rupert's cube In geometry, Prince Rupert's cube (named after Prince Rupert of the Rhine) is the largest cube that can pass through a hole cut through a unit cube,

neg: Prince Rupert's cube the hole through which a cube larger than the original one (up to side length formula_2) may pass. The parts of the unit cube that remain, after emptying this hole, form two triangular prisms and two irregular tetrahedra, connected by thin bridges at the four vertices of the square. Each prism has as its six vertices two adjacent vertices of the cube, and four points along the edges of the cube at distance 1/4 from these cube vertices. Each tetrahedron has as its four vertices one vertex of the cube, two points at distance 3/4 from it on two of

neg: Metrication in the United Kingdom Language". His proposal envisaged a system of units in which the base unit of length was defined by a pendulum that had a period of one second, and the base unit of mass was defined by a cube of rainwater having sides equal to the base unit of length. He recycled the existing names of units of measure, so that there were 10 lines in an inch, 10 inches in a foot, and so on. A century later, his concept of defining unit mass in terms of a cube of water with edges of unit length was one of the

neg: Snub cube plus signs, where "t" ≈ 1.83929 is the tribonacci constant. Taking the even permutations with an odd number of plus signs, and the odd permutations with an even number of plus signs, gives a different snub cube, the mirror image. Taking all of them together yields the compound of two snub cubes. This snub cube has edges of length "α", a number which satisfies the equation and can be written as To get a snub cube with unit edge length, divide all the coordinates above by the value "α" given above. The "snub cube" has two special orthogonal projections, centered,

neg: Hypercube a "p"-generalized "n"-cube are: formula_20. This is "p" vertices and "pn" facets. Hypercube In geometry, a hypercube is an "n"-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length. A unit hypercube's longest diagonal in "n" dimensions is equal to formula_1. An "n"-dimensional hypercube is also called an n"-cube or an n"-dimensional cube. The term "measure polytope" is also used, notably in the work of H. S.

neg: Truncated cube Truncated cube In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces (6 octagonal and 8 triangular), 36 edges, and 24 vertices. If the truncated cube has unit edge length, its dual triakis octahedron has edges of lengths 2 and 2 + . The area "A" and the volume "V" of a truncated cube of edge length "a" are: The "truncated cube" has five special orthogonal projections, centered, on a vertex, on two types of edges, and two types of faces: triangles, and octagons. The last two correspond to the B and A

neg: Tychonoff cube in general topology. Let formula_1 denote the unit interval formula_2. Given a cardinal number formula_3, we define a Tychonoff cube of weight formula_4 as the space formula_5 with the product topology, i.e. the product formula_6 where formula_7 is the cardinality of formula_8 and, for all formula_9, formula_10. The Hilbert cube, formula_11, is a special case of a Tychonoff cube. The Axiom of Choice is assumed throughout. Tychonoff cube In mathematics, more specifically in general topology, the Tychonoff cube is the generalization of the unit cube from the product of a finite number of unit intervals to the product of an

neg: Hilbert cube is homeomorphic to the product of countably infinitely many copies of the unit interval [0, 1]. In other words, it is topologically indistinguishable from the unit cube of countably infinite dimension. If a point in the Hilbert cube is specified by a sequence formula_2 with formula_3, then a homeomorphism to the infinite dimensional unit cube is given by formula_4. It is sometimes convenient to think of the Hilbert cube as a metric space, indeed as a specific subset of a separable Hilbert space (i.e. a Hilbert space with a countably infinite Hilbert basis). For these purposes, it is best not

neg: Hypercube Hypercube In geometry, a hypercube is an "n"-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length. A unit hypercube's longest diagonal in "n" dimensions is equal to formula_1. An "n"-dimensional hypercube is also called an n"-cube or an n"-dimensional cube. The term "measure polytope" is also used, notably in the work of H. S. M. Coxeter (originally from Elte, 1912), but it has now been superseded.

neg: Cube (algebra) Cube (algebra) In arithmetic and algebra, the cube of a number is its third power: the result of the number multiplied by itself twice: It is also the number multiplied by its square: This is also the volume formula for a geometric cube with sides of length , giving rise to the name. The inverse operation of finding a number whose cube is is called extracting the cube root of . It determines the side of the cube of a given volume. It is also raised to the one-third power. Both cube and cube root are odd functions: The cube

neg: Tychonoff cube Tychonoff cube In mathematics, more specifically in general topology, the Tychonoff cube is the generalization of the unit cube from the product of a finite number of unit intervals to the product of an infinite, even uncountable number of unit intervals. The Tychonoff cube is named after Andrey Tychonoff, who first considered the arbitrary product of topological spaces and who proved in the 1930s that the Tychonoff cube is compact. Tychonoff later generalized this to the product of collections of arbitrary compact spaces. This result is now known as Tychonoff's theorem and is considered one of the most important results

neg: Bedlam cube Bedlam cube The Bedlam cube is a solid dissection puzzle invented by British puzzle expert Bruce Bedlam. The puzzle consists of thirteen polycubic pieces: twelve pentacubes and one tetracube. The objective is to assemble these pieces into a 4 x 4 x 4 cube. There are 19,186 distinct ways of doing so, up to rotations and reflections. The Bedlam cube is one unit per side larger than the 3 x 3 x 3 Soma cube, and is much more difficult to solve. Two of the BBC's 'Dragons' from "Dragons' Den", Rachel Elnaugh and Theo Paphitis, were to invest in the

neg: Ancient Mesopotamian units of measurement all quantities were written as metrological symbols and never as numerals followed by a unit symbol. For example there was a symbol for one-sheep and another for one-day but no symbol for one. About 600 of these metrological symbols exist, for this reason archaic Sumerian metrology is complex and not fully understood. In general however, length, volume, and mass are derived from a theoretical standard cube, called 'gur', filled with barley, wheat, water, or oil. The mass of a gur-cube, called 'gun' is defined as the weight a laden ass can carry. However, because of the different specific gravities of

neg: Prince Rupert's cube for the largest square that lies within a unit cube. More generally, show how to find the largest rectangle of a given aspect ratio that lies within a unit cube. As they show, the optimal rectangle must always pass through the center of the cube, with its vertices on edges of the cube. Based on this, they show, depending on the desired aspect ratio, that the optimal rectangle must either lie on a plane that cuts diagonally through four corners of the cube, or it must be formed by an isosceles right triangle on one corner of the cube and

neg: Containerization units (TEU, or sometimes "teu"). An equivalent unit is a measure of containerized cargo capacity equal to one standard (length) × (width) container. As this is an approximate measure, the height of the box is not considered. For instance, the "high cube" and the "half height" containers are also called one TEU. 48' containers have been phased out over the last ten years in favor of 53' containers. The maximum gross mass for a dry cargo container is , and for a container (including the high cube) it is . Allowing for the tare mass of the container, the maximum

neg: Prince Rupert's cube be passed through a hole made in another cube "of the same size" without splitting the cube into two pieces. If two points are placed on two adjacent edges of a unit cube, each at a distance of 3/4 from the point where the two edges meet, then the distance between the two points will be These two points, together with a second set of two points placed symmetrically on the opposite face of the cube, form the four vertices of a square that lies entirely within the unit cube. This square, extruded in both directions perpendicularly to itself, forms

neg: Soma cube Soma cube The Soma cube is a solid dissection puzzle invented by Piet Hein in 1933 during a lecture on quantum mechanics conducted by Werner Heisenberg. Its name is alleged to be derived from the fictitious drug "soma" consumed as a pastime by the establishment in Aldous Huxley's dystopic novel "Brave New World". Seven pieces made out of unit cubes must be assembled into a 3×3×3 cube. The pieces can also be used to make a variety of other 3D shapes. The pieces of the Soma cube consist of all possible combinations of three or four unit cubes, joined at

neg: Fourier transform converges to in as tends to infinity, by the boundedness of the Hilbert transform. Naively one may hope the same holds true for . In the case that is taken to be a cube with side length , then convergence still holds. Another natural candidate is the Euclidean ball . In order for this partial sum operator to converge, it is necessary that the multiplier for the unit ball be bounded in . For it is a celebrated theorem of Charles Fefferman that the multiplier for the unit ball is never bounded unless . In fact, when , this shows

neg: Proof of impossibility Euclidean. A good example is the irrational number the square root of 2. It is simply the length of the hypotenuse of a right triangle with legs both one unit in length, and it can be constructed with straightedge and compass. But it was proved centuries after Euclid that Euclidean numbers cannot involve any operations other than addition, subtraction, multiplication, division, and the extraction of square roots. Both trisecting the general angle and doubling the cube require taking cube roots, which are not constructible numbers by compass and straightedge. formula_2 is not a Euclidean number ... and therefore it is

neg: Minus Cube Minus Cube The Minus Cube () is a 3D mechanical variant of the n-puzzle which was manufactured in the Soviet Union. It consists of a bonded transparent plastic box containing seven small cubes, each glued together from two U-shape parts: one white and one coloured. The length of one side of the interior of the box is slightly more than twice the length of the side of a small cube. There is an empty space the size of one small cube inside the box and the small cubes are moveable inside the box by tilting the box causing a cube

neg: Minus Cube Cube is 12 times more difficult to solve. However, if one does not confine oneself to these two types of the Minus Cube, there are 48 Minus Cube variants which can be solved. Minus Cube The Minus Cube () is a 3D mechanical variant of the n-puzzle which was manufactured in the Soviet Union. It consists of a bonded transparent plastic box containing seven small cubes, each glued together from two U-shape parts: one white and one coloured. The length of one side of the interior of the box is slightly more than twice the length of the side of

neg: Stericated 5-cubes Stericated 5-cubes In five-dimensional geometry, a stericated 5-cube is a convex uniform 5-polytope with fourth-order truncations (sterication) of the regular 5-cube. There are eight degrees of sterication for the 5-cube, including permutations of runcination, cantellation, and truncation. The simple stericated 5-cube is also called an expanded 5-cube, with the first and last nodes ringed, for being constructible by an expansion operation applied to the regular 5-cube. The highest form, the steriruncicantitruncated 5-cube, is more simply called an omnitruncated 5-cube with all of the nodes ringed. The Cartesian coordinates of the vertices of a "stericated 5-cube" having edge length 2 are

neg: Cissoid of Diocles used to find "u" and "v" so that " a" is to "u" as "u" is to "v" as "v" is to "b" i.e. "a"/"u"="u"/"v"="v"/"b", as discovered by Hippocrates of Chios. As a special case, this can be used to solve the Delian problem: how much must the length of a cube be increased in order to double its volume? Specifically, if "a" is the side of a cube, and "b"=2"a", then the volume of a cube of side "u" is so "u" is the side of a cube with double the volume of the original cube. Note however that

neg: Doubling the cube given length times . In music theory, a natural analogue of doubling is the octave (a musical interval caused by doubling the frequency of a tone), and a natural analogue of a cube is three repetitions of the same interval. In this sense, the problem of doubling the cube is solved by the major third in equal temperament. This is a musical interval that is exactly one third of an octave. It multiplies the frequency of a tone by , the side length of the Delian cube. Doubling the cube Doubling the cube, also known as the Delian problem, is

neg: Soma cube and 3×4×5 units. The Bedlam cube is a 4×4×4 sided cube puzzle consisting of twelve pentacubes and one tetracube. The Diabolical cube is a puzzle of six polycubes that can be assembled together to form a single 3×3×3 cube. Soma cube The Soma cube is a solid dissection puzzle invented by Piet Hein in 1933 during a lecture on quantum mechanics conducted by Werner Heisenberg. Its name is alleged to be derived from the fictitious drug "soma" consumed as a pastime by the establishment in Aldous Huxley's dystopic novel "Brave New World". Seven pieces made out of unit cubes must

neg: Exponentiation of a square with side-length is . The expression is called "the cube of "b"" or ""b" cubed" because the volume of a cube with side-length is . When it is a positive integer, the exponent indicates how many copies of the base are multiplied together. For example, . The base appears times in the repeated multiplication, because the exponent is . Here, is the "base", is the "exponent", and is the "power" or, more specifically, "3 raised to the 5th power". The word "raised" is usually omitted, and sometimes "power" as well, so can also be read "3 to

neg: Truncated octahedron Truncated octahedron In geometry, the truncated octahedron is an Archimedean solid. It has 14 faces (8 regular hexagonal and 6 square), 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a zonohedron. It is also the Goldberg polyhedron G(1,1), containing square and hexagonal faces. Like the cube, it can tessellate (or "pack") 3-dimensional space, as a permutohedron. Its dual polyhedron is the tetrakis hexahedron. If the original truncated octahedron has unit edge length, its dual tetrakis cube has edge lengths and . A truncated octahedron is constructed from a regular octahedron with

neg: Archytas and long leg are proportionally similar as the triangle increase in size, which is essentially today's version of similarity of triangles. Archytas then applied the mean proportionals for a given length of a cube. If the volume of the original cube is written as V = x, where "x" represents the length of a side, we let k and k represent the proportionality constants, and the cube is then doubled so that a side length is now 2x, a mean proportional between the two can be written as . With the proportionals finished, Archytas completed the solution to his similar

neg: Pentagonal icositetrahedron If its dual snub cube has unit edge length, its surface area and volume are: The "pentagonal icositetrahedron" has three symmetry positions, two centered on vertices, and one on midedge. Isohedral variations with the same chiral octahedral symmetry can be constructed with pentagonal faces having 3 edge lengths. This variation shown can be constructed by adding pyramids to 6 square faces and 8 triangular faces of a snub cube such that the new triangular faces with 3 coplanar triangles merged into identical pentagon faces. This polyhedron is topologically related as a part of sequence of polyhedra and tilings of pentagons

neg: Cube (algebra) if and only if one can arrange solid unit cubes into a larger, solid cube. For example, 27 small cubes can be arranged into one larger one with the appearance of a Rubik's Cube, since . The difference between the cubes of consecutive integers can be expressed as follows: or There is no minimum perfect cube, since the cube of a negative integer is negative. For example, . Unlike perfect squares, perfect cubes do not have a small number of possibilities for the last two digits. Except for cubes divisible by 5, where only 25, 75 and 00 can be

neg: Klee–Minty cube Klee–Minty cube The Klee–Minty cube or Klee–Minty polytope (named after Victor Klee and ) is a unit hypercube of variable dimension whose corners have been perturbed. Klee and Minty demonstrated that George Dantzig's simplex algorithm has poor worst-case performance when initialized at one corner of their "squashed cube". In particular, many optimization algorithms for linear optimization exhibit poor performance when applied to the Klee–Minty cube. In 1973 Klee and Minty showed that Dantzig's simplex algorithm was not a polynomial-time algorithm when applied to their cube. Later, modifications of the Klee–Minty cube have shown poor behavior both for other basis-exchange pivoting

neg: Tic-tac-toe variants on one or multiple boards. The game Quixo is played on a 5 by 5 board of cubes with two players or teams. On a player's turn, they select a blank cube or a cube with their symbol on it that is at the edge of the board. If a blank cube was selected, the cube is turned to be the player's symbol (either a X or O). The game ends when one player gets 5 in a row. Unrestricted n-in-a-row is played on an infinite tic-tac-toe board where the goal is for one player to get n in a

neg: Avogadro constant with extremely high purity and with few lattice defects. This method defines the Avogadro constant as the ratio of the molar volume, "V", to the atomic volume "V": The unit cell of silicon has a cubic packing arrangement of 8 atoms, and the unit cell volume may be measured by determining a single unit cell parameter, the length of one of the sides of the cube, "a". In practice, measurements are carried out on a distance known as "d"(Si), which is the distance between the planes denoted by the Miller indices {220}, and is equal to "a"/. The 2006 CODATA

neg: Klee–Minty cube according to a 1994 paper by Fukuda and Namiki. Both the simplex algorithm and the criss-cross algorithm visit exactly 3 additional corners of the three-dimensional cube on average. The first two links have both an algebraic construction and a picture of a three-dimensional Klee–Minty cube: Klee–Minty cube The Klee–Minty cube or Klee–Minty polytope (named after Victor Klee and ) is a unit hypercube of variable dimension whose corners have been perturbed. Klee and Minty demonstrated that George Dantzig's simplex algorithm has poor worst-case performance when initialized at one corner of their "squashed cube". In particular, many optimization algorithms for linear

neg: Elongated triangular orthobicupola = 4 tetrahedra, so total we have 20 tetrahedra. What is the volume of a tetrahedron? Construct a tetrahedron having vertices in common with alternate vertices of a cube (of side formula_1, if tetrahedron has unit edges). The 4 triangular pyramids left if the tetrahedron is removed from the cube form half an octahedron = 2 tetrahedra. So formula_2 The hexagonal prism is more straightforward. The hexagon has area formula_3, so formula_4 Finally formula_5 numerical value: formula_6 The elongated triangular orthobicupola forms space-filling honeycombs with tetrahedra and square pyramids.<br> Elongated triangular orthobicupola In geometry, the elongated triangular orthobicupola is one

neg: Dehn invariant the cube is zero. More generally, the Dehn invariant of any parallelepiped is also zero. Only one of the two angles of the tetrahedron and octahedron can be included, as the other one is a rational combination of the one that is included and the angle of the cube. The Dehn invariants of each of the other Platonic solids will be a vector in formula_14 formed by multiplying the unit vector for that solid's angle by the length and number of edges of the solid. No matter how they are scaled by different edge lengths, the tetrahedron, icosahedron, and dodecahedron

neg: Cubic crystal system voids located at the midpoints of the edges of the unit cell as well as one octahedral hole in the very center of the cell, for a total of four net octahedral voids. One important characteristic of a crystalline structure is its atomic packing factor. This is calculated by assuming that all the atoms are identical spheres, with a radius large enough that each sphere abuts on the next. The atomic packing factor is the proportion of space filled by these spheres. Assuming one atom per lattice point, in a primitive cubic lattice with cube side length "a", the sphere

neg: Cubic crystal system Cubic crystal system In crystallography, the cubic (or isometric) crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals. There are three main varieties of these crystals: Each is subdivided into other variants listed below. Note that although the "unit cell" in these crystals is conventionally taken to be a cube, the primitive unit cell often is not. The three Bravais lattices in the cubic crystal system are: The primitive cubic system (cP) consists of one lattice point

neg: Cubic crystal system the larger gas molecules are trapped in the polyhedral cages. Cubic crystal system In crystallography, the cubic (or isometric) crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals. There are three main varieties of these crystals: Each is subdivided into other variants listed below. Note that although the "unit cell" in these crystals is conventionally taken to be a cube, the primitive unit cell often is not. The three Bravais lattices in the cubic crystal system are:

neg: Cantic 5-cube Cantic 5-cube In geometry of five dimensions or higher, a cantic 5-cube, cantihalf 5-cube, truncated 5-demicube is a uniform 5-polytope, being a truncation of the 5-demicube. It has half the vertices of a cantellated 5-cube. The Cartesian coordinates for the 160 vertices of a cantic 5-cube centered at the origin and edge length 6 are coordinate permutations: with an odd number of plus signs. It has half the vertices of the cantellated 5-cube, as compared here in the B5 Coxeter plane projections: This polytope is based on the 5-demicube, a part of a dimensional family of uniform polytopes called demihypercubes

neg: Hydrostatics normal to any contacting surface. If a point in the fluid is thought of as an infinitesimally small cube, then it follows from the principles of equilibrium that the pressure on every side of this unit of fluid must be equal. If this were not the case, the fluid would move in the direction of the resulting force. Thus, the pressure on a fluid at rest is isotropic; i.e., it acts with equal magnitude in all directions. This characteristic allows fluids to transmit force through the length of pipes or tubes; i.e., a force applied to a fluid in a

neg: Prince Rupert's cube by the two opposite points, as in the case of Prince Rupert's problem. If the aspect ratio is not constrained, the rectangle with the largest area that fits within a cube is the one that has two opposite edges of the cube as two of its sides, and two face diagonals as the other two sides. Alternatively, one may ask for the largest formula_4-dimensional hypercube that may be drawn within an formula_5-dimensional unit hypercube. The answer is always an algebraic number. For instance, the problem for formula_6 asks for the largest cube within a four-dimensional hypercube. After Martin Gardner posed

neg: Partial cube Partial cube In graph theory, a partial cube is a graph that is an isometric subgraph of a hypercube. In other words, a partial cube is a subgraph of a hypercube that preserves distances—the distance between any two vertices in the subgraph is the same as the distance between those vertices in the hypercube. Equivalently, a partial cube is a graph whose vertices can be labeled with bit strings of equal length in such a way that the distance between two vertices in the graph is equal to the Hamming distance between their labels. Such a labeling is called a

neg: Cubic crystal system on each corner of the cube. Each atom at a lattice point is then shared equally between eight adjacent cubes, and the unit cell therefore contains in total one atom ( × 8).The body-centered cubic system (cI) has one lattice point in the center of the unit cell in addition to the eight corner points. It has a net total of 2 lattice points per unit cell ( × 8 + 1).The face-centered cubic system (cF) has lattice points on the faces of the cube, that each gives exactly one half contribution, in addition to the corner lattice points, giving

neg: Coherence (units of measurement) Units (1960) was designed around the principle of coherence. In SI, which is a coherent system, the unit of power is the watt, which is defined as one joule per second. In the US customary system of measurement, which is non-coherent, the unit of power is the horsepower, which is defined as 550 foot-pounds per second (the pound in this context being the pound-force); similarly the gallon is not equal to a cubic yard (nor is it the cube of any length unit of the system). The earliest units of measure devised by humanity bore no relationship to each other.

neg: Prince Rupert's cube to a space diagonal of the cube. The projection of the cube onto a plane perpendicular to this diagonal is a regular hexagon, and the best hole parallel to the diagonal can be found by drawing the largest possible square that can be inscribed into this hexagon. Calculating the size of this square shows that a cube with side length slightly larger than one, is capable of passing through the hole. Approximately 100 years later, Dutch mathematician Pieter Nieuwland found that a better solution (in fact, the optimal solution) may be achieved by using a hole with a different angle

neg: Truncated 5-cubes truncated vertex. The Cartesian coordinates of the vertices of a "truncated 5-cube" having edge length 2 are all permutations of: The truncated 5-cube is constructed by a truncation applied to the 5-cube. All edges are shortened, and two new vertices are added on each original edge. The "truncated 5-cube", is fourth in a sequence of truncated hypercubes: The "bitruncated 5-cube" may be constructed by bitruncating the vertices of the 5-cube at formula_3 of the edge length. The Cartesian coordinates of the vertices of a "bitruncated 5-cube" having edge length 2 are all permutations of: The "bitruncated 5-cube" is third in

neg: Cubical complex Cubical complex In mathematics, a cubical complex or cubical set is a set composed of points, line segments, squares, cubes, and their "n"-dimensional counterparts. They are used analogously to simplicial complexes and CW complexes in the computation of the homology of topological spaces. An elementary interval is a subset formula_1 of the form for some formula_3. An elementary cube formula_4 is the finite product of elementary intervals, i.e. where formula_6 are elementary intervals. Equivalently, an elementary cube is any translate of a unit cube formula_7 embedded in Euclidean space formula_8 (for some formula_9 with formula_10). A set formula_11 is a

neg: Tychonoff space are precisely those spaces which can be embedded in compact Hausdorff spaces. More precisely, for every Tychonoff space "X", there exists a compact Hausdorff space "K" such that "X" is homeomorphic to a subspace of "K". In fact, one can always choose "K" to be a Tychonoff cube (i.e. a possibly infinite product of unit intervals). Every Tychonoff cube is compact Hausdorff as a consequence of Tychonoff's theorem. Since every subspace of a compact Hausdorff space is Tychonoff one has: Of particular interest are those embeddings where the image of "X" is dense in "K"; these are called Hausdorff compactifications

neg: Cube cube has the largest volume among cuboids with the same total linear size (length+width+height). For a cube whose circumscribing sphere has radius "R", and for a given point in its 3-dimensional space with distances "d" from the cube's eight vertices, we have: Doubling the cube, or the "Delian problem", was the problem posed by ancient Greek mathematicians of using only a compass and straightedge to start with the length of the edge of a given cube and to construct the length of the edge of a cube with twice the volume of the original cube. They were unable to solve

neg: Runcinated 5-cubes Runcinated 5-cubes In five-dimensional geometry, a runcinated 5-cube is a convex uniform 5-polytope that is a runcination (a 3rd order truncation) of the regular 5-cube. There are unique 8 degrees of runcinations of the 5-cube, along with permutations of truncations, and cantellations. Four are more simply constructed relative to the 5-orthoplex. The Cartesian coordinates of the vertices of a "runcinated 5-cube" having edge length 2 are all permutations of: The Cartesian coordinates of the vertices of a "runcitruncated 5-cube" having edge length 2 are all permutations of: The Cartesian coordinates of the vertices of a "runcicantellated 5-cube" having edge length

neg: K-cell (mathematics) = ("x"...,"x") in whose coordinates satisfy the inequalities "a" ≤ "x" ≤ "b" is a "k"-cell. Every "k"-cell is compact. A "k"-cell of dimension "k" ≤ 3 is especially simple. For example, a 1-cell is simply the interval ["a","b"] with "a" < "b". A 2-cell is the rectangle formed by the Cartesian product of two closed intervals, and a 3-cell is a rectangular solid. The sides and edges of a "k"-cell need not be equal in (Euclidean) length; although the unit cube (which has boundaries of equal Euclidean length) is a 3-cell, the set of all 3-cells with equal-length edges

neg: De Bruijn's theorem de Bruijn, was unable to pack bricks of dimension formula_1 into a formula_2 cube. The cube has a volume equal to that of formula_3 bricks, but only formula_4 bricks may be packed into it. One way to see this is to partition the cube into formula_3 smaller cubes of size formula_6 colored alternately black and white. This coloring has more unit cells of one color than of the other, but with this coloring any placement of the formula_1 brick must have equal numbers of cells of each color. Therefore, any tiling by bricks would also have equal numbers of cells

neg: Trouble Maker (duo) Trouble Maker (duo) Trouble Maker () was a duo formed by Cube Entertainment in 2011, composed of Kim HyunA and Jang Hyun-seung. In November 2011, Cube Entertainment labelmates Hyun-seung (former member of Beast) and Hyuna (former member of 4Minute and Wonder Girls) formed the duo Trouble Maker. Hyuna had previously released two singles, but both described the sub-unit as something different to either of their respective groups. The sub-unit was officially billed as 'JS & Hyuna', which Hyun-seung went on to reveal on his Twitter included his new stage name for the subunit, Jay Stomp. On November 25, the unit

neg: Rectified 10-cubes 10-polytopes with BC symmetry. Cartesian coordinates for the vertices of a rectified 10-cube, centered at the origin, edge length formula_1 are all permutations of: Cartesian coordinates for the vertices of a birectified 10-cube, centered at the origin, edge length formula_1 are all permutations of: Cartesian coordinates for the vertices of a triirectified 10-cube, centered at the origin, edge length formula_1 are all permutations of: Cartesian coordinates for the vertices of a quadrirectified 10-cube, centered at the origin, edge length formula_1 are all permutations of: Rectified 10-cubes In ten-dimensional geometry, a rectified 10-cube is a convex uniform 10-polytope, being a rectification

neg: Quadray coordinates unit of volume, although the infrastructure does not demand using this setting. The four quadrays may be linearly combined to provide integer coordinates for the inverse tetrahedron (0,1,1,1), (1,0,1,1), (1,1,0,1), (1,1,1,0), and for the cube, octahedron, rhombic dodecahedron and cuboctahedron of volumes 3, 4, 6 and 20 respectively, given the starting tetrahedron of unit volume. For example, given A, B, C, D as (1,0,0,0), (0,1,0,0), (0,0,1,0) and (0,0,0,1) respectively, the vertices of an octahedron with the same edge length and volume four would be A + B, A + C, A + D, B + C, B + D, C

neg: Rectified 5-cubes with edge length formula_1 is given by all permutations of: E. L. Elte identified it in 1912 as a semiregular polytope, identifying it as Cr as a second rectification of a 5-dimensional cross polytope. The "birectified 5-cube" may be constructed by birectifing the vertices of the 5-cube at formula_1 of the edge length. The Cartesian coordinates of the vertices of a "birectified 5-cube" having edge length 2 are all permutations of: These polytopes are a part of 31 uniform polytera generated from the regular 5-cube or 5-orthoplex. Rectified 5-cubes In five-dimensional geometry, a rectified 5-cube is a convex uniform 5-polytope,

neg: Truncated 6-cubes Cartesian coordinates of the vertices of a "truncated 6-cube" having edge length 2 are the permutations of: The "truncated 6-cube", is fifth in a sequence of truncated hypercubes: The Cartesian coordinates of the vertices of a "bitruncated 6-cube" having edge length 2 are the permutations of: The "bitruncated 6-cube" is fourth in a sequence of bitruncated hypercubes: The Cartesian coordinates of the vertices of a "tritruncated 6-cube" having edge length 2 are the permutations of: These polytopes are from a set of 63 Uniform 6-polytopes generated from the B Coxeter plane, including the regular 6-cube or 6-orthoplex. Truncated 6-cubes In

neg: Madelung constant higher order Madelung constants or so-called electrostatic lattice constants. In their case, instead of the nearest neighbor distance formula_15 another standard length like the cube root of the unit cell volume formula_16 is appropriately used for purposes of normalization. For instance, the Madelung constant then reads The proper calculation of electrostatic lattice constants has to consider the crystallographic point groups of ionic lattice sites; for instance, dipole moments may only arise on polar lattice sites, i. e. exhibiting a "C", "C", "C" or "C" site symmetry ("n" = 2, 3, 4 or 6). These second order Madelung constants turned out

neg: Pure type system "types" and "terms" and collapse the type hierarchy, as is the case with the calculus of constructions, but this is not generally the case, e.g. the simply typed lambda calculus allows only terms to depend on terms. Pure type systems were independently introduced by Stefano Berardi (1988) and Jan Terlouw (1989). Barendregt discussed them at length in his subsequent papers. In his PhD thesis, Berardi defined a cube of constructive logics akin to the lambda cube (these specifications are non-dependent). A modification of this cube was later called the L-cube by Geuvers, who in his PhD thesis extended the Curry–Howard

neg: Cantellated 5-cubes Cantellated 5-cubes In six-dimensional geometry, a cantellated 5-cube is a convex uniform 5-polytope, being a cantellation of the regular 5-cube. There are 6 unique cantellation for the 5-cube, including truncations. Half of them are more easily constructed from the dual 5-orthoplex The Cartesian coordinates of the vertices of a "cantellated 5-cube" having edge length 2 are all permutations of: In five-dimensional geometry, a bicantellated 5-cube is a uniform 5-polytope. The Cartesian coordinates of the vertices of a "bicantellated 5-cube" having edge length 2 are all permutations of: The Cartesian coordinates of the vertices of an cantitruncated 5-cube having an edge

neg: Hexagram six long roots and six short roots. A six-pointed star, like a regular hexagon, can be created using a compass and a straight edge: A regular hexagram can be constructed by orthographically projecting any cube onto a plane through three vertices that are all adjacent to the same vertex. The twelve midpoints to edges of the cube form a hexagram. For example, consider the projection of the unit cube with vertices at the eight possible binary vectors in three dimensions formula_1 onto the plane formula_2. The midpoints are formula_3, and all points resulting from these by applying a permutation to

neg: Doubling the cube cube involve neusis, the cissoid of Diocles, the conchoid of Nicomedes, or the Philo line. Archytas solved the problem in the 4th century BC using geometric construction in three dimensions, determining a certain point as the intersection of three surfaces of revolution. False claims of doubling the cube with compass and straightedge abound in mathematical crank literature (pseudomathematics). Origami may also be used to construct the cube root of two by folding paper. There is a simple neusis construction using a marked ruler for a length which is the cube root of 2 times another length. Then AG is the

neg: Rectified 6-cubes Rectified 6-cubes In six-dimensional geometry, a rectified 6-cube is a convex uniform 6-polytope, being a rectification of the regular 6-cube. There are unique 6 degrees of rectifications, the zeroth being the 6-cube, and the 6th and last being the 6-orthoplex. Vertices of the rectified 6-cube are located at the edge-centers of the 6-cube. Vertices of the birectified 6-cube are located in the square face centers of the 6-cube. The rectified 6-cube may be constructed from the 6-cube by truncating its vertices at the midpoints of its edges. The Cartesian coordinates of the vertices of the rectified 6-cube with edge length

neg: Square root of 5 one, when traversing the cube "surface" (the shortest distance when traversing through the "inside" of the cube corresponds to the length of the cube diagonal, which is the square root of three times the edge). The number can be algebraically and geometrically related to and , as it is the length of the hypotenuse of a right triangle with catheti measuring and (again, the Pythagorean theorem proves this). Right triangles of such proportions can be found inside a cube: the sides of any triangle defined by the centre point of a cube, one of its vertices, and the middle point

neg: Pentagonal icositetrahedron a square to a height that the new triangles of the raised tetrahedra become coplanar to the triangles which do share an edge with a square. The result is the pentagonal icositetrahedron. Denote the tribonacci constant by "t", approximately 1.8393. (See snub cube for a geometric explanation of the tribonacci constant.) Then the pentagonal faces have four angles of cos () ≈ 114.8° and one angle of cos (2 − "t") ≈ 80.75°. The pentagon has three short edges of unit length each, and two long edges of length ≈ 1.42. The acute angle is between the two long edges.

neg: Stericated 5-cubes all permutations of: The stericated 5-cube is constructed by a sterication operation applied to the 5-cube. The Cartesian coordinates of the vertices of a "steritruncated 5-cube" having edge length 2 are all permutations of: The Cartesian coordinates of the vertices of a "stericantellated 5-cube" having edge length 2 are all permutations of: The Cartesian coordinates of the vertices of an stericantitruncated 5-cube having an edge length of 2 are given by all permutations of coordinates and sign of: The Cartesian coordinates of the vertices of an steriruncitruncated penteract having an edge length of 2 are given by all permutations of

neg: Rectified 7-cubes all permutations of: Cartesian coordinates for the vertices of a birectified 7-cube, centered at the origin, edge length formula_1 are all permutations of: Cartesian coordinates for the vertices of a trirectified 7-cube, centered at the origin, edge length formula_1 are all permutations of: Rectified 7-cubes In seven-dimensional geometry, a rectified 7-cube is a convex uniform 7-polytope, being a rectification of the regular 7-cube. There are unique 7 degrees of rectifications, the zeroth being the 7-cube, and the 6th and last being the 7-cube. Vertices of the rectified 7-cube are located at the edge-centers of the 7-ocube. Vertices of the birectified

neg: Critical mass from an exhaust pipe. Initially the fumes appear black, then gradually you are able to see through them without any trouble. This is not because the total scattering cross section of all the soot particles has changed, but because the soot has dispersed. If we consider a transparent cube of length "L" on a side, filled with soot, then the optical depth of this medium is inversely proportional to the square of "L", and therefore proportional to the areal density of soot particles: we can make it easier to see through the imaginary cube just by making the cube larger.

neg: Fat object "d"-dimensional cube with side-length 2"a" has volume (2"a"). It is enclosed in a "d"-dimensional ball with radius "a√d" whose volume is "V"("a√d"). Hence for every "d"-dimensional object: For even dimensions ("d"=2"k"), the factor simplifies to: formula_7. In particular, for two-dimensional shapes "V"=π and the factor is: √(0.5 π)≈1.25, so: From similar considerations: A "d"-dimensional ball with radius "a" is enclosed in a "d"-dimensional cube with side-length 2"a". Hence for every "d"-dimensional object: For even dimensions ("d"=2"k"), the factor simplifies to: formula_10. In particular, for two-dimensional shapes the factor is: 2/√π≈1.13, so: From similar considerations: Multiplying the above relations gives the

neg: Sudoku Cube completed in China by American Classic Toy Inc, a company belonging to Horowitz. The product is sold in the United States in retailers such as Barnes & Noble and FAO Schwarz. There are 12 types of Sudoku Cube, which differ in difficulty and are aimed at different age ranges. In a standard Rubik's Cube, the player must match up colours on each side of the cube. In the Sudoku Cube, the player must place the numbers one to nine on each side with no repetition. This is achieved by rotating the sides of the cube. Variations of the Sudoku Cube

neg: Pentellated 6-cubes Pentellated 6-cubes In six-dimensional geometry, a pentellated 6-cube is a convex uniform 6-polytope with 5th order truncations of the regular 6-cube. There are unique 16 degrees of pentellations of the 6-cube with permutations of truncations, cantellations, runcinations, and sterications. The simple pentellated 6-cube is also called an expanded 6-cube, constructed by an expansion operation applied to the regular 6-cube. The highest form, the "pentisteriruncicantitruncated 6-cube", is called an "omnitruncated 6-cube" with all of the nodes ringed. Six of them are better constructed from the 6-orthoplex given at pentellated 6-orthoplex. The omnitruncated 6-cube has 5040 vertices, 15120 edges, 16800 faces (4200

neg: Larry D. Nichols cube assembled from eight unit cubes with magnets on their inside faces, allowing the cubes to rotate in groups of four around three axes. The object of the puzzle was to mix the colors on the faces of the cube and then restore them. After making many preliminary models, in 1968 a working prototype was constructed, and on April 11, 1972, U.S. patent 3,655,201 was issued covering the Nichols' Cube. The patent focused on the 2×2×2 puzzle but mentioned the possibility of larger versions. Nichols' patents became the subject of a court hearing, between his employer, and the Ideal Toy

neg: Magic cube section diagonal also sum up to the cube's magic number, the cube is called a perfect magic cube; otherwise, it is called a semiperfect magic cube. The number "n" is called the order of the magic cube. If the sums of numbers on a magic cube's broken space diagonals also equal the cube's magic number, the cube is called a pandiagonal cube. In recent years, an alternate definition for the perfect magic cube has gradually come into use. It is based on the fact that a pandiagonal magic square has traditionally been called perfect, because all possible lines sum correctly.