"involve either half turns or a full rotation. It has six stationary centers, each with a different color, that always stay opposite from another specific center. White is always opposite of yellow. Blue is always opposite of green. Red is always opposite of purple. Each center determines what color that side will be. The centers are exactly based on the original Rubik's Cube. They are connected by a three dimensional cross with screws and springs allowing for less tension. Many cube solvers use the same notation when completing a cube, whether they are working on the original Rubik's Cube, the"
"Cube root In mathematics, a cube root of a number ""x"" is a number ""y"" such that ""y"" = ""x"". All real numbers (except zero) have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. For example, the real cube root of 8, denoted , is 2, because 2 = 8, while the other cube roots of 8 are −1 + ""i"" and −1 − ""i"". The three cube roots of −27""i"" are The cube root operation is not distributive with addition or subtraction. In"
"the cube root of numbers having many digits in the ""Aryabhatiya"" (section 2.5). Cube root In mathematics, a cube root of a number ""x"" is a number ""y"" such that ""y"" = ""x"". All real numbers (except zero) have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. For example, the real cube root of 8, denoted , is 2, because 2 = 8, while the other cube roots of 8 are −1 + ""i"" and −1 − ""i"". The three cube roots of −27""i"""
"some contexts, particularly when the number whose cube root is to be taken is a real number, one of the cube roots (in this particular case the real one) is referred to as the ""principal cube root"", denoted with the radical sign . The cube root operation is associative with exponentiation and distributive with multiplication and division if considering only real numbers, but not always if considering complex numbers: for example, the cube of any cube root of 8 is 8, but the three cube roots of 8 are 8, −4 + 4""i"", and −4 − 4""i"". The cube roots"
"is a pattern in the rightmost digit: adding and subtracting with 1 or 3. Starting from zero: There are two steps to extracting the cube root from the cube of a two digit number. Say you are asked to extract the cube root of 29791. Begin by determining the one's place (units) of the two digit number. You know it must be one, since the cube ends in 1, as seen above. Note that every digit corresponds to itself except for 2, 3, 7 and 8, which are just subtracted from ten to obtain the corresponding digit. The second step"
"8-cube In geometry, an 8-cube is an eight-dimensional hypercube (8-cube). It has 256 vertices, 1024 edges, 1792 square faces, 1792 cubic cells, 1120 tesseract 4-faces, 448 5-cube 5-faces, 112 6-cube 6-faces, and 16 7-cube 7-faces. It is represented by Schläfli symbol {4,3}, being composed of 3 7-cubes around each 6-face. It is called an octeract, a portmanteau of tesseract (the ""4-cube"") and ""oct"" for eight (dimensions) in Greek. It can also be called a regular hexdeca-8-tope or hexadecazetton, being an 8-dimensional polytope constructed from 16 regular facets. It is a part of an infinite family of polytopes, called hypercubes. The"
"V-Cube 8 The V-Cube 8 is an 8×8×8 version of Rubik's Cube. Unlike the original puzzle (but like the 4×4×4 and 6×6×6 cubes), it has no fixed facets: the center facets (36 per face) are free to move to different positions. The design was covered by Panagiotis Verdes' patent from 2007 but Verdes Innovations SA did not produce it for sale until 2014. Other 8×8×8 cubes are produced by the Chinese companies ShengShou, MoYu, and YuXin. Methods for solving the 3×3×3 cube work for the edges and corners of the 8×8×8 cube, as long as one has correctly identified the"
"8 8 (eight) is the natural number following 7 and preceding 9. 8 is: A number is divisible by 8 if its last three digits, when written in decimal, are also divisible by 8, or its last three digits are 0 when written in binary. There are a total of eight convex deltahedra. A polygon with eight sides is an octagon. Figurate numbers representing octagons (including eight) are called octagonal numbers. A polyhedron with eight faces is an octahedron. A cuboctahedron has as faces six equal squares and eight equal regular triangles. A cube has eight vertices. Sphenic numbers always"
"however, easy to show that most numbers are not perfect cubes because ""all"" perfect cubes must have digital root 1, 8 or 9. That is their values modulo 9 may be only −1, 1 and 0. Moreover, the digital root of any number's cube can be determined by the remainder the number gives when divided by 3: Every positive integer can be written as the sum of nine (or fewer) positive cubes. This upper limit of nine cubes cannot be reduced because, for example, 23 cannot be written as the sum of fewer than nine positive cubes: The equation has"
"of a number ""x"" are the numbers ""y"" which satisfy the equation For any real number ""x"", there is ""one"" real number ""y"" such that ""y"" = ""x"". The cube function is increasing, so does not give the same result for two different inputs, plus it covers all real numbers. In other words, it is a bijection, or one-to-one. Then we can define an inverse function that is also one-to-one. For real numbers, we can define a unique cube root of all real numbers. If this definition is used, the cube root of a negative number is a negative number."
"8-8 duoprism In geometry of 4 dimensions, a 8-8 duoprism or octagonal duoprism is a polygonal duoprism, a 4-polytope resulting from the Cartesian product of two octagons. It has 64 vertices, 128 edges, 80 faces (64 squares, and 16 octagons), in 16 octagonal prism cells. It has Coxeter diagram , and symmetry , order 512. The uniform ""8-8 duoprism"" can be constructed from [8]×[8] or [4]×[4] symmetry, order 256 or 64, with extended symmetry doubling these with a 2-fold rotation that maps the two orientations of prisms together. These can be expressed by 4 permutations of uniform coloring of the"
"a sequence of expanded hypercubes: Octagon In geometry, an octagon (from the Greek ὀκτάγωνον ""oktágōnon"", ""eight angles"") is an eight-sided polygon or 8-gon. A ""regular octagon"" has Schläfli symbol {8} and can also be constructed as a quasiregular truncated square, t{4}, which alternates two types of edges. A truncated octagon, t{8} is a hexadecagon, t{16}. The sum of all the internal angles of any octagon is 1080°. As with all polygons, the external angles total 360°. If squares are constructed all internally or all externally on the sides of an octagon, then the midpoints of the segments connecting the centers"
"Truncated 8-cubes In eight-dimensional geometry, a truncated 8-cube is a convex uniform 8-polytope, being a truncation of the regular 8-cube. There are unique 7 degrees of truncation for the 8-cube. Vertices of the truncation 8-cube are located as pairs on the edge of the 8-cube. Vertices of the bitruncated 8-cube are located on the square faces of the 8-cube. Vertices of the tritruncated 7-cube are located inside the cubic cells of the 8-cube. The final truncations are best expressed relative to the 8-orthoplex. Cartesian coordinates for the vertices of a truncated 8-cube, centered at the origin, are all 224 vertices"
"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"
"dual of an 8-cube can be called a 8-orthoplex, and is a part of the infinite family of cross-polytopes. Cartesian coordinates for the vertices of an 8-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) with -1 < x < 1. The elements of the regular polytopes can be expressed in a configuration matrix. Rows and columns reference vertices, edges, faces, and cells, with diagonal element their counts (f-vectors). The nondiagonal elements represent the number of row elements are incident to"
"8-demicube In geometry, a demiocteract or 8-demicube is a uniform 8-polytope, constructed from the 8-hypercube, octeract, with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes. E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM for an 8-dimensional ""half measure"" polytope. Coxeter named this polytope as 1 from its Coxeter diagram, with a ring on one of the 1-length branches, and Schläfli symbol formula_1 or {3,3}. Cartesian coordinates for the vertices of an 8-demicube centered at the origin are alternate halves of the 8-cube: with an odd"
"888 (number) 888 (eight hundred [and] eighty-eight) is the natural number following 887 and preceding 889. 888 is a repdigit (a number all of whose digits are equal), and a strobogrammatic number (one that reads the same upside-down on a seven-segment calculator display). 888 = 700227072 is the smallest cube in which each digit occurs exactly three times, and the only cube in which three distinct digits each occur three times. 888 the smallest multiple of 24 whose digit sum is 24, and as well as being divisible by its digit sum it is divisible by all of its digits."
"solution expressions involve the real cube root of a real number, while if all three solutions are real numbers then they may be expressed in terms of the complex cube root of a complex number. Quartic equations can also be solved in terms of cube roots and square roots. The calculation of cube roots can be to traced back to Babylonian mathematicians from as early as 1800 BCE. In the fourth century BCE Plato posed the problem of doubling the cube, which required a compass-and-straightedge construction of the edge of a cube with twice the volume of a given cube;"
"root is coprime with 10 are somewhat easier since the digits are scrambled in consistent ways, as we shall see in the next section. An easy task for the beginner is extracting cube roots from the cubes of 2 digit numbers. For example, given 74088, determine what two digit number, when multiplied by itself once and then multiplied by the number again, yields 74088. One who knows the method will quickly know the answer is 42, as 42 = 74088. Before learning the procedure, it is required that the performer memorize the cubes of the numbers 1-10: Observe that there"
"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,"
"Octacontagon In geometry, an octacontagon (or ogdoëcontagon or 80-gon from Ancient Greek ὁγδοήκοντα, eighty) is an eighty-sided polygon. The sum of any octacontagon's interior angles is 14040 degrees. A ""regular octacontagon"" is represented by Schläfli symbol {80} and can also be constructed as a truncated tetracontagon, t{40}, or a twice-truncated icosagon, tt{20}, or a thrice-truncated decagon, ttt{10}, or a four-fold-truncated pentagon, tttt{5}. One interior angle in a regular octacontagon is 175°, meaning that one exterior angle would be 4°. The area of a regular octacontagon is (with ) and its inradius is The circumradius of a regular octacontagon is Since"
"if we denote the cube roots of unity by any field containing two distinct roots of will contain the quotient between two distinct cube roots of unity. Such a quotient is a primitive cube root of unity—either ω or formula_28. It follows that a splitting field of will contain ω, as well as the real cube root of 2; conversely, any extension of containing these elements contains all the roots of . Thus Note that applying the construction process outlined in the previous section to this example, one begins with formula_30 and constructs the field formula_31. This field is not"
"Figure-eight knot (mathematics) In knot theory, a figure-eight knot (also called Listing's knot or a Cavendish knot) is the unique knot with a crossing number of four. This makes it the knot with the third-smallest possible crossing number, after the unknot and the trefoil knot. The figure-eight knot is a prime knot. The name is given because tying a normal figure-eight knot in a rope and then joining the ends together, in the most natural way, gives a model of the mathematical knot. A simple parametric representation of the figure-eight knot is as the set of all points (""x"",""y"",""z"") where"
"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"
"(except for the two knots mentioned) is 6. The Alexander polynomial of the figure-eight knot is the Conway polynomial is and the Jones polynomial is The symmetry between formula_7 and formula_8 in the Jones polynomial reflects the fact that the figure-eight knot is achiral. Figure-eight knot (mathematics) In knot theory, a figure-eight knot (also called Listing's knot or a Cavendish knot) is the unique knot with a crossing number of four. This makes it the knot with the third-smallest possible crossing number, after the unknot and the trefoil knot. The figure-eight knot is a prime knot. The name is given"
"8-simplex honeycomb In eighth-dimensional Euclidean geometry, the 8-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 8-simplex, rectified 8-simplex, birectified 8-simplex, and trirectified 8-simplex facets. These facet types occur in proportions of 1:1:1:1 respectively in the whole honeycomb. This vertex arrangement is called the A8 lattice or 8-simplex lattice. The 72 vertices of the expanded 8-simplex vertex figure represent the 72 roots of the formula_1 Coxeter group. It is the 8-dimensional case of a simplectic honeycomb. Around each vertex figure are 510 facets: 9+9 8-simplex, 36+36 rectified 8-simplex, 84+84 birectified 8-simplex, 126+126 trirectified 8-simplex, with the"
"this required the construction, now known to be impossible, of the length . A method for extracting cube roots appears in ""The Nine Chapters on the Mathematical Art"", a Chinese mathematical text compiled around the 2nd century BCE and commented on by Liu Hui in the 3rd century CE. The Greek mathematician Hero of Alexandria devised a method for calculating cube roots in the 1st century CE. His formula is again mentioned by Eutokios in a commentary on Archimedes. In 499 CE Aryabhata, a mathematician-astronomer from the classical age of Indian mathematics and Indian astronomy, gave a method for finding"
"cube root of 29791 is 31. Another example: This process can be extended to find cube roots that are 3 digits long, by using arithmetic modulo 11. These types of tricks can be used in any root where the order of the root is coprime with 10; thus it fails to work in square root, since the power, 2, divides into 10. 3 does not divide 10, thus cube roots work. To approximate a common log (to at least one decimal point accuracy), a few log rules, and the memorization of a few logs is required. One must know: From"
"Hero of Alexandria devised a method for calculating cube roots in the 1st century CE. Methods for solving cubic equations and extracting cube roots appear in ""The Nine Chapters on the Mathematical Art"", a Chinese mathematical text compiled around the 2nd century BCE and commented on by Liu Hui in the 3rd century CE. The Indian mathematician Aryabhata wrote an explanation of cubes in his work Aryabhatiya. In 2010 Alberto Zanoni found a new algorithm to compute the cube of a long integer in a certain range, faster than squaring-and-multiplying. Cube (algebra) In arithmetic and algebra, the cube of a"
"nested cube roots such as can represent rational numbers as well. Again, by realizing that the whole expression appears inside itself, we are left with the equation If we solve this equation, we find that ""x"" = 2. More generally, we find that is the positive real root of the equation ""x"" − ""x"" − ""n"" = 0 for all ""n"" > 0. For ""n"" = 1, this root is the plastic number ""ρ"", approximately equal to 1.3247. The same procedure also works to get as the real root of the equation ""x"" + ""x"" − ""n"" = 0 for"
"be −2, , or . This is related with the concept of monodromy: if one follows by continuity the function ""cube root"" along a closed path around zero, after a turn the value of the cube root is multiplied (or divided) by formula_10 Cube roots arise in the problem of finding an angle whose measure is one third that of a given angle (angle trisection) and in the problem of finding the edge of a cube whose volume is twice that of a cube with a given edge (doubling the cube). In 1837 Pierre Wantzel proved that neither of these"
"is to determine the first digit of the two digit cube root by looking at the magnitude of the given cube. To do this, remove the last three digits of the given cube (29791 → 29) and find the greatest cube it is greater than (this is where knowing the cubes of numbers 1-10 is needed). Here, 29 is greater than 1 cubed, greater than 2 cubed, greater than 3 cubed, but not greater than 4 cubed. The greatest cube it is greater than is 3, so the first digit of the two digit cube must be 3. Therefore, the"
"8-simplex In geometry, an 8-simplex is a self-dual regular 8-polytope. It has 9 vertices, 36 edges, 84 triangle faces, 126 tetrahedral cells, 126 5-cell 4-faces, 84 5-simplex 5-faces, 36 6-simplex 6-faces, and 9 7-simplex 7-faces. Its dihedral angle is cos(1/8), or approximately 82.82°. It can also be called an enneazetton, or ennea-8-tope, as a 9-facetted polytope in eight-dimensions. The name ""enneazetton"" is derived from ""ennea"" for nine facets in Greek and ""-zetta"" for having seven-dimensional facets, and ""-on"". The elements of the regular polytopes can be expressed in a configuration matrix. Rows and columns reference vertices, edges, faces, and cells,"