"Cube. Three 3x3 cubes are very similar to the original: the Gear Cube Extreme, the Gear Cube Ultimate, and the Even Less Gears Cube. Gear Cube Extreme: Unlike the original Gear Cube with 12 total gears, the Gear Cube Extreme (also known as the Anisotropic Cube) has only eight gears. The top and bottom rows remain gear cogs, whereas the middle layer edge cogs have been removed. Taking away the middle layer cogs and replacing them with standard edges allows for different movement between rows. Due to the added straight edges, the cube can now make 90° turns on two"
"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"
"There are 24 edge pieces which show two coloured sides each, and eight corner pieces which show three colours. Each corner piece or pair of edge pieces shows a unique colour combination, but not all combinations are present (for example, there is no piece with both red and orange sides, if red and orange are on opposite sides of the solved Cube). The location of these cubes relative to one another can be altered by twisting the layers of the cube, but the location of the coloured sides relative to one another in the completed state of the puzzle cannot"
"Snub cube In geometry, the snub cube, or snub cuboctahedron, is an Archimedean solid with 38 faces: 6 squares and 32 equilateral triangles. It has 60 edges and 24 vertices. It is a chiral polyhedron, that is, it has two distinct forms, which are mirror images (or ""enantiomorphs"") of each other. The union of both forms is a compound of two snub cubes, and the convex hull of both sets of vertices is a truncated cuboctahedron. Kepler first named it in Latin as cubus simus in 1619 in his Harmonices Mundi. H. S. M. Coxeter, noting it could be derived"
"which are all gear cogs. Each edge piece has two different colors on it, the colors of the two sides it lies between. It also has eight corners, four on the top layer and four on the bottom layer. Each corner connects three different sides and possesses those three colors. Also, the cube has a total of 41,472 possible permutations. Each side is permitted to turn in only 180° increments or half turns. Due to the gears not making a full rotation along with the side it's attached to, it prohibits 90° turns. This is why all of the algorithms"
"edges is 2 less than the sum of the numbers of vertices and faces. For example, a cube has 8 vertices and 6 faces, and hence 12 edges. In a polygon, two edges meet at each vertex; more generally, by Balinski's theorem, at least ""d"" edges meet at every vertex of a ""d""-dimensional convex polytope. Similarly, in a polyhedron, exactly two two-dimensional faces meet at every edge, while in higher dimensional polytopes three or more two-dimensional faces meet at every edge. In the theory of high-dimensional convex polytopes, a facet or ""side"" of a ""d""-dimensional polytope is one of its"
"and eight corner pieces which show three colors. Each piece (or quartet of edge pieces) shows a unique color combination, but not all combinations are present (for example, there is no edge piece with both red and orange sides, since red and orange are on opposite sides of the solved Cube). The location of these cubes relative to one another can be altered by twisting the layers of the Cube 90°, 180° or 270°, but the location of the colored sides relative to one another in the completed state of the puzzle cannot be altered: it is fixed by the"
"each cube). If an edge connects two vertices (Red and Green) and the number of the edge is three, then it means that the third cube has Red and Green faces opposite to each other. To find a solution to this problem we need the arrangement of four faces of each of the cubes. To represent the information of two opposite faces of all the four cubes we need a directed subgraph instead of an undirected one because two directions can only represent two opposite faces, but not whether a face should be at the front or at the back."
"another four gears out and replaces them with standard edges. As the amount of gears get fewer and fewer, the difficulty level rises. So four sides can make 90° turns whereas the fifth and sixth sides are using the gear cogs giving only one option being a 180° turn. Gear Ball: This cube has two sets of edges. The outer edges are gear shaped with six teeth, and a black barrier on it. The inner edges are smaller and not circular, and only have too teeth. In addition, the corners have a three-pronged barrier. This cube is manufactured by Mefferts,"
"then prying one of its edge cubes away from the other two layers. Consequently, it is a simple process to ""solve"" a Cube by taking it apart and reassembling it in a solved state. There are six central pieces which show one coloured face, twelve edge pieces which show two coloured faces, and eight corner pieces which show three coloured faces. Each piece shows a unique colour combination, but not all combinations are present (for example, if red and orange are on opposite sides of the solved Cube, there is no edge piece with both red and orange sides). The"
"boundary of an ""n""-cube is For example, the boundary of a 4-cube (n=4) contains 8 cubes (3-cubes), 24 squares (2-cubes), 32 lines (1-cubes) and 16 vertices (0-cubes). This identity can be proved by combinatorial arguments; each of the formula_8 vertices defines a vertex in a ""m""-dimensional boundary. There are formula_9 ways of choosing which lines (""sides"") that defines the subspace that the boundary is in. But, each side is counted formula_10 times since it has that many vertices, we need to divide by this number. This identity can also be used to generate the formula for the ""n""-dimensional cube surface"
"sides alternating colors, so there are three colors, paired by opposite sides. Each symmetry form has a different Wythoff symbol. A cube has eleven nets (one shown above): that is, there are eleven ways to flatten a hollow cube by cutting seven edges. To color the cube so that no two adjacent faces have the same color, one would need at least three colors. The cube is the cell of the only regular tiling of three-dimensional Euclidean space. It is also unique among the Platonic solids in having faces with an even number of sides and, consequently, it is the"
"""snub cube"", one of the Archimedean solids. It has 24 vertices and 60 edges, and is an Archimedean graph. Snub cube In geometry, the snub cube, or snub cuboctahedron, is an Archimedean solid with 38 faces: 6 squares and 32 equilateral triangles. It has 60 edges and 24 vertices. It is a chiral polyhedron, that is, it has two distinct forms, which are mirror images (or ""enantiomorphs"") of each other. The union of both forms is a compound of two snub cubes, and the convex hull of both sets of vertices is a truncated cuboctahedron. Kepler first named it in"
"The truncated triangular trapezohedron is another polyhedron which can be formed from cube edge truncation. The ""truncated cube"", is second in a sequence of truncated hypercubes: In the mathematical field of graph theory, a truncated cubical graph is the graph of vertices and edges of the ""truncated cube"", one of the Archimedean solids. It has 24 vertices and 36 edges, and is a cubic Archimedean graph. 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"
"aforementioned ambiguity in some configurations. In ""marching tetrahedra"", each cube is split into six irregular tetrahedra by cutting the cube in half three times, cutting diagonally through each of the three pairs of opposing faces. In this way, the tetrahedra all share one of the main diagonals of the cube. Instead of the twelve edges of the cube, we now have nineteen edges: the original twelve, six face diagonals, and the main diagonal. Just like in ""marching cubes"", the intersections of these edges with the isosurface are approximated by linearly interpolating the values at the grid points. Adjacent cubes share"
"9-cube In geometry, a 9-cube is a nine-dimensional hypercube with 512 vertices, 2304 edges, 4608 square faces, 5376 cubic cells, 4032 tesseract 4-faces, 2016 5-cube 5-faces, 672 6-cube 6-faces, 144 7-cube 7-faces, and 18 8-cube 8-faces. It can be named by its Schläfli symbol {4,3}, being composed of three 8-cubes around each 7-face. It is also called an enneract, a portmanteau of tesseract (the ""4-cube"") and ""enne"" for nine (dimensions) in Greek. It can also be called a regular octadeca-9-tope or octadecayotton, as a nine-dimensional polytope constructed with 18 regular facets. It is a part of an infinite family of"
"by rotating the matrix elements by 180 degrees. formula_1 Cartesian coordinates for the vertices of a hepteract 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) with -1 < x < 1. 7-cube In geometry, a 7-cube is a seven-dimensional hypercube with 128 vertices, 448 edges, 672 square faces, 560 cubic cells, 280 tesseract 4-faces, 84 penteract 5-faces, and 14 hexeract 6-faces. It can be named by its Schläfli symbol {4,3}, being composed of 3 6-cubes around each 5-face. It can be called"
"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"
"gears externally in order for the mechanics to be seen. At first sight, it looks quite formidable, nevertheless it is actually simpler to solve than the original Rubik's Cube. There are two objectives when solving the cube. The first goal is taking the mixed up puzzle back to its original cubic state. The second goal is to actually solve the puzzle by arranging each side back to their own beginning colors. The Gear Cube has a design like no other because the puzzle is made almost completely out of gears. The cube has 12 edge pieces, four on every layer,"
"7-cube In geometry, a 7-cube is a seven-dimensional hypercube with 128 vertices, 448 edges, 672 square faces, 560 cubic cells, 280 tesseract 4-faces, 84 penteract 5-faces, and 14 hexeract 6-faces. It can be named by its Schläfli symbol {4,3}, being composed of 3 6-cubes around each 5-face. It can be called a hepteract, a portmanteau of tesseract (the ""4-cube"") and ""hepta"" for seven (dimensions) in Greek. It can also be called a regular tetradeca-7-tope or tetradecaexon, being a 7 dimensional polytope constructed from 14 regular facets. It is a part of an infinite family of polytopes, called hypercubes. The dual"
"versatile in being able to contain multiple smaller cuboids, e.g. sugar cubes in a box, boxes in a cupboard, cupboards in a room, and rooms in a building. A cuboid with integer edges as well as integer face diagonals is called an Euler brick, for example with sides 44, 117 and 240. A perfect cuboid is an Euler brick whose space diagonal is also an integer. It is currently unknown whether a perfect cuboid actually exists. The number of different nets for a simple cube is 11, however this number increases significantly to 54 for a rectangular cuboid of 3"
"show three colors. Each piece (or sextet of edge pieces) shows a unique color combination, but not all combinations are present (for example, there is no edge piece with both red and orange sides, since red and orange are on opposite sides of the solved Cube). The location of these cubes relative to one another can be altered by twisting the layers of the Cube 90°, 180° or 270°, but the location of the colored sides relative to one another in the completed state of the puzzle cannot be altered: it is fixed by the distribution of color combinations on"
"one corner, one edge and one face centre each. The puzzle has eight corner pieces, which can be permutated in 8! different ways, and twelve edge pieces, which can be permutated in 12! ways. The position of the face centre pieces is fixed, therefore there is only one possible permutation for the face centres. Since all cubies only consist of one colour (unlike on the Rubik's Cube), they have no visible orientation and therefore this is not taken into account. The exact number of combinations is not yet reached because of several constraints: As some pieces are identical to each"
"corner pieces and 24 face center pieces. Each corner piece has 3 colors, and each face center piece has only a single color. Unlike the Rubik's Cube, its faces do not rotate; rather, the pieces are scrambled by rotating around a cube edge. When twisting the puzzle, a 180° turn exchanges two corner pieces and swaps two pairs of face center pieces, but preserves the cube shape. The entire puzzle can be scrambled in this way. However, it is also possible to twist an edge by ~71°, such that the base of two groups of a corner piece and a"
"a version with edges of equal length and in a canonical version where all edges touch the same midsphere. (They only look noticeably different for solids containing triangles.) The shown duals are dual to the canonical versions. The chamfered tetrahedron (or alternate truncated cube) is a convex polyhedron constructed as an alternately truncated cube or chamfer operation on a tetrahedron, replacing its 6 edges with hexagons. It is the Goldberg polyhedron G(2,0), containing triangular and hexagonal faces. The chamfered cube is a convex polyhedron with 32 vertices, 48 edges, and 18 faces: 12 hexagons and 6 squares. It is constructed"
"If the shape is considered as a union of five cubes yielding a simple nonconvex solid without self-intersecting surfaces, then it has 360 faces (all triangles), 182 vertices (60 with degree 3, 30 with degree 4, 12 with degree 5, 60 with degree 8, and 20 with degree 12), and 540 edges, yielding an Euler characteristic of 182-540+360 = +2. Its convex hull is a regular dodecahedron. It additionally shares its edge arrangement with the small ditrigonal icosidodecahedron, the great ditrigonal icosidodecahedron, and the ditrigonal dodecadodecahedron. The compound of ten tetrahedra can be formed by taking each of these five"
"24. Now that the analog triangle has been constructed, the number of elements of any dimension that compose an arbitrarily dimensioned cube (called a hypercube) can be read from the table in a way analogous to Pascal's triangle. For example, the number of 2-dimensional elements in a 2-dimensional cube (a square) is one, the number of 1-dimensional elements (sides, or lines) is 4, and the number of 0-dimensional elements (points, or vertices) is 4. This matches the 2nd row of the table (1, 4, 4). A cube has 1 cube, 6 faces, 12 edges, and 8 vertices, which corresponds to"
"and second layer edges simultaneously, with each corner paired up with a second-layer edge piece, thus completing the first two layers (F2L). This is then followed by orienting the last layer, then permuting the last layer (OLL and PLL respectively). Fridrich's solution requires learning roughly 120 algorithms but allows the Cube to be solved in only 55 moves on average. A now well-known method was developed by Lars Petrus. In this method, a 2×2×2 section is solved first, followed by a 2×2×3, and then the incorrect edges are solved using a three-move algorithm, which eliminates the need for a possible"
"rows and as well as 180° turns on four rows. Gear Cube Ultimate: This cube is built exactly like the Gear Cube Extreme with only eight gears. The only difference is that it added Tony Fisher style stickers to the four inner pieces on each side, making it one step more difficult to correctly orient the six sides. A common issue while solving this cube is having the entire puzzle complete except for the newly added stickers being mismatched with different sides. Even Less Gears: The only 3x3 type of Gear Cube with a total of four gears. It takes"
"eight internal cells) are each included in seven different winning lines; the other 48 cells (24 face cells and 24 edge cells) are each included in four winning lines. The corner cells and the internal cells are actually equivalent via an automorphism; likewise for face and edge cells. The group of automorphisms of the game contains 192 automorphisms. It is made up of combinations of the usual rotations and reflections that reorient or reflect the cube, plus two that scramble the order of cells on each line. If a line comprises cells A, B, C and D in that order,"
"twenty-six unique miniature cubes, also called ""cubies"" or ""cubelets"". Each of these includes a concealed inward extension that interlocks with the other cubes while permitting them to move to different locations. However, the centre cube of each of the six faces is merely a single square façade; all six are affixed to the core mechanism. These provide structure for the other pieces to fit into and rotate around. So there are twenty-one pieces: a single core piece consisting of three intersecting axes holding the six centre squares in place but letting them rotate, and twenty smaller plastic pieces which fit"
"The six edges in each matching sextet are distinguishable, since corresponding edges are mirror images of each other. Any permutation of the edges in each set is possible, including odd permutations, giving 24! arrangements for each set or 24! total, regardless of the position or orientation any other pieces. Assuming the cube does not have a fixed orientation in space, and that the permutations resulting from rotating the cube without twisting it are considered identical, the number of permutations is reduced by a factor of 24. This is because the 24 possible positions and orientations of the first corner are"
"10-cube In geometry, a 10-cube is a ten-dimensional hypercube. It has 1024 vertices, 5120 edges, 11520 square faces, 15360 cubic cells, 13440 tesseract 4-faces, 8064 5-cube 5-faces, 3360 6-cube 6-faces, 960 7-cube 7-faces, 180 8-cube 8-faces, and 20 9-cube 9-faces. It can be named by its Schläfli symbol {4,3}, being composed of 3 9-cubes around each 8-face. It is sometimes called a dekeract, a portmanteau of tesseract (the ""4-cube"") and ""deka-"" for ten (dimensions) in Greek, It can also be called an icosaxennon or icosa-10-tope as a 10 dimensional polytope, constructed from 20 regular facets. It is a part of"