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A square whose side length is a triangular number can be partitioned into squares and half-squares whose areas add to cubes. From Gulley (2010).The n th coloured region shows n squares of dimension n by n (the rectangle is 1 evenly divided square), hence the area of the n th region is n times n × n.
[9] [10] [11] There exist non-convex polyhedra that do not have nets, and it is possible to subdivide the faces of every convex polyhedron (for instance along a cut locus) so that the set of subdivided faces has a net. [5] In 2014 Mohammad Ghomi showed that every convex polyhedron admits a net after an affine transformation. [12]
The cube of a number n is denoted n 3, using a superscript 3, [a] for example 2 3 = 8. The cube operation can also be defined for any other mathematical expression, for example (x + 1) 3. The cube is also the number multiplied by its square: n 3 = n × n 2 = n × n × n. The cube function is the function x ↦ x 3 (often denoted y = x 3) that
The cube is non-composite polyhedron, meaning it is a convex polyhedron that cannot be separated into two or more regular polyhedrons. The cube can be applied to construct a new convex polyhedron by attaching another. [40] Attaching a square pyramid to each square face of a cube produces its Kleetope, a polyhedron known as the tetrakis ...
Ratio of a circle's circumference to its radius. Equal to : 1900 to 1600 BCE [2] Square root of 2, Pythagoras constant [4] 1.41421 35623 73095 04880 [Mw 2] [OEIS 3] Positive root of = 1800 to 1600 BCE [5] Square root of 3, Theodorus' constant [6]
All 11 unfoldings of the cube. A polyhedral net for the cube is necessarily a hexomino, with 11 hexominoes (shown at right) actually being nets. They appear on the right, again coloured according to their symmetry groups. A polyhedral net for the cube cannot contain the O-tetromino, nor the I-pentomino, the U-pentomino, or the V-pentomino.
A net = is said to be frequently or cofinally in if for every there exists some such that and . [5] A point is said to be an accumulation point or cluster point of a net if for every neighborhood of , the net is frequently/cofinally in . [5] In fact, is a cluster point if and only if it has a subnet that converges to . [6] The set of all ...
An example of a 3 × 3 × 3 magic cube. In this example, no slice is a magic square. In this case, the cube is classed as a simple magic cube.. In mathematics, a magic cube is the 3-dimensional equivalent of a magic square, that is, a collection of integers arranged in an n × n × n pattern such that the sums of the numbers on each row, on each column, on each pillar and on each of the four ...