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In mathematics, the centered polyhedral numbers are a class of figurate numbers, each formed by a central dot, surrounded by polyhedral layers with a constant number of edges. The length of the edges increases by one in each additional layer.
The centered cube number for a pattern with n concentric layers around the central point is given by the formula [1] + (+) = (+) (+ +). The same number can also be expressed as a trapezoidal number (difference of two triangular numbers), or a sum of consecutive numbers, as [2]
Visually, 1729 can be found in other figurate numbers. It is the tenth centered cube number (a number that counts the points in a three-dimensional pattern formed by a point surrounded by concentric cubical layers of points), the nineteenth dodecagonal number (a figurate number in which the arrangement of points resembles the shape of a ...
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.
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In mathematics, the centered icosahedral numbers also known as cuboctahedral numbers are a sequence of numbers, describing two different representations for these numbers as three-dimensional figurate numbers. As centered icosahedral numbers, they are centered numbers representing points arranged in the shape of a regular icosahedron.
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