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Pascal's pyramid's first five layers. Each face (orange grid) is Pascal's triangle. Arrows show derivation of two example terms. In mathematics, Pascal's pyramid is a three-dimensional arrangement of the trinomial numbers, which are the coefficients of the trinomial expansion and the trinomial distribution. [1]
a number represented as a discrete r-dimensional regular geometric pattern of r-dimensional balls such as a polygonal number (for r = 2) or a polyhedral number (for r = 3). a member of the subset of the sets above containing only triangular numbers, pyramidal numbers, and their analogs in other dimensions. [1]
For example, a triangle (the 2-dimensional simplex) one 2-dimensional element (itself), three 1-dimensional elements (lines, or edges), and three 0-dimensional elements (vertices, or corners); this corresponds to the third row 1, 3, 3, 1 of Pascal's triangle. This fact can be explained by combining Pascal's rule for generating the triangle with ...
The basic quantities describing a sphere (meaning a 2-sphere, a 2-dimensional surface inside 3-dimensional space) will be denoted by the following variables r {\displaystyle r} is the radius, C = 2 π r {\displaystyle C=2\pi r} is the circumference (the length of any one of its great circles ),
This formula, sometimes called Tartaglia's formula, is essentially due to the painter Piero della Francesca in the 15th century, as a three-dimensional analogue of the 1st century Heron's formula for the area of a triangle.
Start by labeling p 1, p 2 and p 3 as the corners of the Sierpiński triangle, and a random point v 1. Set v n+1 = 1 / 2 (v n + p r n), where r n is a random number 1, 2 or 3. Draw the points v 1 to v ∞. If the first point v 1 was a point on the Sierpiński triangle, then all the points v n lie on the Sierpiński triangle.
Triangles have many types based on the length of the sides and the angles. A triangle whose sides are all the same length is an equilateral triangle, [3] a triangle with two sides having the same length is an isosceles triangle, [4] [a] and a triangle with three different-length sides is a scalene triangle. [7]
Centers of tetrahedra or higher-dimensional simplices can also be defined, by analogy with 2-dimensional triangles. [13] Some centers can be extended to polygons with more than three sides. The centroid, for instance, can be found for any polygon. Some research has been done on the centers of polygons with more than three sides. [14] [15]
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