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In 7-dimensional geometry, a 7-simplex is a self-dual regular 7-polytope. It has 8 vertices , 28 edges , 56 triangle faces , 70 tetrahedral cells , 56 5-cell 5-faces, 28 5-simplex 6-faces, and 8 6-simplex 7-faces.
A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number , other examples being square numbers and cube numbers . The n th triangular number is the number of dots in the triangular arrangement with n dots on each side, and is equal to the sum of the n natural ...
For example, a 7-simplex is (1,1) 8 = (1,2,1) 4 = (1,4,6,4,1) 2 = (1,8,28,56,70,56,28,8,1). The number of 1-faces (edges) of the n-simplex is the n-th triangle number, the number of 2-faces of the n-simplex is the (n − 1) th tetrahedron number, the number of 3-faces of the n-simplex is the (n − 2) th 5-cell number, and so on.
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]
Pascal triangle simplex numbers: Image title: Derivation of triangular, tetrahedral, pentatope and other simplex numbers from Pascal's triangle by CMG Lee, following Wolfram MathWorld notation. Width: 100%: Height: 100%
7-simplex numbers In number theory , a pentatope number is a number in the fifth cell of any row of Pascal's triangle starting with the 5-term row 1 4 6 4 1 , either from left to right or from right to left.
The third tetrahedral number equals the fourth triangular number as the nth k-simplex number equals the kth n-simplex number due to the symmetry of Pascal's triangle, and its diagonals being simplex numbers; similarly, the fifth tetrahedral number (35) equals the fourth pentatope number, and so forth
Pages in category "Simplex numbers" ... Triangular number This page was last edited on 3 April 2023, at 01:34 (UTC). Text is available under the Creative ...