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. Pascal's Pyramid is the three-dimensional analog of the two-dimensional Pascal's triangle, which contains the binomial numbers and relates to the binomial expansion and the binomial distribution. The binomial and trinomial numbers, coefficients, expansions, and distributions are subsets of the multinomial constructs with the same names. Pascal's Pyramid is more precisely called "Pascal's tetrahedron", since it has four triangular surfaces. (The pyramids of ancient Egypt had five surfaces: a square base and four triangular sides.)
Pascals triangle is a very simple structure allowing easy access to foiling.
(a+b)^0 1 (a+b)^1 a b (a+b)^2 a^2 2ab b^2 (a+b)^3 a^3 3a^2b 3ab^2 b^3 (a+b)^4 a^4 4a^3b 6a^2b^2 4ab^3 b^4 (a+b)^5 a^5 5a^4b 10a^3b^2 10a^2b^3 5ab^4 b^5Read more about Pascal's Pyramid: Overview of The Tetrahedron, Trinomial Expansion Connection, Trinomial Distribution Connection, Addition of Coefficients Between Layers, Ratio Between Coefficients of Same Layer, Relationship With Pascal's Triangle, Parallels To Pascal's Triangle and Multinomial Coefficients, Usage
Famous quotes containing the words pascal and/or pyramid:
“Let man consider what he is in comparison with all existence; let him regard himself as lost in this remote corner of nature; and from the little cell in which he finds himself lodged, I mean the universe, let him estimate at their true value the earth, kingdoms, cities, and himself. What is a man in the infinite?”
—Blaise Pascal (1623–1662)
“So universal and widely related is any transcendent moral greatness, and so nearly identical with greatness everywhere and in every age,—as a pyramid contracts the nearer you approach its apex,—that, when I look over my commonplace-book of poetry, I find that the best of it is oftenest applicable, in part or wholly, to the case of Captain Brown.”
—Henry David Thoreau (1817–1862)