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These pages are about the discrete treatment of positive integers: 1, 2, 3, 4, 5 . . . ; "finite or symbolic" (Zeilberger) and a pattern of sieved integers adding sieved-ly to powers.
This exploration attempts to expand a feel for the diagonals of Pascal's Triangle (also called figurate
numbers or polytope numbers or "linear spans of sums of
permutations" [Aguiar,
p. 2]) as well as the
The
Euler/Pascal Cube of accumulating series has, on
its faces, both Pascal's Triangle and Euler's Triangle as well as a face of 1's.
The cube is formed of values
Series accumulate from front to back and relate top to bottom and left to right (in two logically reciprocating layouts) and are fully explained by two overarching equations. One equation is an extension of Euler's Triangle, the other is an extension of Worpitzky's Identity. These series and their relationships seem to be meaningful for binomial theory, combinatorial theory, number theory, commutative theory, computational theory, topological theory, logical theory, probability, statistics, calculus ... etc.
Interrelatedness of series ultimately gives insight into what dimension is--in numerical terms. It is the basis of an elementary grasp of Fermat's Last Theorem. Here is a summary and proof by contradiction of the FLT logic. The relationships inform some general historical statements made by Pascal, Fermat, Bernoulli and others. Here are some figurate explanations of powers and possible approaches to FLT.
This Euler/Pascal cube discretely visualizes why Edward Lorenz might make this statement in his paper Does a climate exist, "The writer feels that . . . the difference equation captures much of the mathematics, even if not the physics, of the transitions from one regime of flow to another, and indeed, of the whole phenomenon of instability" (Gleick, p. 169). And why Richard Feynman might employ perturbation theory in an approach to nonlinear problems (which involved solving a linear problem within a nonlinear problem then applying an expansion to leftover parts). Even quantum theory has a Euler/Pascal cubic-layout sense--Kenneth Wilson's renormalization theory for phase transitions honors concepts of self-similarity and scaling for calculations about systems. "Variability in the standard measures of mass or length meant that a different sort of quantity was remaining fixed . . . Allowing mass to vary depending on scale meant that mathematicians could recognize similarity across scales" (Gleick, p.160-3). The Euler/Pascal cube visualizes variability of a (constantly) xth case across many scales of powers' series.
Perhaps the reality of the Euler/Pascal cube's numerical relationships (including its dimensions) will generally facilitate more clarity and insight in our usage of and references to "dimension".
Author's Profile:
Cecilia wonders what dimensions actually are. She also thinks harmony in number is just as interesting as dissonance which (is emphasized in prime-number studies). She is an accidental-mathematician and an intentional musician, moved from Washington D.C. to Lincoln Nebraska.
Cecilia AT noticingnumbers.net
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Send mail to
Cecilia@noticingnumbers.net with
questions or comments about this web site.
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rows of Pascal's Triangle (also called binomial numbers or
combinatorial numbers or simplex numbers), and their relationship with Eulerian numbers (from
of power series,
shells /nexus numbers to the powers, and summations (... of summations of
summations ... ) of powers.