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PASCAL
TRIANGLE
Contents:
Mathematical Connections
Blaise Pascal
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Blaise Pascal
was born at Clermont on June,19,1623. In 1653 Pascal invented
the arithmetical triangle.Pascal`s arithmetical triangle is
the basic number formula in nature. As a mathematician Pascal
is best known in connection with his correspondence with
Fermat in 1654. in which he laid down the principles of the
theory of probabilities. He died in Paris on August 19,
1662... |
Formulae for Fib(x-n) and Fib(x+2n)
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It is possible
to derive operator`s equations for Fibonacci numbers . There
are systems of equations where all the coefficients are the
numbers of the Pascal Triangle ... Now available in PDF format
for which you will need the ACROBAT READER .
Simple exercises in the
operational calculus
The formulas in the
present paper are all simple exercises in the operational
calculus ( once one knows it ). Formula involving a product of
Fibonacci and Lucas Numbers:
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Bernoulli numbers and the Pascal
triangle
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The Bernoulli
numbers play an important role in mathematics. They first
appeard in Ars Conjectandi, a famous, and posthumously
published, treatise in 1713, by Jakob Bernoulli. Bernoulli`s
numbers appear in analysis, number theory and differential
topology. There is a remarkable connection between Bernoulli`s
numbers and Pascal`s
Triangle... |
Triangular numbers
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The triangular
number is a polygonal number: a number that can be represented
by a regular geometric arrangement of equally spaced points.
As the name suggests triangular numbers can be visualised as a
triangle of points... The triangular numbers can be found in
the third diagonal of Pascal`s Triangle
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Tetrahedral numbers
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These numbers
correspond to placing discrete points in the cofiguration of a
tetrahedron ( triangular base pyramid ). Tetrahedral numbers
are pyramidal numbers and are the sum of consectuive
triangular numbers. The first few are 1,4,10,20,35,56,84,120
...The tetrahedral number is a figurate number : a number that
can be represented by a regular geometric arrangement of
equally spaced points.As the name suggests tetrahedral numbers
can be visualised as a tetrahedron of points
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Pentatope numbers
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Pentatope is the name of a specific geometric figure that human beings cannot directly visualize because it does not exist as a 3-dimensional object. Pentatope Numbers (4-tetrahedron numbers) can be found in the fifth column of Pascal`s triangle, which are of course the sum of the tetrahedral numbers.... |
Atomic structure and the Pascal
Triangle
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Nuclei with
either numbers of protons or neutrons equal to Z,N =
2,8,20,28,50,82 or 126 exhibit certain properties, which are
analogous to closed shell properties in atoms, including
anomalously low masses, high natural abundances and high
energy first excited states. The connection between magic
numbers on one side and pyramidal numbers and tetrahedral
numbers and triangular numbers, on the other, is highly
amusing... |
Cell Division and the Pascal Triangle
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As we have
seen, by geometrical model of cell division, each cycle leads
to the new cells of the same generation without surviving of
the old cell. The cell A0 becomes older by the law
t=n*T
where n is the
number of the cycle andT is the period of of the
cycle.But, by the previous model the age of the cells is
always T. That is in contradaction with the natural law
and with the reproduction of the more - cellular organisms
...
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Titius - Bode Law
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The so named
Bode's Law was published as a scientific law by Johann Elert
Bode, German astronomer and Director of the Observatory of
Berlin. But the study was originally made by Christian
Freiherr von Wolf ( 1679 - 1754 ), German mathematician and
philosopher, and divulged at 1766 by Johann Daniel Dietz, also
known as Titius of Wittenberg ( 1729 - 1796 ), German teacher
of physics of the University of Wittenberg, who verified the
validity of the Wolf 's calculation ...
See also :
Kepler`s Third Law
The Law of Venus
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Contents: Leonardo Fibonacci
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Leonardo
Fibonacci was born in Pisa, Italy, around 1175. He was the
gratest European mathematician of the Middle Ages. He was the
first to introduce the Hindu - Arabic number system into
Europe. By charting the populations of rabbits Fibonacci
discovered a number series from which one can derive the
Golden Section. French mathematician Edouard Lucas named this
series Fibonacci numbers and found their numerous significant
applications. Leonardo Fibonacci died in Pisa soon after 1240
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FIBONACCI IDENTITIES FROM MATRICES AND VECTORS
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We can express the recurrence
relation F(n+2)= F(n+1) + F(n) in terms of the action of a 2 X 2 matrix.
A matrix is an ordered set of numbers listed rectangular form. If a matrix A has n rows
and m columns then we say it's a n X m matrix. The two-dimensional vector V is an ordered
pair of elements, called scalars, of a field : V = (a,b). If U = (a,b) is written as (a,b)
then U is a 1 X 2 matrix which we shall call a row - vector. Similarly, if U = (a,b) is
written vertically, then U becomes a 2 X 1 matrix which we shall call a column - vector ... |
Mathematical Constants and the Fibonacci
Numbers
The relations between Fibonacci and Lucas
numbers
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The Lucas
numbers are formed in the same way as the Fibonacci numbers :
by adding the latest two to get the next one but instead of
starting at 1 and 1 ( the Fibonacci numbers ), we start with 1
and 3 ( the Lucas numbers ). Each Lucas number is the sum of
two proceeding numbers ...Now available in PDF format for
which you will need the ACROBAT READER
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Pythagoras Theorem and Fibonacci
Numbers
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At the heart of
Pythagoras` teachings was the vision of the underlying
harmony of the universe. This harmony had to be
abstracted from the confusion of visible things
and daily events. As a matter of fact this harmony existed in
the abstract - in the same way as numbers and mathematical
formulas are abstractions. Pythagoras believed in secrecy and
communalism, so it is almost impossible distguishing his work
from the work of his followers. Pythagoras and his followers
contributed to music, astronomy and
mathematics.... |
Fibonacci Function Calculator
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Fibonacci
function calculator. Now available for all real values in the
range -1475 < real value < +1475 . The first derivate
calculation of Fibonacci function is possible too ...
Lucas Function Calculator
is also available. The first derivate calculation of Lucas
function is possible too...
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The Function Derivative Calculator
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In order to
compute the derivative f '(x) try the Derivative
Calculator... Click on Numerical Derivative Calculator -
Data to open a Javascript program designed to calculate
numerical derivative based on 15 measured data...
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Computing a Sum
Divergent Series of the Fibonacci
Numbers
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"In his first
paper on the Calculus (1669), Newton proudly introduced the
use of infinite series to expedite the processes of the
calculus... As Newton, Leibnitz, the several Bernoullis,
Euler, d'Alembert, Lagrange, and other 18th-century men
stuggled with the strange problem of infinite series and
employed them in analysis, they perpetuated all sorts of
blunders, made false proofs, and drew incorrect conclusions;
they even gave arguments that now with hindsight we are
obliged to call ludicrous." ...
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Alternating Series of the Fibonacci
Numbers
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We now proceed
to investigate series with terms of arbitary signs and begin
with the so-called alternating series whose terms are
alternately positive and negative...Let us find the
alternating sum of all Fibonacci numbers! We shall now
introduce operator of the finite differences that connects the
function ... |
Contents: Golden Section
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The Golden
Section appears repeatedly in growth patterns in nature.
Throughout history, The Golden Ratio has fascinated
mathematicians and artists for centuries. Lucas Pacioli in his
Divina proportione wrote about the Golden Section also
called the divine proportion. Pacoli`s work influenced
Leonardo da Vinci and Albrecht Durer. The Golden Section is
manifested in the structure of the human body. There are
interesting series of equations for Fibonacci numbers and
Golden Section ...
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The relation between Fibonacci numbers and
Golden Section
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The Golden
Section is the number often encountered when taking the ratios
of distances in simple geometric figures. The Fibonacci
numbers appear in arrangements of leaves because the Fibonacci
numbers form the best whole number approximations to the
Golden Section. There is a remarkable connection between
Golden Section on one side and Fibonacci and Lucas numbers on
the other ... |
Lucas numbers and the Golden Section
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Lucas is best
known for his results in number theory: in particular he
studied the Fibonacci sequence and the associated Lucas
sequence is named after him.Lucas is also well known for his
invention of the Tower of Hanoi puzzle and other mathematical
recreations. The Tower of Hanoi puzzle appeared in 1883 under
the name of M. Claus. Notice that Claus is an anagram of
Lucas! His four volume work on recreational mathematics
(1882-94) has become a classic.
Lucas died as the result of a freak accident at a banquet when
a plate was dropped and a piece flew up and cut his cheek. He
died of erysipelas a few days later. The main numerical
sequence considered by Lucas is the sequence of numbers 1, 3,
4, 7, 11, 18, 29, 47, ...
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Fibonacci Numbers in Nature
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The Fibonacci numbers are Nature's numbering system. They appear everywhere
in Nature, from the leaf arrangement in plants, to the pattern of the
florets of a flower, the bracts of a pinecone, or the scales of
a pineapple. The Fibonacci numbers are therefore applicable to the
growth of every living thing, including a single cell, a grain of
wheat, a hive of bees, and even all of mankind.Fibonacci slides.
...
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The Year of Creation, 5508, and the Golden Section
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In Byzantine Julian calendar was used, and the counting was conducted from the 5508 BC which was the Year of Creation as adopted in the 7th Century at Constantinople by the Eastern Orthodox Church.This creation of the world and the beginning of Being took place in the spring period, in the month of March, 5508 B.C. On Friday, the sixth day of creation at noon, man was created.
The Year of Creation, 5508, is related to the Golden Section, Phi...The Number 666 is related to phi
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Human Body and the Golden Section
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It is a very
proveable fact that our human bodies are phi-designed as the golden
section template is intimatelyseen throughout our whole human form
ratio's. This absolutely proves that we like the macrocosm (the
planets, and stars) or the microcosm (of atomic and subatomic
particles) all were created using the PHI design
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The Golden Section in Art and Architecture
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Web
Gallery.Golden section in art and architecture
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Leonardo Da Vinci and Platonic Solids
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The constructions of the Platonic solids are included
in Book XIII of Euclid's Elements. Propositions 13 through 17
describe the construction of the tetrahedron, octahedron,
cube, icosahedron, and dodecahedron in that order. Plato was
mightily impressed by these five definite shapes that
constitute the only perfectly symmetrical arrangements of a
set of (non-planar) points in space, and late in life he
expounded a complete "theory of everything" (in the treatise
called Timaeus) based explicitly on these five solids. It's
uncertain who first described all five of these shapes - it
may have been the early Pythagoreans - but some sources
(including Euclid) indicate that Theaetetus (another friend of
Plato's) wrote the first complete account of the five regular
solids. Plato conceived the four classical elements as atoms
with the geometrical shapes of four of the five platonic
solids .
...
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Virtual Gallery of Leonardo Da Vinci
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Leonardo Da Vinci Web
Art Gallery
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Drawings of Leonardo Da Vinci
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In his pictures, drawings and notebooks Leonardo expressed the code of nature ( Da Vinci code ). It is in connections with the mathematics of the Golden section and also with his research of the numerology.
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Milija Belic ART GALLERY
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Milija Belic Web
Art Gallery
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Virtual Library of Useful URLs
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Links to the
other sites on Pascal Triangle, Fibonacci numbers and Golden
Section |
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