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Math Exploration

Pascal’s
Triangle, named after Blaise Pascal who was a famous French mathematician and
philosopher, displays many patterns within it that are useful in everyday math
situations. Pascal’s Triangle can be used to solve problems involving
probability. It can also be used to discover how many different combinations of
things are possible. Another situation in which Pascal’s Triangle proves to be
useful is in differentiation from first principles, Pascal’s Triangle can also
be used to find coefficients which aid in simplifying nearly all problems.

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Though named
after Blaise Pascal, the origin of this pattern actually dates back hundreds of
years before his birth. The Persians and the Chinese seem to each have
discovered the pattern independently during the eleventh century. A Chinese
mathematician by the name of Chia Hsien was documented using the triangle to
find the square and cube roots of various numbers. A Persian mathematician,
Omar Khayyam, was also documented using something that could be seen as
Pascal’s Triangle. Later on in China, several Chinese mathematicians continued
to work on the study of this topic to solve for equations higher than a cubic
value. Yang Hui, one of these Chinese mathematicians, provided the earliest
recorded research of Pascal’s Triangle. Later, Zhu

Shijie demonstrated the use of
Pascal’s Triangle through a visual representation in the year 1303. In this
representation, Shijie spoke about the pattern as if it was an ancient concept.
This explains the idea that this pattern was known and used long before Blaise
Pascal was even born, though he did popularize it.

Pictured on the next page, from left to
right: Blaise Pascal, Omar Khayyam, and Chia

Hsien

Years after it first appeared in Persia and China, the
triangle came to be known as

Pascal’s Triangle with Blaise Pascal’s
completion of the Traité du triangle
arithmétique in

1654.  Making use of the many already known binomial
coefficients, French mathematician, Blaise Pascal, developed many of the
pattern’s ideas and applications within these writings.  Although Pascal is best known for his work
with the arithmetic triangle, he made many other contributions to mathematics
thirty-nine years of life, Blaise Pascal also discovered an important geometry
theorem, produced work regarding cycloids, created an early form of a
calculating machine, created the early foundation of the study of probability,
and began the research and interest in the topic of calculus.  Pascal’s contributions to mathematics,
especially of the popularization of his triangle, were undoubtedly brought
forth from the mind of an extremely intelligent and free-thinking man.

Pascal’s
Triangle is known for its patterns. The patterns that exist within the triangle
are what make it such a great tool. For instance, in each diagonal line of the
triangle, the numbers follow a pattern. The first diagonal line is “1” s, the
second diagonal line is just numerically ordered numbers – i.e. 1, 2, 3…
etcetera. The third diagonal line makes up the triangular number sequence. This
sequence comes from a pattern of dots that make up triangles as they grow. The
pattern is 1, 3, 6, 10, 15, 21, 28… etcetera. All it takes to understand this
sequence is to visualize the idea that a triangle is made up of lines of dots,
and to continue the sequence one must continuously add lines of dots.

A formula can
also be used to determine triangular numbers. (I will include this formula
later; I just don’t know how to do the notation of it on my computer yet).
Another obvious pattern found in Pascal’s Triangle is the fact that it is
symmetrical. If you drew a line directly down the middle of the triangle, the
numbers on either side of the line would match up. Pascal’s Triangle is a
mirror image of itself. Yet another pattern that exists within the triangle is
hidden in the horizontal lines. If you find the sum of each horizontal line,
you will find that they double each time. The sum of each horizontal line is
added to itself to create the sum of the next horizontal line.

Pascal’s
Triangle can also be connected to the Fibonacci Sequence. Fibonacci’s sequence
can be found in Pascal’s Triangle. The Fibonacci Sequence was created by an
Italian mathematician who actually went by the name of Leonardo da Pisa, son of
Guglielmo Bonnacio, a merchant from Pisa. He created the sequence in 1175. The
sequence can also be formed in a more direct way, very similar to the method
used to form the Triangle, by adding two consecutive numbers in the sequence to
produce the next number. The creates the sequence:

1,1,2,3,5,8,13,21,34,
55,89,144,233, etc. . . .. Fibonacci’s sequence is pretty much all around us.
In nature, the amount of petals found on a flower is usually a Fibonacci
number, and the way in which a sea shell spirals as it grows progresses at the
same rate as the Fibonacci sequence. In architecture, art, and music you find a
constant called the “golden mean,” or phi, which is 1.61803 and
corresponds to the ratio that exists between two consecutive Fibonacci numbers
— the higher that the numbers in the sequence are, the closer that they match
the golden mean. A rectangle with a ratio of 1:1.61803 has been considered to
be aesthetically perfect for a long time. The front of the Parthenon in Athens,
for example, forms a rectangle of those proportions.

The Fibonacci Sequence can be found in the
Golden Rectangle, the lengths of the segments of a pentagram, and in nature,
and it describes a curve which can be found in string instruments, such as the
curve of a grand piano. The Fibonacci Sequence is a sequence of numbers in
which each term is the sum of the two terms that came before it. Within
Pascal’s Triangle, the sum of the numbers on diagonals become the Fibonacci
Sequence. The further that the sequence continues, the closer the ratio of a
term to the one before it gets to the Golden Ratio. The Golden Ratio is an
irrational number.

Another pattern found in Pascal’s
Triangle is Triangular Numbers. Triangular Numbers are just one type of
polygonal numbers. The triangular numbers can be found in the diagonal starting
at row 3 as shown in the diagram. The first triangular number is 1, the second
is 3, the third is 6, the fourth is 10, and so on. Another type of polygonal
numbers found in Pascal’s Triangle is Square Numbers. They are found in the
same diagonal as the triangular numbers. A Square Number is the sum of the two numbers
in any circled area in the diagram. (The colors are different only to
distinguish between the separate “rubber bands”). The nth
square number is equal to the nth triangular number plus the (n-1)th
triangular number. (Remember, any number outside the triangle is 0). The
interesting thing about these 4-sided polygonal numbers is that their name
explains them perfectly. The very first square number is 02.
The second is 12, the third is 22 (4), the
fourth is 32 (9), and so on.

Pascal’s
Triangle is commonly found in the study of physics. An external force being
applied to the surface of a liquid increases the liquid pressure at the surface
of the liquid. This increase in liquid pressure in transmitted equally in all
directions and to the walls of the container in which it is filled. This result
is called Pascal’s law which is stated: “Pressure applied at any point of a
liquid enclosed in a container, is transmitted without loss to all other parts
of the liquid”.

Pascal’s Triangle can also be
found in various other real life situations. These include, automobiles,
hydraulic brake systems, hydraulic jacks, hydraulic presses, and hydraulic
machines.

The braking systems found in cars, buses, and other vehicles
work thanks to Pascal’s law.

The hydraulic brakes allow for
equal amounts of pressure to be distributed throughout the liquid. When the
brake pedal is pushed, it exerts force on the master cylinder, which increases
the amount of liquid pressure that it has in it. The liquid pressure is then
distributed equally throughout the liquid in the metal pipes and then to all
the pistons of the other cylinders. Due to the increase in liquid pressure, the
pistons in the cylinders move outward, pressing the brake pads with the brake
drums. The force of friction between the brake pads and the brake drums stops
the wheels.

Works Cited

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