More About Pascal's Triangle

Blaise Pascal was not the first man in Europe to study the binomial coefficients, and never claimed to be; indeed, both Blaise Pascal and his father Etienne had been in correspondence with Father Marin Mersenne, who published a book with a table of binomial coefficients in 1636. Many authors discussed the ideas with respect to expansions of binomials, answers to combinatorial problems and figurative numbers, numbers relating to figures such as triangles, squares, tetrahedrals and pyramids.

Where Pascal's Triangle Really Started

The so called "Pascal" triangle was known in China as early as 1261. In 1261 the triangle appears to a depth if six in Yang Hui and to a depth of eight in Zhu Shijiei in 1303. Yang Hui attributes the triangle to Jia Xian, who lived in the eleventh century. They used it the same way we do, as a means of generating the binomial coefficients. It wasn't until the eleventh century that a method of solving quadratic and cubic equations was recorded, although they seemed to have existed since the first millennium. At this time Jia Xian generalized the square and cube root procedures to higher roots by using the array of numbers known today as the Pascal triangle and also extended and improved the method into one usable for solving polynomial equations of any degree.

Kyle Hardin

Without pascals triangle, math would be like a day with out sunshine. Many people underestimate the magnitude pascals triangle has on the world today. It has influenced me and changed my perspective of algebra.

Kyle Hardin

This reminds me of pascal triangle and shows many reflections which are often used in math along with other tessalations of the mind.

pascals triangle confuses me greatly even though my teacher, mrs. petit, has explained it to me many times. sometimes it helps me do hard problems in math and sometimes it doesnt

This is pascals triangle. it is used very often in math.

This is pascuals Triangle.

a tesselation is caused by a shape being repeated over and over again in different dirrections with no gaps or spaces

this is a picture of a tesselation. the angels are reflected across the x and y axis.

Angel Investing and Teselations

An angel investor or angel (also known as a business angel or informal investor) is an affluent individual who provides money for a business start-up, usually in exchange for ownership equity. A small but increasing number of angel investors organize themselves into angel groups or angel networks to share research and pool their investment capital. Angels typically invest their own funs. Angel investing deals with money, which deals with numbers, which deals with math, because in math, you use numbers. That is how angels are mixed in with math. :)

A tessellation is created when a shape is repeated over and over again covering a plane without any gaps or overlaps. Another word for tessellation is a tiling. The dictionary says that the word "tessellate" means to form or arrange small squares in a checkered or Mosaic pattern. The word "tessellate" comes from the Ionic version of the Greek word "tesseres," which in English means "four." The first tilings were made from square tiles. A regular polygon has 3 or 4 or 5 or more sides and angles, all equal. A regular tessellation means a tessellation made up of congruent regular polygons.

Angels

The angels in this picture represent math by reflection. The angels reflect across the X and Y axis and/or are upside down and flip in a different way. Reflection is when an image is produced by reflecting as by a mirror. This picture shows that with the angels. Many are reflected, rotated, or flipped showing the same image, just in a different manner.

This picture is of a tessellation. It is very pretty. There are no gaps between the angels. That is because it is a tessellation. There aren't supposed to be any gaps between the angels or other shapes used in the making of tessellations.

The butterflies in this picture are set in a way where if you cut the picture into 4 quarters they would be the same.

The butterflies are symmetrycal. They seem to contrast against each other and set them off.

Just like a butterfly's wings, my love for math never tires. There are so many new problems you can do just like the scales on the wings. This changed me greatly.

Inoticed that the picture resembled butterflies and was a catalyst to my mathmatical thoughts going through my brain. The wings reminded me of linear pairs and changed my view of them drasticly.

The wings of the blue butterflies are perpendicular to each other. there are 2 pairs of parrallel wings.

Butterfly

The wings of the butterflies are symetrical to one another. There is a square in the middle of the butterflies. It looks like a cross turned on it's size.

A Butterfly Lesson

There are many ways you can put butterflies into a math lesson, but I personally like this one the best. In this particular picture, you must solve the math problems to figure out the color. If you get them all right, you can color the picture!

Butterflies Mixed in With Math?!?

Who knew that butterflies could be used in math! Butterflies are used everyday to teach young children about the line of symetry. Kindergarten, first, and second graders use butterfly wings to learn about the math concept of symetry.

Fractals-Julia Sets

Julia sets are in the complex plane, where the horizontal axis represents real numbers, and the vertical axis represents imaginary numbers. Each Julia set is determined by its constant value, c, which is a complex number. A common formula used is Z=Z^2 +C.

Applications of Fractals

One of the earliest applications of fractals was in computer graphical modeling. Storing a database of terrain features is difficult, but storing an algorithm which can generate pictures which look relatively realistic on the fly is easy. Other applications include random number generation.

The Simplest Fractal

The Cantor Set is one of the most simplistic fractals, inroduced by Georg Cantor in 1883. It is formed by removing the middle one third of a straight line and repeating the process for the resulting lines for infinite times.

More About Fractals

In math, any of a class of complex geometric shapes that commonly have fractional dimension, is a fractal. Fractals are distict from simple figures of classical geometry. These are the square, circle, sphere, and etc. They are capable of describing many irregularly shaped objects. The term "fractal" was derived from the Latin word "fractus" which means fragmented or broken.

Fractals

The geometry of artificial things, objects, such as surfaces and other regular forms can be portrayed and depicted very accurately with mathematical equations. The objects we found in nature such as clouds, mountains, bark or lightnings are on the other hand usually irregular or fractured in their forms. Anyway, also some sort of normality, regularity and mathematical principles are charasterictical to them.

Different Fractals

Mathematical fractals are infinite and endless. Fractals of the nature are always finite and at least in some ammount irregular.

Features of a Fractal

A fractal has many different features. A fractal has a fine structure at arbitrarily small scales. It is also too irregular to be described in traditional Euclidean geometric language. Fractals are self-simular and have a simple and recursive definition.

History Of Fractals

The mathematics behind fractals began to start in the 17th Century when mathematician and philosopher Leibniz considered recursive self-similarity. It was not until 1872 that a function appeared whose graph would be considered a fractal today. This is when Karl Weierstrass gave an example of a function with the non-intuitive property of being everywhere continuous but nowhere differentiable.