Pascal's Triangle

Pascal's Triangle is more than just a big triangle with numbers. There are two major areas where we use Pascal's Triangle, in Algebra and Probability. It is used in Probability to find Combinations. Such as how many ways you can pick so many things out of the total given. The triangular numbers and Fibonacci numbers can be found in Pascal's Triangle. The triangular numbers are easier to find: starting with the third one on the left side go down to your right and you get 1, 3, 6, 10, etc.


Renee

Pascals Triangle

This is a pictur of the triangle that I was talking about. Its pretty amaizing if you ask me.

Matt

Picture of the Week Pascal's Triangle

One of the most interesting numerical patterns in this world in my opinion is Pascal's Triangle. A triangle that uses geometric arangement to tell binomial coefficeints in the shape of a triangle. This triangle was named after Blaise Pascal, but this is unfair considering that others before him had studied the same patterns.Personally I can say that this triangle has came in very handy when I was learning the long division of algebraic equations..

--Jenny

Pascals Triangle

This triangle is known as Pascals Triangle and is a very useful tool in math. It uses binomial expansion and can always be depended upon to help yourself out.

Matt

Pascal's Triangle

In Pascal's triangle Blaise Pascal used binomial expansion. The outer edge of diagonals are only 1's. The next layer of diagonals are all the natural numbers. The inside of the triangle is made from adding the other numbers. The pattern resembles Sierpinkski's triangle.

-Kristin

Picture of the Week Jan. 25

In Mathmatics patterns are always something that you will see, as 1st graders eleven years ago we made patterns out of little blocks to learn our additions tables(Or atleast I did). As we've grown older the patterns we see in Mathmatics have gotten more and more complex. In tenth grade we learned about Sierpinski's Triangle that in a short explanation is multiple triangles scaled down. While we also learned Koch's snowflake and had to eventually draw one of our own. In tenth grade geomotry we also learned about Pascals Triangle. Which is this week's picture of the week . This Triangle is an interesting, complex, well worth knowing, triangle that uses numbers to tell us the binomials of some algebraic mathmatical problems.

--Jenny

Picture of the Week-Jan. 25

The pictue is of Pascal's triangle. It is named after the mathematician Blaise Pascal. He used the binomial theorem and expansion to create many more things. The first number in the first row is 0 + 1 = 1, whereas the numbers 1 and 3 in the third row are added to produce the number 4 in the fourth row. This pattern continues on infinitely.

Renee

Picture of the Week - Jan. 25

This triangle is Pascal's triangle. The rows in the triangle are number starting with zero. In row one, 1+1=2. The 1 and 2 in row three create sum of 3 in the fourth row. The pattern continues forever. In this example, it only goes to row five. The set of numbers that form the triangle were discovered before Pascal, but Pascal was the first to organize them into the triangle.

-Kristin

Tessellations

Regular tessellations are only made up of equilateral triangles, squares, and hexagons. Wallpapers and floor tiling are the most common everyday use of tessellations. Geodesic domes are designed by tessellating the sphere with triangles that are as close to equilateral triangles as possible.

Kristin

Picture of the Week - Jan. 18

A tessellation is a collection of plane figures that fills the plane with no overlaps and no gaps. Tessellations are seen throughout art history, from ancient architecture to modern art.

-Kristin

Tessellations

Many transformations can be used to make a tessellation. I agree with Jenny when she said that last weeks picture was just an example of what this weeks picture was going to be. Mrs. Petit kind of threw me off with the different characters. Angels represent the good, while the demons represent the bad in our society.

-Renee

Picture of the Week-Jan. 21

Well obviously, we are all guessing that the picture is a tessellation. This is a picture of a honeycomb. This is an example of a tessellated natural structure. Many items used to make a tessellation are symmetrical.

-Renee

About the Picture 3

I came to the final conclusion that it is a tessellation and thats the only thing I could think of it being. When I looked at the picture, it tessellated my mind and thats what gave me the final push to make my desicion.

Matt

Picture of the Week: 3!!!!!!

When I look at this picture it really brings out the demons in me. I feel like a better person after staring at this because it puts me into a swirl of angels and keeps me away from the demons.

Matt

About the Other Post

This picture has some translational and rotational symmetry. This is a great example of how they are used in the real world of math and physics.

Matt

Picture of the Week- Janurary 11.

This is a picture of an innocent butterfly that was waded up into a symmetrical masterpiece. This picture brings tears to my eyes when I look at it and throws me into a world of wonder.

Matt

What is a Tessellation.

Well in my last post I guessed that Mrs. Petit's picture of the week was a Tessellation. Now I think I'm going to discuss what that is and even if I am wrong about what the picture is I am hoping Mrs. Petit will give credit for trying.A tessellation is a combination of one or more shapes that, when repeated, cover a surface with no gaps and no overlaps. We created one of these in Pre-AP Geometry my Sophomore year when we decorated a Valentine's Day card. I used a lightning bolt while some of my fellow classmates used crosses, boots, and even a penguin. Well, the word tessellation came from the Latin word "tessella", meaning a small square tile often used in ancient Roman mosaics. A tessellation is often called a tiling, which you can see in a modern day checkerboard. With some research I also learned that it is possible to Tessellate non-plane surfaces, such as a sphere. My picture is a spherical version of Mrs. Petit's picture of the week.

--Jenny

Picture of the Week January 18 2010

Well, with this weeks Picture of the week Mrs. Petit really out did herself.I have stared at this picture for hours and all I can see are some angels and demons that appear.They seem to be rotated in numerous directions, and make a pattern, but just what in the world could it be. My guess is this is a Tesselation that used rotation, translation, and reflection to create a complex picture of angels and demons.Thanks Mrs. Petit I knew you were working your way up to this with last weeks picture.

--Jenny

In this picture, you can find many lines of symmetry. There are patterns like this seen in everyday life. Symmetry is everywhere. Rugs and paintings are examples of household items that could have symmetry. Symmetry can be used in everyday life, geometry, other math, science, tecnology, and art.

-Kristin

Symmetry generally means a use of proportion, balance, or patterned self-similarity. The most familiar type of symmetry to most people is in geometry. Mirror symmetry (like in the picture of the week) is symmetry that is reflected on an axis. Another way of thinking would be that if the picture were folded over an axis, the two halves would look identical. Isosceles triangles, quadrilaterals, and isosceles trapezoids are shapes with symmetry.

-Kristin

Picture of the Week-Jan. 11

I would have to agree with Jenny. Symmetry is something we don't realize that we use so often. This picture of symmetry uses many different geometric shapes and transformations. These things add to the visual effect of the picture as a whole. It makes it more interesting I'll just say.

-Renee

Uses of Symmetry!

In everyday life we don't really think about if we are using symmetry. We are though. Think about the books we read though the words on the pages arent' the same. The pages are the same on either side of the book. Alot of rugs use symmetry and rotations also.

--Jenny

Picture of the week January 11, 2010

When you first look at this picture of the week it looks just like a butterfly with mishapen wings. There is so much more to this pictures story though. When you look deep into this picture you see Symmetry, rotations, reflections, and it appears to be centered as if it was on a coordinate plane. Symmetry is refering to the shapes ability to look the same, across an axis. This picture shows rotational and translational symmetry.

--Jenny

Sierpinski Triangle

Also called the Sierpinski gasket. Named after the Polish mathematician who described it in 1915. It is a basic example of a self-similar set. It is a mathematically generated pattern that can be reproducible at any magnification or reduction.

Kristin

Menger Sponge Fractal.

In mathmatics the Menger Sponge is a fractal curve.It is a universal curve in the fact that it has a topological demension. Each face in a Menger Sponge is a Serpenski Carpet. The basic idea that is behined the sponge is it looks like a Rubiks cube.

Jenny

Koch Snowflake

The Koch snowflake is a mathematical curve and one of the earliest fractal curves to have been described. The Koch curve has an infinite length because each time the steps above are performed on each line segment of the figure there are four times as many line segments, the length of each being one-third the length of the segments in the previous stage. The Koch curve is a special case of the Cesaro curve.


-Renee

Example of Fractals

Sierpinski Carpet

It was first described in 1916. The carpet is a generalization of the Cantor set to two dimensions. It is described as a universal curve. The technique can be applied to repetitive tiling arrangement; triangle, square, hexagon being the simplest.

Matt

More Fractals

More applications include:

• Generation of patterns for camouflage, such as MARPAT
• Computer and video game design
• Generation of new music
• Seismology

Matt

Fractals

Fractals are used in many ways.
Some applications include:

• Signal and Image Compression
• Classifications and histopathology slides in medicine
• Digital Sundial
• T-shirts and Fashion
• Generation of various art forms

Kristin

Fractals

The math behind fractals began in the 17th century when the mathematician Gottfried Leibniz considered recursive self similarity. Benoît Mandelbrot coined the term "fractal." A mathematical fractal is based on an equation that undergoes iteration.


Jenny

Picture of the Week-1/07/10

The picture is of a fractal. A fractal is a geometric shape that can be split into parts, each part is a reduced size copy of the whole. It has a fine structure at a small scale. It has a simple and recursive definition.

-Renee