The Basics of Fractals

For our upcoming Math Club meeting on Mathy Holidy Ornaments, we'll be learning about the structure of fractals.  These animations found through Wikipedia give a visual representation of the interations involved in creating a fractal.

The famous Sierpinski triangle deletes the triangle formed by connecting the midpoints of each triangle.

source

A fractal is made up of copies of itself, as shown by zooming in on a "completed" Sierpinksi Triangle.


As described on the Wikepedia entry for Fractals, the above image shows the Koch Snowflake, "a fractal that begins with an equilateral triangle and then replaces the middle third of every line segment with a pair of line segments that form an equilateral 'bump'."

The animation above shows how fractals can be used to simulate moutainous terrain in addition to other natural structures.

 

Pascal's Pyramid

 
Math Club 5/15/2012 was all about Pascal's Pyramid, the 3D version of the famous triangle. We constructed 6 layers of the pyramid and looked at some patterns analogous between the pyramid and triangle.

Pascal's Pyramid is built as a regular tetrahedron, with the three lateral faces mimicking Pascal's Triangle. More interesting is the inner structure of the pyramid, where each interior number is the sum of the three numbers just above it. (Whereas the triangle is constructed with each entry being the sum of the two numbers just above.)

Sums of Rows and Layers

The rows of Pascal's Triangle have many interesting properties. When looking at Pascal's Pyramid, it's the layers that hold similar properties.

One example is the sum of each row of Pascal's Triangle is a power of 2, where the sum of each layer of Pascal's Pyramid is a power of 3!

Powers of 11 and 111

The digits in each row of Pascal's Triangle represent powers of 11, although for rows 5 and greater, "carrying" the tens digit is necessary.

The similar pattern in Pascal's Pyramid is probably easier to see than to describe. The powers of 111 are evident in the layers of Pascal's Pyramid.

 Binomial and Trinomial Expansion

The connection between the Binomial Theorem and Pascal's Triangle is a pretty common Algebra 2 topic.

And the rewards are even greater using Pascal's Pyramid for a trinomal expansion! An example is shown.

 Sierpinski's Triangle and Sierpinski's Pyramid

Although we didn't build enough layers of Pascal's Pyramid to provide visual evidence, we could imagine that removing the even numbers would make a Sierpinski Pyramid in the same way a Sierpinski Triangle is formed in the 2D version.

Tri-College Math Contest Information

Tri-College Math Contest

Concordia College

Monday, February 27^th

(We will meet at the Old Main building at 9:10.)

9:10 Report to Old Main (don’t be late)

9:30-10:15 Individual Exam

10:25-10:45 Team Exam

10:45-11:30 Lunch/Break

11:30-12:00 Awards Ceremony

*Cars may park in any lot on campus, but not in reserved parking spots.  Plan to come around 9:00 to find a parking spot.

*The lunch break can be spent in the Knutson Campus Center (building 12 on the map) – they have a café, Caribou Coffee, and concessions.

*Bring pencils and a calculator (no TI-89) – calculators allowed on Team Test only.

*No penalty for guessing. Answer every question.

*Casual dress.  Packer gear if you have it.

Call or text Mrs. Backlund at 701-200-2037 ASAP if you have an emergency

and need to be late or absent.

I need all 16 of you to make up two teams of 8.

Second Math Club Meeting: CALCULATOR PROGRAMMING

Ms. Courtney Johnson presented a beautiful introduction to calculator programming.  Calculator programs were integrated into her high school curriculum in Breckenridge, MN.  We learned how to use the functions:

Store
Output
Disp
ClrHome
Input
Prompt
If
Then
End
Stop
As well as some more advanced topics like:
Else
Menu
Lbl
Goto
Pause
For
We tried out programs that calculated slope between points, determined even and odd numbers, listed factors, GCF, and LCM of numbers. The ideas were challenging, yet understandable with a bit of trial, error, and practice.