### Math•Art

#### Art Club Family Art Night Students

Here is the information and downloads you'll need to transform a great architectural landmark of Paris into the polynomial equation that represents its shape.

You may recognize this structure as the world famous St. Louis Arch.  The St. Louis Arch has a very smooth and mathematically friendly shape.
The mathematical shape of the St. Louis Arch is

y = -0.2x² + 3x + 0.5

We were able to create this mathematical equation from the red dots you see on the grid paper that was superimposed on the picture of the St. Louis Arch.
How can dots on a grid paper be turned into a mathematical equation, you ask?  Well, it's pretty simple with a little help from Microsoft Excel®.  All of this will be explained in the steps below.

Any smooth shape can be converted into a polynomial (mathematical) equation.  For example, the shape of the Coka-Cola® bottle is recognized throughout the world.  But did you know it too has a very mathematical shape?  In fact, the mathematical shape of the Coca-Cola bottle is

y = ─7x6 + 6x5 ─ 0.004x4 + 0.07x3 ─ 0.7x2 + 3x + 6

WOW!  That's one heck of a polynomial equation!
By the way, a "polynomial equation" is a mathematical equation that contains the variable " x " raised up to higher powers like x2, x3, x4, x5, and so on.  You'll learn a lot about polynomials in my 8th grade algebra class.
So as you can see, even the most complex of shapes can be transformed into polynomial equations.

#### Steps for Transforming a Great Architectural Works of Paris into Polynomial Equation

STEP 1
Select one of the great architectural works found in or around Paris France.
Choose one of the landmarks from the list below and look on the Internet for a nice picture of it.

Landmarks in Paris, France
1. Musee D’Orsay
2. Louvre Museum
3. Les Invalides
4. Opera Garnier
5. Centre Georges-Pompidou
6. Notre Dame in Paris
7. C.N.I.T. building in Paris
8. Sorbonne University in Paris
9. The Pantheon in Paris
10. Arc de Triomphe
11. Luxembourg Palace and Gardens
12. Palais-Royal in Paris
13. Ecole Militaire
14. Cartier Foundation in Paris
15. Palais de Chaillot – Trocadero Museum
16. San Chapelle in Paris
17. Parc de la Villette – Cite des Sciences
18. French Naval Ministry at the Place de la Concorde
19. Grand Palais
20. Pettit Palais
21. Place du Tertre and Sacre-Coeur

Landmarks in France, Outside of Paris
1. Chateau de Versailles and Gardens
2. Fontainebleau Palace in France
3. Chartres Cathedral

STEP 2
Once you locate a good picture of the landmark you are interested in, download a copy of it and save it to your directory.

STEP 3
Right click the following graph paper choose "Save Target As..." to save this image in the same directory where you saved a copy of your favorite landmark.

STEP 4
Open the Microsoft Paint program and then open the file of your Paris landmark.
Your Paris landmark should be large enough to fill the whole edit page of Microsoft Paint.
It's OK if the image is larger than the screen.  I can help you re-size it to the perfect size of Paint's edit window.
If it doesn't fill the whole screen, you should go find another image of your Paris landmark, save it, and try again.

STEP 5
Open another copy of Microsoft Paint and then open the graph paper file you downloaded and saved.

STEP 6
Select the entire graph paper (tip: use "Ctrl-A") and copy it to the clipboard (tip: use "Ctrl-C").

STEP 7
Tab over to the other Microsoft Paint where your Paris landmark is displayed and paste (tip: use "Ctrl-V") the grid paper onto the picture of your Paris landmark.  DON'T TOUCH ANOTHER THING BEFORE YOU READ THE NEXT STEP!

STEP 8
(see me, Mr. Tomczak, for help with this step)
While the grid paper is still "floating" from the previous pasting step (STEP 7), move the graph paper so the origin, the (0,0) point, lines up with the left beginning side your Paris landmark.

STEP 8
Now you're ready to start "• • • dotting • • •" the shape of the structure so it can be digitized.

STEP 9
Start from the left and make your first red dot right on the Y-axis (horizontal axis).
Then carefully add red dots along the outer structure of your shape until you go clear across to the other side of your image.

STEP 10
Rename your • • • dotted • • • image and save it under a different file name.
Make sure you are saving as you are going along.

STEP 11
Right click this Microsoft Excel spreadsheet and choose "Save Target As..." to save this spreadsheet to your directory.
In this spreadsheet you will enter the (x,y) values of the red dots.
This is called digitizing the image.

STEP 12
Open the Excel spreadsheet you just downloaded.  It should look like the following:

STEP 13
Now read the (x,y) values of the red dots off your picture and type them into the spreadsheet.
The spreadsheet is set up with about 100 points.
You can add more to the bottom of the list if you need to.
Your spreadsheet should begin to look like this: