Chapter 1:

Geometric Shapes

As you look around you, there are many geometric shapes that can be seen - in plant forms, in fabric designs, in art and in architecture. You learned about some of the geometric shapes while you were reading the linked "outside" web page on constructions (see the Introduction page of MathArt Connections for that link).

You will also see many geometric shapes in plant forms and in fabrics for shirts and dresses. The following pictures show the hexagonal shape of a lily, geometric figures in clothing fabric samples, and triangles in an orchid:

(photos from http://www.iflowers.com/, a fascinating site where you can order flowers online, and fabrics from South Dakota Antiques at http://www.tias.com/stores/sda/)

The page linked below is on the Math Forum website, which you will find is a very valuable resource for mathematics. I would like you to take some time right now to explore the Math Forum website. Click on the link below and then explore all the links in "The Forum Features".

http://mathforum.com/dr.math/faq/formulas/faq.figuredef.html

"The Geometry Pages" is a another website that I created as a resource for my students, and you will find a description of many types of polygons in Chapter 3 of the Geometry pages, by following the link below:

http://mathforum.org/~sanders/

To summarize: a polygon is a geometric figure with any number of sides. Polygons are named by how many sides they have, so a triangle has three sides ("tri" is the Greek word for 3). Here are the mathematical names for some of the more common polygons, which we will be using in this class:

3 sides: triangle

5 sides: pentagon

6 sides: hexagon

8 sides: octagon

10 sides: decagon

12 sides: dodecagon

Mathematics can be found everywhere in the world around us, in everything from natural forms to human creations. You can find examples of many geometric shapes, especially polygons, in art and graphic design. Let's look at some examples of these. The following web page will show you some beautiful examples of mathematical artwork:

http://www.cs.berkeley.edu/~sequin/ART/

There are some particularly nice examples of geometric art at this next page:

http://www.graphica.com/gallery/

Piet Mondrian is one of the most famous artists to paint in a style called Pure Abstraction. To see some of his work, click on the link below:

http://www.ibiblio.org/wm/paint/auth/mondrian/

His work is described as follows: "Shapes and colors have always had their own emotional force: the designs on ancient bowls, textiles, and furnishings are abstract, as are whole pages of medieval manuscripts. But never before in Western painting had this delight in shape as such, in color made independent of nature, been taken seriously as a fit subject for the painter. Abstraction became the perfect vehicle for artists to explore and universalize ideas and sensations."

But geometry is not found only in abstract paintings! Every artist uses the basic principles of mathematics when drawing or painting any type of artwork, even the human figure. Visit the web page below, and read about the geometry of drawing:

Test Question #2: As explained on the "draw sketch" website above,what are the five elements of shape?

You have read that the triangle is the strongest, most rigid shape. This is why the triangle is used so often architecture, from bridges to buildings. Many bridges are built using an engineering concept called the "triangular truss"- a geometric design composed of triangles. The following web page will tell you why triangles are used in construction:

http://www.ce.ufl.edu/activities/trusslab/stable.html

The triangular truss can span great distances, as we can see in this bridge:

Engineers and architects used a complex triangular truss to build the geodesic dome of the US Pavilion exhibition hall at Expo '67 in Montreal, Canada in 1967 (photo from http://www.GreatBuildings.com/ )

Of course, examples of squares and rectangles appear in most buildings, as it is the most common shape in architecture. Why is this? Because this rectangles, each with its 90 degree angles, fits together perfectly! Doors fit into their frames, windows fit into the walls, and one wing of a building fits up against another wing of the building if all are rectangles:

(National Center for Atmospheric Research from http://www.GreatBuildings.com/)

Besides being part of the basic shapes of buildings, geometric figures are often part of the ornamentation of buildings. The ornamental tile work: in this building, L'Institut du Monde Arabe in Paris, France, is a good example. (photo from http://www.GreatBuildings.com/)

You will find many other examples at the following website:

http://www.islamicarchitecture.org/architecture/palaces/thealhambra.html

Polygons are geometric figures formed by straight lines. But there is a wide variety of mathematical shapes that are curved, rather than straight. The most well-known curve is called the Circle, and we see circles everywhere around us - from carnival rides to candies!

(photos from http://www.royaleastershow.co.nz/html and http://cakedecoco.com/)

The circle is only one of many mathematical curves called The Conics. You will find out more about the circle, and all the conics, by clicking on the following link:

http://mathforum.org/~sanders/geometry/

The Guggenheim Museum in New York City, designed by Frank Lloyd Wright, is a building based on circles. (photo from http://www.GreatBuildings.com/)

Beautiful geometric graphics can be made from circles, as you can see in this example of a mathematical shape formed entirely by circles:

The following link will take you to a web page that will show you how to construct this cardioid:

http://mathforum.org/~sanders/connectinggeometry/Cardioid.html

The ellipse is another interesting mathematical curve:

http://mathforum.org/~sanders/geometry/GP18Ellipse.html

. . . and the parabola has many interesting applications:

http://mathforum.org/~sanders/geometry/GP19Parabola.html

The Gateway Arch in St. Louis, Missouri designed by Eero Saarinen in a1947 competition was constructed from 1961 to 1966. This memorial arch, monument, and observation tower is a 630 foot high graceful sweeping tapered curve of stainless steel. It is the tallest memorial in the US. Although it may appear to be a Parabola, it is actually not - it is a close relative of the Parabola, another mathematical curve called a "Catenary". The arch is 630 feet wide at the base. (photo from http://www.GreatBuildings.com/gbc.html))

You will find interesting applications of some of the other mathematical curves at the web page linked below:

http://mathforum.org/~sanders/geometry/GP21MoreCurves.html

Many plant forms contain mathematical shapes: the seeds at the center of a sunflower grow in a geometric spiral (outlined in green below):

It may seem that polygons are quite different from curves, and in many ways they are. But sometimes the distinction between straight and curved lines is not very clear. We can see this in the beautiful graphic below, called a "Fractal":

This geometric shape appears curved, but is made up of many very small straight lines. The shape is based on a fractal called The Mandelbrot Set, named after its creator Benoit B. Mandelbrot who coined the name "Fractal" in 1975 from the Latin fractus or "to break".

So what is a Fractal? Mathematically, a Fractal is a geometric shape formed by repetition of a pattern, and can be made up of lines, curves, triangles, or other geometric shapes. The best way to really understand what Fractals is to look at an example.

One of the simplest Fractals is called the Sierpinski gasket, again named after its creator, a mathematician named Sierpinski. The following steps will show you how to construct this Fractal:

Step 1) Construct an equilateral triangle.

Step 2) Construct the midpoints of all 3 sides and join them.

Step 3) Construct the midpoints of the sides of all the 3 outer new triangles.

Step 4) Again, construct the midpoints of the sides of all the 3 outer new triangles, creating a beautiful geometric graphic!

You can continue this process indefinitely. This is a fractal, a geometric figure formed by continuing a process and repeating it over and over again. You can color this fractal in many different ways, creating beautiful artwork, such as the one below:

This next link will take you to a web page with a school project based on the Sierpinski Gasket.

http://tcanright.internations.net/carpet0001a.html

Here is another beautiful fractal , called The Koch Snowflake:

http://www.scidiv.bcc.ctc.edu/Math/Snowflake.html

Test Question #5: What is the interesting fact that is true about the length compared to the area of the Koch Snowflake?

"Extending beyond the typical perception of mathematics as a body of sterile formulas, Fractal geometry mixes art with mathematics to demonstrate that equations are more than just a collection of numbers." from the National Center for Supercomputing Applications.

But we can appreciate the beauty of the Fractals without the specific mathematics behind it, such as the group of beautiful Fractal images below:

The following website will tell you more about Fractals:

http://www.glyphs.com/art/fractals/what_is.html

And what do Fractals have to do with mathematics? The natural beauty of the Fractal gives students incentive to explore coordinate systems, counting schemes, pattern development, integer arithmetic, the concept of infinity, and other topics in the mathematics and science curriculum. Learn more about this at the web page below:

http://archive.ncsa.uiuc.edu/Edu/fractal/Ftopic.html

. . . And here is some more really beautiful artwork based on Fractals: be sure to visit Fractal Gallery 1, 2 and 3:

http://www.xs4all.nl/~wolter/index.html

You will find Fractal images based on circles only at:

http://xahlee.org/SpecialPlaneCurves_dir/InversionGallery_dir/inversionGallery.html

(All three of the links above came from the web pages beginning at: http://archive.ncsa.uiuc.edu/Edu/fractal/fractal_Home.html)

Chapter 1 Project

The project for this lesson is to create a graphic design, using geometric shapes. You might want to construct a Fractal (Sierpinski Gasket, a Koch Snowflake, or another type of Fractal from your reading) OR an "Op Art" geometric graphic as described in the Geometry Pages: http://mathforum.org/~sanders/geometry/ (chapter 3). If you would like to see some examples of graphics created by students like yourselves, please go to the Student Work section from the Table of Contents.

First do some research on the Internet, and find some Fractals or Op Art. Read some information about Fractals or Op Art, and explore any aspects that interest you. Design and construct a geometric graphic based on a Fractal. of Op Art.

• You should use only geometric shapes in your design (triangles, pentagons, hexagons, circles, half-circles etc.)
• Draw the graphic using compass and ruler, in pen on unlined white paper , then color it using colored pens or pencils OR construct the graphic using The Geometer's Sketchpad, hide all the points, print it out and then color it with colored pens or pencils. You might want to review the instructions on the introductory page at Introduction to Course.
• If you use pencil, paper and compass, lay out your design with light pencil lines, then complete the graphic using compass and ruler, in pen, then color it using colored pens
• If you use Sketchpad, hide all the points before you print it. (Click on one point. Then use the Edit menu to Select All Points. Then use the Display menu to Hide the points.)
• For both of these options, complete your geometric graphic using a black pen for the outlines, and colored pens or colored pencils for shading.

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