# High School Materials

## Geometry in Art and in the Real World in General

Geometry is said to study "the properties, measurement, and relationships of points, lines, angles, surfaces, and solids". Our approach in this course is to study those lines, surfaces and other geometric objects and show how they appear everywhere in the world around us.

We will explore geometry as it is found in art, in road signs, in patterns we find in buildings, and in may other places.

To the student: You will be asked to do some reading, so that you can learn some of the material at your own pace. We try to include plenty of pictures to help you understand what we are talking about. Feel free to ask your teacher if something is not quite clear though!

### Applications of geometry in the real world

But before we delve into all the geometry, let's take a look at some people who use geometry. You will see it's more common than you might think.

Designers use geometry when they develop patterns used in buildings.

The two pillars in Baarn

The pillars you see here were designed by the graphic designer /artist M.C. Escher. The designs were creted using some techniques in geometry we will be talking about this semester. It looks pretty tricky, but if you know enough geometry you can make designs like this too. Below are some close-ups that show the square tiles that create the pattern. Geometry will explain how to create the flower designs and the animals.

Facade of the Lyceum in The Hague

This high school in The Hague, the Netherlands has a beautiful facade with a design involving winged horses. In the Escher museum a page from the architect's office can be seen which shows the dimensions of the design. The front panel measures 497 x 787.5 (centimeters) and the sides are 497 x 307.5 (centimeters).

 Detail from the left side Notes from the architect

Notice that the underlying geometric shape is again the square. Here Escher used geometry to transform the squares so that they turned into winged horses. If you look at the drawing made by the architect you will notice that that drawing too is all geometry. It's a rectangle in the center and two other 4-sided shapes besides that.

There are several architects who are known for their geometric designs. One is for instance Frank Lloyd Wright.

Wright designed houses, and some of the features of the houses like the window shown here. As you can see, the whole design is based on geometric shapes.

An entirely different area that uses geometry is the movie industry. Animated movies like Toy Story rely heavily on geometry, as do computer generated images used in special effects.

One example given in computer science is for instance the so-called Stanford Bunny. The idea is that if you want your computer to draw a bunny, it needs fairly simple instructions. One way to "build" your bunny on the computer is to create it from triangles (geometric shapes are easy to feed into the computer). A bunny made up or some 350 triangles would look something like this:

If you want a better looking bunny, then you use more triangles. There are even more sophisticated techniques that work even better and those are geometric in nature as well.

Kasimir Malevich, Suprematism, 1915.

And finally there is geometric art. This painting by Malevich is just one of many examples. You might notice about 3 squares and many differnt size rectangles that make up the painting. The colors create more interest, but all the shapes used are geometric in nature.

The painting is made in Oil on Canvas. It is now in the Stedelijk Museum in the Netherlands.

## Course Materials

The online materials are supplemented with worksheets from Key to Geometry by Hawley, Suppes, Gearhart, and Rasmussen (Key Curriculum Press) Product Information Other useful resources are:

• Read the introduction above. This will show you some examples of how geometry is used in the real world.

## Part I             Points, Lines, Segments and Polygons

 Basic Concepts Points, Lines, Segments, Angles etc. Squares, Rectangles, Parallelograms and Other Polygons Online Worksheets For This Section Lines and Line Segment Worksheet Angles and their Properties Worksheet Recognizing Polygons Worksheet

In Key to geometry the following pages cover the material:

• Lines: page 3-5
• Line segments and polygons: 6-18
• Intersection Points: 19-20
• Lines, segments and polygons again: 21 - 25
• Lines determined by 2 points: 26-27
• Lines and line segments: 28-32
• Vocabulary Quiz: 33
• Naming and finding triangles: 34-38

### Symmetry

There are two types of symmetry we will look at.