Sean Richards

Sept. 13^th, 1999

West Chester University

The objective of this lesson plan is to open student’s eyes to the actual size and diversity of the subject of mathematics. This will be done through the study of the development of perspective drawing. This will push forward the truth that mathematics is not just about numbers. It is a combination of many streams of thought that over time rise and fall to create both breakthroughs and revolutions in science, art, engineering, and architecture to name a few. To show the students that there is significant history to mathematics, as well as a strong motivation for the developments we study.

It is strongly recommended to have the students understand parallel lines and the intersection of planes in space. Since we are dealing with a 3-dimensional world in this lesson, knowing how planes and lines intersect and interact with one another is vital. From this it will be much easier to approach a 3-dimensional scenario where parallel lines intersect!

-One set of diagrams and illustrations that can be transferred to transparencies. These transparencies help clarify some of the more complex ideas.

-Two 4 x 4 grids. One grid is a ‘flat’ the other is a ‘perspective.’

-This lesson has included with it a set of pictures that should be made into transparencies for easy viewing. The process is fairly simple at a copy center and will add greatly to the student’s appreciation of what is being taught. A selection of a couple pictures should be enough. The setting of 200% magnification is best for filling an 8 ½ x 11 page with a 4 x 5 picture.

Step 1: Hand out the ‘flat’ 4x5 grid. The students should take this home and draw something on it. Artistic talent varying it will be important to have spare sheets ready the following day. Inevitably students will draw things that are either too complex or too simple to make the lesson enjoyable. Looking over them to check the difficulty will be very important.

Step 2: Hand out the ‘perspective’ grid. This grid is tilted and in perspective. Tell the students to pay careful attention to the intersection points of their drawing and the ‘flat’ grid. Take their image and copy it onto the ‘perspective’ grid. Once they have completed transferring the image, they have drawn a floor in. It is important to make sure the image from the first grid is simple enough to transfer. Practice it yourself a few times to see what is realistic and what is not. It is important to let the students run wild a little if they want to, and to let them challenge their respective abilities. It may be necessary to hand out an extra set of grids to give them the option to start over if they find it is going to be too difficult.

The lesson focuses on the development of perspective drawing as both an art and a science. It explains how throughout time math has motivated these revolutions and how mathematics can be a very trendy subject. To motivate this we start with the history of art, painting, and architecture focusing on their limitations as a result of 2-dimensional representation. This gives the starting point of Brunelleschi and his perspective design, as a result of collaboration with a mathematician and academic named Tuscanelli, and allows a class to follow the general results of such a discovery in the way we view things today.

Pre-renaissance art had a great lack in perspective. The art was flat and didn’t do the real world any justice, or at least the world as we perceive it. Stained glass windows are another great example of this style of art. The art from stained glass windows took much of its technique from the illustrations in early hand copied editions of the Bible. In Arabic art depiction of plant or animal is strictly forbidden and much of the work involved patterns and pure geometric shapes. If we were to search for the origins of perspective drawing this would eliminate one possible civilization. The Chinese and other West Asian cultures have had beautiful perspectives of landscapes. Most people will remember the rolling hills with mountains in the background. Even careful study of these though will show that they are a series of 2-dimensional drawings layered on each other. At the time this leaves Greece and Egypt as possibilities. Egypt’s hieroglyphs and art have a lot in common, this means that they are primarily 2-dimensional. And that strikes one more possible civilization. Last but not least is the Greek/Roman culture. People are most familiar with the resilient statues of these civilization or rather two civilizations. Greeco-Roman architecture contains some advanced observations of perspective, but the technique and art of placing a 3-dimensional image on a 2-dimensional canvas did not develop in this time. Well that was our last one, of that time period. So it must have developed later, much later in fact.

This brings us to the fact that perspective drawing really wasn’t developed until the 15^th century. The first popularly known perspective drawing was made by a builder by the name of Brunelleschi. At the time he was living in Florence, and entering into a competition. He needed his entry to really stand out. He needed something fresh and new that would give his entry that edge. You see Brunelleschi was not just a builder, he was an academic of sorts. And he was competing for the right to build a dome on a church. We should really say cathedral. Now Brunelleschi was not just a builder. Really at the time no one stuck to just one trade, at least the really talented people. It brings up the term Rennesaince man, these people are thought of as not the best at any one trade, but jacks of all trades. Remarkebly I find this to be an underestimation. They were made masters of many trades through their diversity. Brunelleschi attended a school called the Abacus school. It’s an odd name since they did not use abacuses at all and in fact these schools were the forerunners of our modern arithmetic. At these schools Arabic arithmetic was taught, and constantly shown to be significantly faster and more efficient. The majority of the Abacus school students were artisans. Artisans that were looking to better themselves through understanding of mathematics. I know it seems odd given all that people have said about math and art people being different thinkers, but it’s all hogwash. Back to our story. At the abacus school Brunelleschi made the acquaintance of Tuscanelli.

Tuscanelli was a more hard-core mathematician. He studied the historic work of Padua, he advised Christopher Columbus (sorry to burst the bubble of all you Bugs Bunny fans), and on top of that was a university graduate. Generally he was a very knowledgeable man that had access to many of the great works of historic mathematics. You see math is trendy. There are hot topics, and there are things that are allowed to disappear into the shadows of history. Inevitably these things are bound to be important, otherwise no one would have bothered with them in the first place. The problem is people are occasionally ahead of their time. They discover great things but aren’t sure how to use them, or no one understands what the braniac in the corner is mumbling about with that evil grin and it gets forgotten for a time. The ground work for perspective was laid in the time of Padua, just no one saw why it was useful or even understood how to do it.

This brings us back to Brunelleschi. To win the contest he collaborated with Tuscanelli and made a perspective drawing. It was a really wild one too! He painted a building. Then cut a hole in the canvas so people could look through. He set the painting up facing the building it was a representation of and moved a mirror in between the building and the painting so it reflected the perfect image of the building and you could hardly tell which was which. As you can imagine people were amazed. Brunelleschi got the job and he lived happily ever after, but help us all for what he did to the art of the time. This perspective idea caught on like wild fire, and to such an extent that it’s origin is all but forgotten by many whom practice it.

Now while you’ve been lecturing and the students have been doing their homework, there have been parallel lines run rampant! They’ve been staring at them while drawing, while transfering, and while walking around in every day life. Here’s the trick. Don’t tell them this: but parallel lines in perspective intersect. They will figure it out if given some time and encouragement to notice things. The process of discovering exactly how will be the result of a group activity/lab.

1) Where in the first sheet are their parallel lines? Are these lines preserved as parallel in the second sheet? Of those lines that are preserved which ones are they?

2) Will these lines meet? Are they still parallel?

Unfortunately the 3^rd dimension is not so simple. More than that, how we perceive the 3^rd dimension is even more complex. This is the perfect time for the overheads. The Tennis court is an example of the same sort of perspective that exists in the work done by the students. Writing directly on the transparency is effective in showing these parallel lines. Be careful to use the non-printed side of the transparency to write on. Pick two parallel lines in the picture. Get a straight edge and draw them in. Be careful and add that intersection point. What would be a good name for this point? Lead them a little. After some suggestions introduce the vanishing point. Add in the rational that if you are looking at these lines they do not continue on, but actually do vanish. Now most of the pictures have a singular point of intersection.

3) What happens if we have two pairs of parallel lines that are not all parallel?

Hand out fresh grids. Have students in pairs/groups draw a pair of parallel line that are not vertical or horizontal, the best are between 45 and 135 degrees. Using intersection points on the grid, transfer the lines to a perspective grid. Now trace out the vanishing point. Draw in the original vanishing point.

4) How many points do we have?

Two points make a line, so we have a line. Now this is a line that the vanishing point lays on. Above that it is where our lines disappear. When we look out across a large field, Plains and Desert people specifically, where do we lose sight of things? In the same way this is called a horizon line for no small reason.

As you can imagine figuring out the entire process of doing this was quite a bit more challenging. The process can be backward engineered from some of the work we have done though. However as always through history and learning we gain the luxuries of the hard work of others. This collaboration of builder and academic really set some fires with the masons of the time, but that’s another story.

Take a given grid, and copy the base line. Place your markers for the divisions across the line. Draw a line vertical of any given length greater than or equal to the base line. Mark the bisector of the line. This is the top of your perspective floor. At the top point of your vertical line draw a line perpendicular that is twice the size of the base line. These endpoints are the key to our perspective. Connect these endpoints with lines to the divisions on the base line. Do so for both sides. Connect the marks on the base line to the bisector of the highest point on your perpendicular. Now you play connect the dots with the intersection points, creating horizontal lines that become increasingly smaller. These lines should be parallel to poth your base line and your perpendicular at the top. The perpendicular is your horizom line. You can change the angle of the perspective by how large you decide to make the vertical line. It does not need to be greater than your base line, but for a first time it makes the process easier to complete.

This is just a brief introduction to perspective and an art or design class can show you much more. I encourage you to chase this as teacher or student, to discover more in the subject and the applications. Many of these applications ignore fully the mathemtaics that are going on in the background. One web sight to get you started is the Art Studio Chalkboard at http://www.saumag.edu/art/studio/chalkboard/lp-intro.html

I have given a historical perspective that works well as an initial day introduction to the material. Using the homework assignments as a night before and night between break works well. The mathematical perspective is a brief introduction to the depth of the material present. There is much more. You should feel free to attack the material in a way you feel most suited to your students, but in case of wandering I hope my suggestions and material are helpful. Good Luck!

Since our vision distorts some of the reality of what is really there, why do we no constantly get confused? By living our whole life in this way we have become used to perspective and process it in ways that are not a conscious effort. This ability to change how one perceives a given object, is what allowed Padua in the time of the Greeco-Roman empire to come up with maps of the world scribed onto a sphere. Yes that whole ‘Columbus proving the world is round’ bit rests on the shoulders of Tuscanelli’s advice and the hard work of Padua around one thousand years earlier.

Pedoe, Dan. Geometry and the Visual Arts, Dover Publications, Inc. New York, 1976.

Rucker, Rudolf v.B. Geometry, Relativity and the Forth Dimension, Dover Publications, Inc. New York, 1977.

Euclid. The Thirteen Books of the Elements, Dover Publications, Inc. New York, 1956.