Mathematical Perspective and Renaissance Art

Renaissance Art --
The Renaissance patrons wanted art that showed joy in human beauty and life’s pleasures. Renaissance art is more lifelike than in the art of the Middle Ages. Renaissance artists studied perspective, or the differences in the way things look when they are close to something or far away. The artists painted in a way that showed these differences. As a result, their paintings seem to have depth.

One point perspective
The simplest form of perspective is one point perspective. It presumes a single Point, which all others move towards. It Is like looking down a straight road as far as you can see, lines which we know are parallel seem to converge on a single point known as the Vanishing Point. We can see this in the picture of the Agora in Athens. (Below)

To see how both one and two point perspective is drawn, mathematically (geometrically), look at the Java Applet at this website: Java Applet -- Perspective

To draw in one point perspective, draw a horizon line and draw a vanishing point anywhere on the horizon. Lines which are parallel in real life are drawn to intersect at the vanishing point.
All lines travel to the vanishing point in both these drawings.
Distant figures appear smaller but have the same shape and proportions as they would close up.

Two point perspective
This is all very fine if you are looking at a thing face on down a corridor, but what if you are facing the edge of something? This is where 2 point perspective comes in.
We are now looking at an object from slightly above and we can see more of the object in question, giving an even stronger sense of its 3-dimensional form.
By lowering the Horizon Line, we can alter our view of the object to make it seem to loom over our viewers. It is important to remember that the horizon line is always at the level of your eyes, regardless of whether you are looking from above or below an object.
Piranesi used this technique to great effect in his drawing "Fantasy on a Magnificent Triumphal Arch"-1765.

Can you find where the vanishing points are?

Geometric Solids -- Nets

In geometry, the "pattern" or "template" that will create a solid figure is called a "net." We've been studying geometric solids, and specifically, rectangular prisms, using product boxes to create "nets." Here, students are measuring their boxes to determine the dimensions -- height, length, and width. They then cut the boxes apart to create the flat net, and then used their measurements to create scaled drawings of their nets.

As part of this unit of study, students calculate the volume and surface area of their boxes, and consider how these two values are related.

They also make nets of cubes, and explore ways that a net can be drawn to create the solid figure, and ways that will not make the solid. Working in this very concrete way, builds understanding of the relationships between all the geometric formulas that are connected to the area and volume of solids.

Geometry Project -- Boxes

We'll soon be starting an exploration in geometry that will be built around work with a box. Each student will need to bring in a box to work with. The best kind of box is usually some sort of food packaging because these are light weight enough to be cut easily, and are of a reasonable size. So keep an eye out for a box that might be emptied in your kitchen or pantry. Cereal boxes, rice or pasta boxes, tea boxes, and the like are perfect. Sometimes the kind of packages that are used for over the counter medicines or soap products are good too. The 64 Math group needs their boxes by Tuesday, Feb. 19, and the 63 Math group will need to bring in boxes by Monday, Feb. 25.

Geometry Puzzle...

Can you arrange a regular pentagon in a plane so that it touches six of its neighbors?

It would look like this...