List the artists and mathematicians who are mentioned in the
following readings and activities and give a brief summary of their contributions:
The use of perspective began during the Renaissance. It changed the way we represented and visualized the world.
We will now investigate some of the mathematical properties of perspective drawing and see how they can help us appreciate art and the world around us.
Experiments with perspective drawing were completed long ago,
when people with interdisciplinary interests (like mathematics
and art) were
perhaps more common. In this 1525 woodcut,
from "Unterweisung der Messung", by Albrecht Durer,
the screw eye on the wall is
the desired viewer's eye, the lute on the left is the object,
the taught string is a light ray, and the picture plane is mounted on
Leonardo Da Vinci and Brook Taylor
researched the question of how to find the viewing distance of a painting,
and Taylor's 1715 work was published in
Linear Perspective: Or, a New Method of Representing Justly All
Manner of Objects as They Appear to the Eye in All Situations London:
Calculating the Viewing Distance for
Interior of Antwerp Cathedral, by Peter Neeffs the Elder, 1651
In the figure below, we see
the trick applied to finding the viewpoint for the
Interior of Antwerp Cathedral painting by Peter Neeffs the Elder.
We first determine the vanishing point V directly in front of us,
which is easy to see, as
it is the intersection of lines which are supposed to be parallel
in the real-world. Some of the lines have been drawn in below
in order to highlight V.
Notice that lines that follow along the edges (coming from us towards
V) of the square tiles of
the floor also intersect at V, in the picture below.
Our second point V' is calculated by following along a diagonal
(indicated on the picture) that follows along the vertices of the
The viewing distance d
is the indicated length,
and the correct viewpoint is directly in front of the main
vanishing point V.
Although it is not possible to tell by viewing this small reproduction of
Interior of Antwerp Cathedral,
the effect of viewing the actual painting
in the Indianapolis Museum of Art gives a surprising sensation of
depth, of being "in" the cathedral.
The viewing distance is only about 24 inches, so most viewers never view
the painting from the best spot for the sensation of depth!
Activity 1: Come One - Come All - to a Better Cube
was drawn assuming that it would be viewed with one eye from a distance
d behind V.
this link of the large drawing
and do as follows as you have a partner read you these directions:
- Use a thumb and a forefinger to measure the distance d between
V and V' which you should see as 2 dots at the very top of your screen.
Keep this distance with your hand.
- Continuing to keep that distance, turn your hand 90 degrees
(perpendicular to the screen) to measure out d units in front of V.
- Place your left eye exactly at that same point (d units in front of V -
not in the center of the page). You will need to get very close
to the computer screen to do this.
- Close your right eye.
- Without changing your position, let your left eye roll down and to the
left and then look at the box. Although it may be too close for
comfortable viewing because you will be so close to the
computer screen, the distortion should go away
and it should look much more like a cube! Most people will be able to
see this by following the directions, but some people may have problems
due to astigmatism...
- Now switch roles and
read the directions to your partner as they complete this activity.