tione,which relied heavily on Francesca’s
work. The illustrations in Pacioli’s book
were done by none other than the famous
scientist, painter, and Renaissance figure
Leonardo da Vinci (1452–1519), who was
himself a great contributor to the study of
perspective. Pacioli was one of da Vinci’s
teachers, and he instructed the eventual
great master a good deal about proportion
and perspective.
What is perspective?
In painting and photography, perspective
gives an image depth, allowing a person to
perceive three dimensions in a two-dimen-
sional picture. In art, this can be achieved
using certain “tools of the trade.” For
example, the horizon line is the eye level
of the artist (and viewer); by raising and lowering the viewpoint, it enables the viewer
to see more or less of the horizontal plane that lies between the viewer and the horizon.
The vanishing point is a point on the horizon line at which parallel lines seem to
meet—the classic example is how train tracks seem to come together in the distance.
The horizontal baseline is parallel to the horizon line and represents the base of the
picture plane. The vanishing point is on the horizon line and is directly opposite (or in
front) of the viewer. All of these points, lines, and angles—and other ways of bringing
depth into a painting—are based on concepts in mathematics.
How is geometryused to create Islamic patterns?
Many of the intricate patterns on Islamic architecture, structures, walkways, and fab-
ric incorporate geometry in the construction of the design. From simple shapes,
Islamic designs evolved into complex geometric shapes involving a high degree of
mathematical symmetry. One of the best places to see such work is in the Alhambra
Palace, a Moorish structure in Granada, Spain, built in the 15th century.
Some Islamic designs were drawn by eye to be aesthetically pleasing; others
involved collaborations between mathematicians and artists. For example, mathemati-
cian and astronomer Abu ̄ al-Wafa ̄ (940–998 CE; born in today’s Iran) worked with arti-
sans in the 10th century, helping to design the ornamental patterns in wood, tile, fab-
ric, and other materials. He discussed such mathematical concepts as constructing a
perpendicular at the endpoint of a line segment, constructing a regular polygon,
dividing segments into equal parts, and even the bisection of angles. Abu ̄ al-Wafa ̄’s 375
MATH IN THE HUMANITIES
This tulip field in Nordwijkerhout, Holland, is one
example of how lines lead to a vanishing point.
Artists employ this concept to create the feeling of
depth in their paintings. Robert Harding World
Imagery/Getty Images.