Over the centuries many artists and artisans have employed plane
tilings in their work. In this talk we will illustrate how many
of these tilings can be used to provide "picture" proofs of
theorems from various areas of mathematics. The artists range
from the Dutch baroque painter Jan Vermeer to the 20th century
graphic artist M. C. Escher; while the theorems include the
Pythagorean Theorem, the arithmetic-geometric-harmonic mean inequality,
the Cauchy-Schwarz inequality, and other less well-known results.
OUTLINE:
I will present several works of art which contain plane tilings, and then show h
ow those tilings lead to "picture" proofs of various theorems. Here is a (
tentative) list of the paintings and the theorems:
A. "Street Musicians at the Doorway of a House" -Jacob Ochtervelt (1634-1682)
1. The Pythagorean Theorem.
B. "A Lady and Two Gentlemen" -Jan Vermeer (1632-1675)
2. If lines from the vertices of a square are drawn to the midpoints
of adjacent sides, then the area of the smaller square so produced
is one-fifth that of the given square.
3. A square inscribed in a semicircle has two-fifths the area of a
square inscribed in a circle of the same radius.
4. The Pythagorean Theorem (again).
5. The arithmetic-geometric-harmonic mean inequality.
6. The "sine of the sum" formula.
C. "No. 67 (Horsemen)" and "No. 18 (Birds)" -M. C. Escher (1898-1972)
7. The area of a quadrilateral Q is equal to one-half the area of
a parallelogram P whose sides are parallel to and equal in
length to the diagonals of Q.
D. Paving tiles in Pompeii, wall tiles from Seville,
and "No. 103 (Fish)" -M.C. Escher
8. If the one-third points on each side of a triangle are
joined to opposite vertices, the resulting triangle is
equal in area to one-seventh that of the initial triangle.
9. The triangle of medians has three-fourths the area of
the original triangle.
E. "A Courtyard of a House in Delft" -Jan de Hooch (1629-1684)
10. The Cauchy-Schwarz Inequality.
TIME REQUIREMENT: Fifteen minutes