Portrait of Dr. Gachet, van Gogh, 1890, and the Golden Rectangle

Successive Golden Rectangles dividing a Golden Rectangle into squares (Portrait of Dr. Gachet, van Gogh, 1890).

A golden rectangle is a rectangle whose side lengths are in the golden ratio, one-to-phi, that is, approximately 1:1.618. A distinctive feature of this shape is that when a square section is removed, the remainder is another golden rectangle, that is, with the same proportions as the first. Square removal can be repeated infinitely, which leads to an approximation of the golden or Fibonacci spiral.

Fibonacci numbers (0,1,1,2,3,5,8,13,21,34...) are a sequence of numbers named after Leonardo of Pisa, known as Fibonacci. The first number of the sequence is 0, the second number is 1, and each subsequent number is equal to the sum of the previous two numbers of the sequence itself.

Portrait of Dr. Gachet is one of the most revered paintings by Dutch artist Vincent van Gogh. There are two authentic versions of this portrait, both painted in June 1890 during the last months of Van Gogh's life. Both show Doctor Gachet sitting at a table and leaning his head onto his right arm, but they are easily differentiated. The portraits were painted in Auvers-sur-Oise close to Paris, and depict Doctor Paul Gachet with a foxglove plant. Gachet took care of Van Gogh during the artist's last months. Gachet was a hobby painter and became good friends with Van Gogh.

List of most expensive paintings
Portrait of Dr. Gachet fetched a record price of $82.5 million ($75 million, plus a 10 percent buyer's commission) in 1990.