San Carlo

In Rome, there is a small 17th-century church called San Carlo alle Quattro Fontane, designed by Francesco Borromini, a contemporary of Gian Lorenzo Bernini and one of the great masters of the Baroque period. Their rivalry produced some of the era’s most inventive designs.

Inside the church, the dome’s pattern consists of a repetition of Greek crosses, octagons, and hexagons. To make the dome appear higher, Borromini gradually reduced the size of these shapes toward the top, creating a striking optical illusion.

This illusion can be reconstructed and improved by projecting the 2D pattern from a cylindrical surface around the dome onto a viewpoint at the center of its base. The intersection of each projection line with the dome determines where the pattern should be drawn.

In the simulation below, you can experiment with this effect. Approaching the center of the dome, the structure appears to lift upward, with the illusion strongest at the origin.

Another fascinating example of optical illusion can be found in Sant’Ignazio di Loyola, where Andrea Pozzo painted a magnificent “false” dome on a flat ceiling — well worth a visit as well!

I completed this project after a high school trip to Rome in 2016, with the help of my excellent math and programming teacher, John Val.

Simulation

Turm der blauen Pferde

Franz Marc painted Der Turm der Blauen Pferde toward the early end of his life, at a moment when his work was moving away from the natural lyricism of his early years toward a more cubist and orphic structure. Shortly afterward, he was drafted to the front, where he died at the Battle of Verdun before orders to be withdrawn reached him.

Just like Marc, the painting took on a tragic fate. It was labelled “Entartete Kunst” (“Degenerate Art”) by the National Socialists, who publicly condemned its modern appearance — although Göring later expropriated it for his private collection. As one can see in the image on the right, the horses on the original painting are almost life-size.

After the Second World War, the painting was lost and its location remains unknown. The canvas likely still exists somewhere in a private collection, and one can only hope it will return to public view someday.

What has survived is a small colour sketch that Marc sent as a postcard to Else Lasker-Schüler on New Year’s Day 1913. The card shows the scene with a clearly visible moon and star, motifs that appear only subtly in the final version. This postcard is on display at the Pinakothek der Moderne in Munich, serving as a quiet reminder of a lost masterpiece.

Turm der blauen Pferde (Tower of Blue Horses) by Franz Marc Postcard with sketch for Turm der blauen Pferde
Visitors view the Degenerate Art exhibition, with Der Turm der Blauen Pferde visible on the right
Visitors view the “Degenerate Art” exhibition (Entartete Kunst) at the Munich Hofgarten in July 1937. Der Turm der Blauen Pferde can be seen on the right.