Purpose of This
Web Page

Ever since ancient times when the Greek philosopher and mathematician Pythagoras of Samos (c. 570–495 BC) spent his afternoons drawing right-angled triangles in the sand with a stick, mathematicians have been flummoxed by the “Pythagorean Paradox” ... that is, the anomaly of a right-angled triangle’s “hypotenuse” (the longest side of a right-angled triangle) being “parallel” with one of the triangle’s legs—as pertains to the particular circumstance of the right-angled triangle that is used in the construction of an “Ellipse,” or in the construction of a “Hyperbola.”

This web page will attempt to demystify the “Pythagorean Paradox” through the use of several graphing illustrations.

Graph Illustrations


Horizontal Ellipse
Screenshot above of an adapted
desmos.com calculator graph

We start with a “horizontal” Ellipse (the light-blue oval above) which has a “vertex” of length five, measured from the center of the Ellipse. The center of this particular Ellipse is also the center of the graph that the Ellipse has been constructed on.

The “focus” of the Ellipse is of length four, also measured from the center of the Ellipse.

Horizontal Hyperbola, with Asymptotes and Rectangular Box

Horizontal Hyperbola
Screenshot above of an adapted
desmos.com calculator graph

Next, we have a “horizontal” Hyperbola (with its two, maroon-colored “branches” displayed above) with a “vertex” of length four, measured from the center of the Hyperbola. The center of this particular Hyperbola is also the center of the graph that the Hyperbola has been constructed on.

The “focus” of the Hyperbola is of length five, measured from the center of the Hyperbola. You have probably already noticed that the positions of the vertex and focus on the graph are reversed from their positions on the graph of the Ellipse.

Ellipse, Hyperbola, Asymptotes, and Circle

Ellipse, Hyperbola, Asymptotes, and Circle
Screenshot above of an adapted
desmos.com calculator graph

And finally, we have both the Ellipse and Hyperbola on the same graph. Also added to the graph is a Circle (with “radius” five) that will come in handy in a second when we add a right-angled triangle to the graph.

Ellipse, Hyperbola, Asymptotes, Circle, Right Triangles and Rectangular Box
(Two Togglable Images)

'Click' the image below
[or 'tab-to,' then press the 'enter' key]
to toggle between two graphing images,
along with their accompanying text underneath.

Screenshot below of an adapted
desmos.com calculator graph

Ellipse Pythagorean Theorem demonstration Hyperbola Pythagorean Theorem demonstration
Photo of Fyodor Bronnikov painting

Above: A Wikimedia photographic reproduction of an 1869 oil painting by Russian painter Fedor Andreevič Bronnikov (1827–1902), entitled:
Pythagoreans Celebrate the Sunrise

Audio button ^ music source

In 1854, a young Fyodor Bronnikov traveled southwest from his hometown of Shadrinsky, Russia—a 3152-mile (5073-kilometer) journey—to Rome, Italy, where he established a studio to pursue his painting career. [Distance map] (Here are more paintings by Fyodor Bronnikov.)

Around 530 BC, a middle-aged Pythagoras traveled west from the Greek island of Samos (his place of birth)—a 932-mile (1500-kilometer) journey—to the port city of Croton on the southeast shore of Italy, where he established a school that dealt with political issues, and that advocated for a simpler, healthier existence, such as by becoming a vegetarian, and by exercising daily. [Distance map]

The city of Croton (now named “Crotone”) was a Greek colony back then. For centuries, Greeks had been leaving their homeland and setting up colonies at locations along the Mediterranean Sea shoreline, including around the Croton area (located on the Ionian Sea, a large bay of the Mediterranean Sea). The Greeks called the area around Croton Italy, which ended up becoming the name for the future country of Italy. [Map and reasons for leaving.]

The harbor of Croton was the best harbor in that area of southeastern Italy in those days; and who knows, maybe that is where the shoreline scenery seen in the oil painting above came from, since Bronnikov’s paintings are historically based. As a matter of fact, this Google Books sample web page has a passage from a book about Pathagoras that says "Behind the harbour the ground rose steeply to a hill ..." in ancient Croton, on which the city was built. And that looks like a hill by a harbor in the painting!

Shoreline of Crotone, Italy

Wikimedia image by AlMare

The photo above was taken in the month of June, looking northeast towards the city of Crotone, Italy. According to this timeanddate.com web page, the sun rises in the northeast in the month of June in Crotone, Italy at around a 60-degree angle north of due east ... and in the oil painting above, the sun looks like it’s coming up at about that angle!

You can decide for yourself if this Google map shows five possible locations that Fyodor Bronnikov may have sketched for the setting for his painting; one location right in Crotone, where there are ancient ruins on a hill; the next location nearby, by a curved shoreline; a third location just a two-plus hour walk southeast of Crotone, where an ancient site from that time period is located; a fourth location near that ancient site featuring a temple built on a hill; and a close-by fifth location near another curved shoreline. Two or more of those locations may have been blended together on Bronnikov’s canvas to create the setting for the painting above.

In the oil painting, behind Pythagoras, are several men playing musical instruments. The man in the white robe standing behind Pythagoras is holding up a white, U-shaped, seven-string instrument called the “Kithara”—the ancient Greek forefather of our modern-day guitar!

Seated to the right of the white-robed man, playing the red Kithara, is the famous Greek Kitharist brought in for the celebration, Euripides; and directly behind the white-robed man, also seated, is the equally famous Eumenides, on rhythm Kithara.

You can hear on this website what a Greek Kithara sounds like.