Conic sections

    
A Poem-Collage Project
Poem by Sarah Glaz  with Collage by Mark Sanders

The poem-collage pair appears in the Bridges 2024 online art gallery: https://gallery.bridgesmathart.org/exhibitions/bridges-2024-exhibition-of-mathematical-art/sarah-glaz-mark-sanders

 
Sarah Glaz
 Sarah Glaz
 University of Connecticut, Storrs,
 Connecticut, USA 
  Website
Mark Sanders
Mark Sanders 
Rushden, Northamptonshire,   
UK 
Website 

History, Mathematics, Poem, Collage
 
   

Apollonius of Perga (262 -190 BCE) was a Greek mathematician and astronomer known in his times as "The Great Geometer." His treatise, Conics, expanded on the work of the two other giants in his field, Euclid and Archimedes, and brought the knowledge of conic sections -- ellipses, parabolas, and hyperbolas -- to the state prior to the invention of the Cartesian coordinate system in the 17th century.

Apollonius was born in Perga, a Greek city located in Pamphylia (today Anatolia, Turkey). Perga is represented in the collage's center by the ruins of its ancient amphitheater and the postage stamp picturing the goddess Artemis, the main deity worshiped in that city. Apollonius was named after the god Apollo, twin brother of Artemis.  Most of the information we have about Apollonius' life comes from the prefaces of the various books of his masterpiece, Conics. Seven out of the eight books of Conics survived either in the original Greek or in Arabic translations. Conics' entire first draft was written in Alexandria, Egypt, where Apollonius received his mathematics education from the followers of Euclid at the Alexandrian Museum, and where he resided for years after, lecturing on mathematics, writing his mathematical treatises and raising a family. Apollonius revised the eight books of Conics for most of his life, sending the final versions as they were completed to various personages in Pergamum. The city of Pergamum, situated in Myasia (today Izmir, Turkey), was a major center of Hellenistic culture and learning. It was home to a library and a Museum similar to the ones in Alexandria, both flourishing under its royal patronage. The first two books of Conics were sent to the mathematician, Eudemus of Pergamum, who might have been Apollonius' teacher prior to his moving to Alexandria. The last several books of Conics were sent to King Attalus I of Pergamum to be placed in Pergamum's library. It is not known how long Apollonius stayed in Pergamum. Apollonius lived during the Golden Age of Greek mathematics, when Greek mathematicians from all the corners of the Hellenistic world kept in touch with each other through frequent correspondence and mutual visits. He may have become acquainted with Pergamum's nobility on one of his visits. Still, sending his books to Pergamum's library rather than Alexandria's library may mean that he moved from Alexandria to Pergamum later in life. Pergamum is represented in the collage by the ruins of its ancient amphitheater at the top left, and Alexandria is featured in the image at the bottom left.

Prior to Apollonius' work, geometers treated conic sections as arising from three types of cones, differing from each other by their vertex angle. One of the innovations Apollonius introduced in Conics was unifying the method of generating conic sections by using only one cone, of any kind, and varying the inclination at which the intersecting plane meets the cone's surface. Both the poem and the images of the cones on the right-hand side of the collage reflect this innovation. The colorful cone is a classic toddler's toy, called Stacking Rings, where doughnut shaped objects need to be looped on a central peg in appropriate order. Watching a child play with this cone toy, we can imagine how a mathematical genius of great magnitude shows its promise, which later will bloom into achievements, very early in life.

The names we use for conic sections today: ellipse, parabola, and hyperbola, were coined by Apollonius in Conics. Most properties of conic sections, including their formulas, were discovered by Apollonius and with the advent of the coordinate system had only to be "translated" from the cumbersome language of ancient Greek geometry into the modern language of coordinate geometry to make them accessible. Apollonius authored several other works, none of which survived, except for a few fragments or mentions in treatises written by his successors. In addition to several other landmark results in geometry, Apollonius is credited with accomplishments in optics and astronomy. He is considered one of the founders of mathematical astronomy. Claudius Ptolemy attributed the theory of epicycles to explain planetary motion to Apollonius. It is also known that he developed an improved sundial, the hemicyclium, which having the hour lines drawn on the surface of a conic section yielded greater accuracy. An image of a hemicyleum sundial is included in the collage's left-hand side, with the sun emerging from behind. It is worth noting that the amphitheater structures above, mirror the shape of the hemicylium. For more details see Mark's Dissecting Apollonius.

Many thanks to Claudine Burns Smith   for help with formatting the poem-collage pair to the Bridges 2024 Art Exhibit specifications.  
                                                                                        

Conic sections poem-collage pair
 
-------------------------------------------
Back  to Home Page