Fra Luca de Pacioli

My Ancestor

Here we have a portrait of Fra Luca da Pacioli (1466 – 1517), Italian Renaissance mathematician and Franciscan friar.

Pacioli is not known for any original discoveries. He's most famous for his books, e.g., his Summa de Arithmetica, Geometria, Proportioni et Proportionalità (Summary of Arithmetic, Geometry, Proportion, and Proportionality), a textbook on mathematical knowledge at the time, and De Divina Proportione (On the Divine Proportion), a book on geometry. In the latter, he expounds on the golden ratio and its relationship to the Platonic solids. It was illustrated by his friend, Leonardo da Vinci; one of these illustrations (of a truncated octahedron) is featured in the title graphic of this web page.

The painting itself contains many interesting mathematical objects. The model hanging in the upper left-hand corner is a truncated rhombicuboctahedron, an Archimedean solid, i.e., a polyhedron whose vertices are all congruent to one another and whose faces are all regular polygons.

In this case, the polygons are squares and equilateral triangles. Notice that the polygon is half-filled with water and that its panes show a reflection of the view through the window behind the painter.

Pacioli is making a demonstration out of Euclid on a piece of slate. A dodecahedron (a Platonic solid composed of regular pentagons, the construction of which is the climax of Euclid's Elements), sits on the book to the right. The young man standing behind Pacioli has not been identified, but he's conjectured to be Albrecht Dürer, a painter of the Northern Renaissance who spent some time in Italy and later wrote a book on geometry and perspective. Some of Dürer's graphic works contain mathematical themes.

Mathematics is in many ways a very tradition-oriented subject. Any mathematician who receives a doctorate was advised in his or her dissertation by a senior mathematician in a master-pupil relationship. An advisor can have a profound influence over their students' mathematical predilections, understanding, and outlook. The Math Genealogy Project allows one to trace any line of "ancestry" from pupil to master as far back as the record goes, or to start with a historical mathematician and trace their mathematical "progeny" down to modern times.

While reading about Pacioli's life, I explored the various lines of his mathematical "descendants." It is unknown who Pacioli's advisor was; his sole student was a man named Domenico da Ferrara, who obtained his doctorate in 1483. Among Ferrara's students was Nicolaus Copernicus (1499), the great pioneer of modern astrophysics. Copernicus' students included Georg Rheticus (1535). Continuing down this line of descent, I came across Carl Friederich Gauss (1799), "Prince of Mathematicians," and some of his most famous successors. At this point I started clicking on names I recognized from my own studies. Here is what I found:

Luca da Pacioli – unknown
Domenico da Ferrara – 1483
Nicolaus Copernicus – 1499
Georg Rheticus – 1535
Sebastian Dietrich – 1544
Andreas Schato – 1562, 1578
Ernestus Hettenback – 1576, 1591
Ambrosius Rhodius – 1600, 1610
Christoph Notnagel – 1630
Johann Quenstedt – 1633, 1634
Michael Walther, Jr. – 1661, 1687
Johann Pasch – 1683
Johann Planer – 1696, 1709
Christian Hausen – 1713
Abraham Kastner – 1739
Johann Pfaff – 1786
Carl Gauss – 1799
Christoph Guderman – 1841
Karl Weierstrass – 1854
Hermann Schwarz – 1864
Leopold Fejer – 1902
Marcel Riesz – 1912
C. Einar Hille – 1918
Irving Segal – 1940
Isadore Singer – 1950
Daniel S. Freed – 1985
Michael Ortiz – 2009

Imagine my surprise when, at the beginning of the twenty-first century, I arrived at…myself, your humble RGC math professor.

This doesn’t make me anything special, however, for Pacioli has (to date) had 109,263 descendants over the centuries!

Michael Ortiz, Ph.D.
Assistant Professor of Mathematics
Department of Natural and Behavioral Sciences

Sul Ross State University Rio Grande College

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