Short answers, pulled from the story.
The Portrait of Luca Pacioli is attributed to the Italian Renaissance artist Jacopo de' Barbari, based on a cartouche on the canvas inscribed IACO.BAR. VIGENNIS. P. 1495. However, the attribution has been questioned, and some scholars argue that Leonardo da Vinci painted at least the rhombicuboctahedron in the image, given his close collaboration with Pacioli on the Divina proportione.
The Portrait of Luca Pacioli is held in the Capodimonte Museum in Naples, southern Italy. The Italian state acquired the painting to prevent its sale to England after it resurfaced in the nineteenth century as property of the Ottaviano branch of the Medicis.
The person standing to the right of Pacioli has not been conclusively identified. Proposed candidates include Guidobaldo da Montefeltro (the Duke of Urbino), Francesco di Bartolomeo Archinto, Galeazzo Sanseverino (a leading figure at the court of Milan), and the German painter Albrecht Durer.
The painting shows Pacioli at a table with a slate, chalk, compass, and a dodecahedron model. A glass rhombicuboctahedron half-filled with water hangs from the ceiling, reflecting the Ducal Palace of Urbino in triple. Pacioli is demonstrating a theorem from an open book by Euclid, and a closed book identified as his 1494 Summary of arithmetic is also visible.
Leonardo da Vinci collaborated with Pacioli for roughly two years after Pacioli moved to Milan in 1496, illustrating Archimedean solids including the rhombicuboctahedron for the Divina proportione. One scholar attributed the rhombicuboctahedron in the portrait to Leonardo's hand, arguing it matches the same geometric figures he drew for that book.
During Durer's first trip to Italy in 1494-1495, he encountered Barbari's theories of proportion. According to Durer's own notes, he met Barbari between 1495 and 1500, and Barbari directed him toward Pacioli's work on mathematics and art. Some of Durer's drawings show affinities with work by Pacioli's master Piero della Francesca, and figures from Leonardo's Treatise on Human Proportion influenced Durer's Christ among the Doctors, painted in Venice in 1506.