Short answers, pulled from the story.
The painting first appeared in a written record during 1631 within the Ducal Palace of Urbino. This inventory listed the artwork as part of the ducal collection before it moved to Florence.
Scholars generally attribute this signature to Jacopo de' Barbari based on that text. The date 1495 appears within the signature block yet the painting is dated around 1500.
A rhombicuboctahedron hangs from the ceiling half filled with water and a physical dodecahedron model rests on the table. These suspended polyhedra serve as visual evidence of geometric study during this era.
No conclusive identification exists for the man standing to the right of Luca Pacioli. One hypothesis suggests the figure is Guidobaldo da Montefeltro Duke of Urbino while another theory points to Francesco di Bartolomeo Archinto or Galeazzo Sanseverino.
Leonardo began collaborating with Luca Pacioli when the mathematician relocated to Milan in 1496. Contacts between Luca Pacioli and his Milanese patrons existed by February 1496 when collaboration began.