This is the kind of research that happens when one is self-isolating through a winter pandemic.
Albrecht Durer's engraving 'Melancholia', was first printed in 1514.
I wrote about the carpentry tools scattered around the edges of the image in my previous post: "Albrecht Durer's 'Melancholia' and His Knowledge of Construction and Practical Geometry", posted on 1/17/2021.
On the left side in the middle is a polyhedron, the 'Cube'^, a 3-d shape.The angel is studying it intently. Why? And what is it?
And why the sphere? A pure geometric shape, white, abstract, perfect, set among real tools we can identify, and a skinny dog. It is the only other abstract thing besides the Cube in the engraving.
I have some suggestions based on Durer's knowledge of and interest in geometry.
Durer wrote about geometry in his book, Underweysung der Mesang mit dem Zirckel und Richtscheyt, published in 1525.* The last 6 words of the title translate as Measurement with Compass and Straightedge.
One of his diagrams:
I copied the pattern of triangles, cut it out, and folded it into a 20 sided polyhedron, (an icosahedron,) fastened it with tape.
Note: The 5 triangles on the right that would complete the shape are not taped together; it fits in my hand.
If the shape were made from wire it would look like the drawings beside the pattern - the view from the sides and then from the top or the bottom. The lines are the edges of the polygons.
Another of Durer's diagrams is 12 joined pentagons which become a polyhedron. To the right are the side and top views. The views are not elevations; they are transparent.
Again, I copied it and cut it out.
It folds up to be much like a soccer ball with edges.
Durer's book contains many of these polyhedrons. He mixes the kinds of polygons to make the 'sphere'. Here are 2 of his simple ones
The models led me to consider the Cube in Melancholia as a study about how to create a sphere from planes, from polygons.
I extended the main sides of the polygon - red lines - so that the shapes became diamonds. Then I extended the edge of the almost invisible left side and found its tip met the main side at the top.
Next I divided the length of 2 sides of the diamonds in half. When I joined the mid points with a red lines I saw that they followed the edge of Durer's polyhedron.
Perhaps Durer was drawing a truncated cube in perspective. I made a diagram: 6 squares which if folded would make a cube. I lopped off 6 corners - red dashed lines.
I added the triangles, printed it, and cut it out.
Cut and fastened with tape, it is flimsy, not solid like Durer's block.
Tipped on its side, it has the planes of Durer's 'Cube' - his 8 sided shape.
Historians tend to title the book, Measurement with Compass and Ruler. In 1525, rods with 10 parts were common, so were boards with straight edges, used for drawing straight lines. Regular, agreed upon dimensions, such as would be on a ruler, did not exist. Note that there is no ruler among the tools Durer includes in his engraving, 'Melancholia'.
^ Durer's polyhedron is not a cube. I use the name as an easy reference.
*Underweysung der Mesang mit dem Zirckel und Richtscheyt can be read online through the Warnock Library in Nuremberg, Germany. A translation by Walter Strauss, published in 1977, is out of print, only available at museum and college libraries which currently are closed.
At any time, but especially in the year of Covid-19, to read a 500 year old book, housed 3800 miles away, while sitting in my office with my cat sprawled out beside me, is remarkable. That I can shift from one page to another and back again as I consider the images is an amazing gift.
8/9/2021 update: Through Interlibrary Loan I have a copy of Walter Strauss' translation here for the next 2 weeks. I can also request it again.