Unfolding the Shadow Line

Who knew what-when? And why did they know it?

Below is an excerpt from Vera Viana's article on 

Archimedean solids in the fifteenth and sixteenth centuries

Nets of Archimedean Solids by Albrecht Dürer

 Albrecht Dürer - Museo Nacional del Prado, Galería online, Public Domain, https://commons.wikimedia.org/w/index.php?curid=17628367

Dürer was a German painter, printmaker, and theorist of the German Renaissance, and he studied the polyhedra. He also studied with Luca Pacioli and Leonardo da Vinci. 

After completing his apprenticeship, Dürer followed the common German custom of taking Wanderjahre—in effect gap years—in which the apprentice learned skills from other masters, their local traditions, and individual styles. 

During his Wanderjahre, I can only assume he traveled with other apprentices, who could have been German carpenters or stone masons. The carpenters of that time would have drawn out their plan view drawing for a roof and then Unfolded the roof surface to obtain other points in the roof surface for scribing. So when Dürer drew out the planar nets for the Platonic Solids or Archimedean Solids, it was something that would have been natural to him from his fellow Wanderjahre adventures. 

I remember studying Dürer about fifteen years ago, and at that time, I was under the impression he was documenting the stereotomy-schiften of the craftsmen. 

Excerpt from the article by Vere Viana

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Dürer adapts this sentence with slight variations for the other regular bodies often referring to the body being shown laid down on the ground (“zugetan nieder in grund gelegt”), torn open (“aufgerissen”), and to the act of raising it or putting it together (“so man die zusamen leget”). As Friedman (2019, p. 34) notes, Dürer never uses the verb to fold or its derivatives and only mentions the act of cutting or tearing apart (“zerschneyden”) when referring to the sphere. Before introducing the “ungeregulirten corporen”, Dürer (1525, after drawing 34) explains:

Regarding the planar nets in Underweysung der Messung, Friedman (2019, p. 50) discusses the possibility that Dürer might have followed the example of Pacioli and Leonardo and that of the French mathematician and philosopher Charles de Bovelles (1479–1567). Bovelles included planar nets of the Platonic Solids and other bodies in De Geometricis Corporibus. Figure 37 shows the net of a solid body that Boveles (1510, p. 377) describes and our interpretation on the right. If we assume that all the faces are regular, Bovelles conceived the elongated pentagonal bipyramid or Johnson Solid J16, which the mathematician Norman Johnson (1966, p. 86) would describe in 1966. The image on the right illustrates five great semicircles of the sphere that circumscribes the prismatic surface. The apexes of the pyramids, however, do not belong to the same sphere.

vertices should touch a hollow sphere 

It would also pave the way, as Friedman and Rougetet (2017, p. 7) sustain, to the German mathematical tradition of describing polyhedra with edge unfolding.

Barbaro describes 32 solid bodies with a planar net that he names spiegatura, a term which, according to Monteleone (2019, p. 77), can be interpreted as an unfolding. It is possible that Barbaro chose this term because of its double meaning, as it seems to derive from the verbs spiegare (to explain) and piegare (to fold). Barbaro (1568, p. 45) further explains that a spiegatura consists of an open figure and that a three-dimensional model can be used to explain how the plan views are obtained:

To describe the bodies, we will follow this order, which, in the first place, will present their unfolding and after, their perfect plan, degraded, and finally, correct, their shadowing. By unfolding, I mean the description of the open figure, from which is made the whole body folding it together to demonstrate the true form, a thing that is truly practical and delightful to transform many bodies into lanterns and other uses of pleasure.Footnote 106

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So all this sounds like the German carpenters' Basiswissen Schiften technique. I'm not sure what the reference to "shadow" is in the reference. However, you can use the Schiften technique to lay down the shadow line for canted rafters.