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
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.
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.
