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I've seen plenty of ancient diagrams representing plane figures.* But I'd like to know how ancient geometers, especially around the time of Euclid, might have represented solids. Did they use diagrams with perspective? Three dimensional models?

*cf. David Fowler, "The Mathematics of Plato's Academy", see the plates between pages 6 and 7

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    $\begingroup$ No need to IMO; read the works of Reviel Netz if you want to know more about this. $\endgroup$ – RP_ May 14 '17 at 18:11
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    $\begingroup$ That sounds like a citation to me. $\endgroup$ – Lee Mosher May 14 '17 at 19:39
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    $\begingroup$ Judging from Fowler, there aren't enough ancient artifacts available to answer this question. $\endgroup$ – Matt F. May 15 '17 at 12:06
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    $\begingroup$ Now that I've looked up the relevant references, that's the sense I get too. Netz essentially relies on Fowler's evidence, plus the speculation that since Greek painters had developed techniques for representing depth via perspective and foreshortening, and since there is a Euclidian treatise on optics theorizing perspective, they'd have had a good sense of how to produce such diagrams. $\endgroup$ – Arnold Brooks May 15 '17 at 18:42
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    $\begingroup$ But the interesting result, as Netz points out, is that diagrams can't be taken to represent quantitative relations once they're drawn this way, which makes me wonder about the status of the constructions in Euclid. Solid diagrams drawn with foreshortening would not be truth-makers of the theorems, in any case. $\endgroup$ – Arnold Brooks May 15 '17 at 18:45
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Greek mathematical diagrams are composed of lines, arcs, and letter labels, and may be used to represent two- or three-dimensional objects. Drawing aids such as ruler and compass were sometimes, but not always, used in their construction, while curves in mathematical papyri were typically drawn freehand. Three-dimensional objects in diagrams were drawn using foreshortening of certain elements, as opposed to "painterly" techniques such as shading or consistent perspective. The technique of foreshortening does not work for spheres, and so for problems involving spheres special techniques were required, such as depicting the objects hidden by a sphere as though they were outside it.

Courtney Roby, in The Encyclopedia of Ancient History (2012).

The encyclopedia entry is based on the analysis of Reviel Netz in The Shaping of Deduction in Greek Mathematics.

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    $\begingroup$ This sounds very authoritative! But how do the actual diagrams compare to this verbal description? Are there any extant ancient Greek diagrams? If there aren't, then on what is this verbal description based — speculation? $\endgroup$ – Quuxplusone May 15 '17 at 1:46
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    $\begingroup$ One of the earliest (if not the earliest?) fragments we have of Euclid's Elements is 2nd century CE (maybe 1st), around 400 years after Euclid. See math.ubc.ca/~cass/Euclid/papyrus for a description. There's a little diagram. It doesn't represent anything 3-dimensional. $\endgroup$ – Marty Oct 6 '18 at 1:23
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    $\begingroup$ Well.. I should have included ostraka too (pottery shards). See ams.org/publicoutreach/feature-column/fc-2011-11 for an exposition. There's a little 2-dim diagram, accompanying a discussion of the icosahedron, from (perhaps) the 2nd centrury BCE. $\endgroup$ – Marty Oct 6 '18 at 1:40
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This is not quite "ancient Greek", it merely dates back to the ninth century...

image

source

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Netz has emphasized that the Greeks of antiquity used their diagrams in a much more topological manner than one might expect, and perspective seemed limited. Archimedes was silent on perspective in spheres and liked to refer to the great circle therein; instead of a sphere in a cylinder, Archimedes' tombstone might have merely been a circle in a square!

In more detail, although the scholars of the 19th and early 20th century were really good at reconstructing authoritative texts of the ancients, along with doing really boss things like dating the time of an author's flourishing to some throwaway line the author might make about an eclipse or comet, these same scholars had the tendency to amend the figures of Euclid/Archimedes/etc. to fit the standards of rigor of the 19th/early 20th century.

Netz is one of the first to note that figures are amenable to the critical, philological method, and thus to answer @Quuxplusone's and @Matt F's questions, even if there are no extant manuscripts (on papyri) in the hand of, say, Archimedes, by comparing and contrasting the manuscripts that are available, one may go back centuries, maybe a millennia, to Archimedes himself (or at least to the scribe at the library of Alexandria). One such manuscript was famously hidden behind a prayer book and lost to the world until rediscovered in a monastery outside of Jerusalem.

As an example of a critical reconstruction of a diagram, consider the figure below from On the Sphere and the Cylinder. $\mathrm{K}$ is a cone and each of the segments $\mathrm{AZ}$, $\mathrm{ZH}$, $\mathrm{HB}$ etc are lines (here showed as arcs). In one branch of the manuscript tree, the bottom $\mathrm{A}$ is missing, most likely due to scribal error. Because the text refers to line $\mathrm{AB}$, one of the copies on that branch added a line between the left $\mathrm{A}$ and the top $\mathrm{B}$, while also fixing the segments straight. But upon reading the proof, the line $\mathrm{AB}$ is intended to be the diameter.

Netz follows the philological principle of "lectio difficilior potior" - the more difficult a reading is, the more likely it is to be correct. Because the manuscript tree most likely branched off relatively early, e.g. in the early 1st millennium, Netz concludes from the study that, going back at least to antiquity and maybe to Archimedes himself, the figure had an $\mathrm{A}$ on the left and an additional $\mathrm{A}$ or maybe a $\mathrm{\Lambda}$ on the bottom, along with arcs for line segments.

Netz Figure

Netz discusses these in The Works of Archimedes: Volume 1, The Two Books On the Sphere and the Cylinder: Translation and Commentary 1st Edition, which is quite scholarly; he also has a popular book, The Archimedes Codex, with his coauthor William Noel, which provides figures available in the extant codices for the image above.

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Below is a dodecahedron, dated to the 1st century, with faces labeled by the Greek alphabet. It's a figure from "A Dodecahedron of Rock Crystal from the Idaean Cave and Evidence for Divination in the Sacred Cave of Zeus," by Angelos Chaniotis. Here's a place to download the full article: https://archiv.ub.uni-heidelberg.de/propylaeumdok/1865/1/Chanotis_Dodecahedron_2006.pdf.

A cursory search shows that we have found at least a few handfuls of carved dodecahedra and icosahedra in Graeco-Roman Egypt. See https://www.metmuseum.org/art/collection/search/551072 for a nice icosahedron. So the Platonic solids were tangible objects a few centuries after Euclid. Scholarship on earlier dodecahedra and icosahedra gets a bit sketchy. As for diagrams, I'd stick with Fowler's uncertainty for now.

A 1st century dodacehedron

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