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January 13

Compass and straightedge equivalent in three dimension?

Looking for any work based on a three dimensional equivalent of compass and straightedge. I *think* the equivalents would be (compass_ given any two points, a sphere can be constructed centered in one point passing through the other, and (straightedge) a plane can be constructed through any three points. Not sure what could be constructed, but curious as to any work in this regard.Naraht (talk) 12:24, 13 January 2020 (UTC)[reply]

We have a little bit on solid constructions at Straightedge_and_compass_construction#Solid_constructions --{{u|Mark viking}} {Talk} 20:23, 13 January 2020 (UTC)[reply]

Not quite what I was looking for, but possibly equivalent.Naraht (talk) 21:24, 13 January 2020 (UTC)[reply]

I think not equivalent: that is about constructions where you can solve arbitrary conics, much more powerful than straightedge and compass. With just straightedge and compass in 3 dimensions, I think the basic theory works out about the same as in 2 dimensions, in that it is about constructing points that are in some tower of quadratic extensions over the rationals. A quick web search doesn't find much on the subject, which suggests there isn't much interesting about it. "Solid" constructions allow constructing points in towers of extensions whose degree is either 2 or 3, rather than just 2. 2601:648:8202:96B0:0:0:0:DF95 (talk) 05:40, 14 January 2020 (UTC) This has more info about so-called solid constructions and explains that they aren't what they sound like. 2601:648:8202:96B0:0:0:0:DF95 (talk) 05:43, 14 January 2020 (UTC)[reply]

The Baragar paper is interesting and relevant to the topic at hand, but the explanation of the term solid in this context seems a bit sketchy. A better (and much more detailed) reference is probably in T.L. Heath'a translation of The Elements in the section "Locus-theorems and loci in Greek geometry" (see [1] p. 329). Keep in mind when reading this that the term "line" as understood by Heath (and other authors of his day and earlier) means what we would call the "curve" in general, something that Baragar seems to overlook. --RDBury (talk) 17:58, 14 January 2020 (UTC)[reply]