Wikipedia:Reference desk/Archives/Mathematics/2022 September 4

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September 4

Constructable numbers from a unit grid?

1) What numbers can be constructed from a pre-existing unit grid using only a compass and straight edge? 2) Extension to 3 dimension. With a three dimensional unit lattice, what numbers can be constructed with a a)"three dimensional compass (given center c, and point r, construct the sphere around c of radius equal to the length from c to r, and b) a plane (generate all points on a plane given points s, t, and u). (I don't think a normal straight edge is needed since for any two points f&g, as long as there are two points m & n that all in a common plane with f&g, simply using two planes would work).Naraht (talk) 19:13, 4 September 2022 (UTC)

We have an article about this: constructible number. Basically, they're the points where each coordinate can be obtained from the integers using only the basic arithmetic operations (addition, subtraction, multiplication and division) and square roots. This shouldn't change in 3 dimensions.--2404:2000:2000:8:91A7:6025:8F24:8D21 (talk) 00:54, 5 September 2022 (UTC)