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If we have unit segment, we can use a compass and ruler to make segments whose length represents many numbers (all rational, sqrt(2)), but there are "unreachable" real numbers.

Are there any such numbers which are "reachable" only in 3D.

In 3D, in addition to usual compass and ruler, we also have 3D compass and 3D ruler:

  • Sphere with a random raduis and center in a given point;
  • Sphere with a center in given point, intersecting another given point;
  • Plane intersecting 3 given points;
  • Usual compass&ruler in any plane;
  • Circle on a sphere surphase through 2 given points (bending 2D ruler);
  • Random point in space, on any plane or sphere or on any line;
  • Intersection of spheres+plane, plane+plane or sphere+sphere;

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Vi0
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1 Answers1

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No, there are no additional solutions. You can construct and solve both linear and quadratic equations in the plane, and every 3D operation you mentioned can be described using either a linear or a quadratic equation, so both systems can construct the same set of points on a given line, or the same set of numbers which is a proper subset of all real numbers.

MvG
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