Do four noncoplanar points determine space?

Question

The title says it all.

In the same way that:

Two distinct points determine a line.

Three noncollinear points determine a plane.

My question is:

Do four noncoplanar points determine 3D space?

Answer 1

Yes, your intuition is correct. Just as two points determine a line in the plane, and three points determine a plane, higher dimensional analogues hold as well. To answer it definitively we will have to choose a framework within which to speak, but in any reasonable choice it will be true. In Euclidean geometry, "any two distinct points determine a line" is taken as an axiom, so requires no further justification. But Euclidean geometry does not model planes in a flat 3d space or anything higher dimensional, so there is no higher dimensional axiom to appeal to.

The theory of affine spaces is a one option. In an affine space, $n$ points are said to be affinely independent if any strict subset of them has a strictly smaller affine span. The dimension of an affine span is defined to be one less than the number of any affinely independent spanning set. With this terminology in place, the statement is true by definition: any $n+1$ points not lying in any affine subspace of dimension lower than $n$ uniquely determine an $n$ dimensional affine subspace.

A similar statement is true in linear algebra, $n$ independent vectors span an $n$ dimensional subspace.

While they are less familiar than affine spaces and vector spaces, I feel incidence structures capture the synthetic flavor of Euclidean geometry better than affine spaces, which are more analytic in flavor. In incidence structures, it is again taken as an axiom.