It is very surprising to me that although the picture of the five ways that three planes can be arranged in 3-dimensional space is in many textbooks (college algebra, linear algebra) the analogous picture with four planes in space is so hard to find. How many arrangements are there? Is there a reference?

I did ask a more general question in MO, where you can read that what we want to count are combinatorial equivalence classes of affine hyperplane arrangements--a very hard problem. Here I'm focusing on the simple question of "are there any drawings out there of all the ways to arrange just 4 planes in 3 affine dimensions." https://mathoverflow.net/questions/307862/counting-hyperplane-arrangements-up-to-combinatorial-equivalence-simple-example

So far no real leads! There is of course lots of great literature on hyperplane arrangements, especially R. Stanley's lectures, the chapter in Grunbaum's book on convex polytopes, Zaslavsky's ground-breaking 1975 work on Facing up to Arrangements, and some entries in the OEIS. For the latter, I'll quote myself from MO:

Some related answers to this question are out there. For instance, http://oeis.org/A241600 counts the number of arrangements of $n$ lines in the affine plane. (Edit: Peter Shor points out at MO that this sequence is defined differently than via combinatorial equivalence.) Some nice pictures are included at http://oeis.org/A241600/a241600.pdf This gives a start to the question in dimension 3. The four arrangements of three lines $(A(3,2)=4)$ can be extended along the third dimension to get arrangements of 3 planes in 3 dimensions, and then of course there is a fifth in which the three planes intersect at a point. These five $(A(3,3)=5)$ are pictured in many beginner algebra texts.

Drawing similar pictures, I think that $A(4,3)\ge 14.$ That is, that there are at least fourteen combinatorial equivalence classes of affine hyperplane arrangements using four planes in $\mathbb{R}^3$. What else is known?

  • $\begingroup$ Interesting coincidence that this question was asked at almost exactly the same time as this question. $\endgroup$
    – joriki
    Aug 10 '18 at 17:54

The minimum number is 5 and the maximum number is 15. In dimension n, the maximum number of regions created by n+1 hyperplanes will be 3n+1−1. the minimum must be n+1.

  • $\begingroup$ 3n+1-1 = 3n. Did you mean 3(n+1)-1? $\endgroup$
    – mr_e_man
    Aug 11 '18 at 2:48
  • 1
    $\begingroup$ This answer is about the question "how many regions can the 4 planes separate 3d space into." My question is "how many ways can the 4 planes be arranged in 3d space, up to combinatorial equivalence." It is a coincidence if the answer turns out to be 15 in both cases. $\endgroup$ Aug 11 '18 at 3:55
  • $\begingroup$ Here is a picture of 14 arrangements of 4 planes in $\mathbb{R}^3$. $\endgroup$ Aug 29 '18 at 21:02

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