Source: Sudipta Das.
There are only 3 solutions to this, N=3,4,5. N=1 produces a trivial 1-dimensional solution N=2 and N=6 produce 2-dimensional solutions. * The apex of the pyramid is above the centroid of the N-gon. * The altitude of a unit equilateral triangle (each face) is sqrt(3)/2 * The pyramid base (N-gon) can be drawn as N isosceles triangles, base legth 1, with angles 90(1-(2/N)),90(1-(2/N)),360/N * The generic altitude of these triangles is 1/(2 tan(180/N)) ** The generic angle of elevation = acos(1/(sqrt(3)tan(180/N))) Base-face elevations: ===================== For N=3 (equilateral triangle base): [tan(60)=sqrt(3)] Angle between base and side is acos(1/3) ~= 70.5288 deg. For N=4 (square base): [tan(45)=1] Angle between base and side is acos(1/sqrt(3)) ~= 54.7356 deg. For N=5 (pentagon base): [tan(36)=sqrt(5 - 2sqrt(5))] Angle between base and side is acos(1/(sqrt(15-6sqrt(5)))) ~= 37.3774 deg. For N=6 (hexagon base): [tan(30)=1/sqrt(3)] Angle between base and side is acos(1) = 0.0000 deg. Higher N yield acos(n>1) so not possible. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Face-face angles: ================= The shortest route between 2 non-adjacent corners, over the surface will meet the intersection of these (equilateral triangle) faces perpendicularly. Therefore this path bisects each face, and has length 2 * sqrt(3)/2. The triangle formed between these lines and the base line produce the angle between the faces. The distance across the base between corners 2 apart is: a^2 = b^2+c^2-2bc.cos(x) [cosine rule] a^2 = 2(1-cos(x)) [since all lengths are unit] Internal angles (x) of regular N-gon = 180(1-(2/N)) N=3: x=60; a^2 = 1 [a = 1] N=4: x=90; a^2 = 2 [a ~= 1.414] N=5: x=108; a^2 = (3+sqrt(5))/2 [a ~= 1.618] [note cos(108) = (1-sqrt(5))/4] N=6: x=120; a^2 = 3 [a ~= 1.732] Rearranging the cosine rule, we also get: x = acos((b^2+c^2-a^2)/2bc). [note: b = c = sqrt(3)/2; b^2+c^2 = 3/2 and 2bc=3/2] N=3: x = acos(1/3) ~= 70.5288 deg. [same as elevation, as expected!] N=4: x = acos(-1/3) ~= 109.4712 deg. N=5: x = acos(-sqrt(5)/3) ~= 138.1897 deg. N=6: x = acos(-1) = 180 deg. [as expected, this is flat!]