This file holds notes on the computation of vertex coordinates for
various solids.

let C36 = cos(36°) = (sqrt(5) + 1) / 4
 ~  .809016994374947424102293417182819058860154589902881431067724311
let S36 = sin(36°) = sqrt((5-sqrt(5))/8)
 ~  .5877852522924731291687059546390727685976524376431459910722724803
let C72 = cos(72°) = (sqrt(5) - 1) / 4 = C36 - .5
 ~  .309016994374947424102293417182819058860154589902881431067724311
let S72 = sin(72°) = sqrt((5 + sqrt(5)) / 8)
 ~  .9510565162951535721164393333793821434056986341257502224473056445
let C144 = cos(144°) = - C36
let S144 = sin(144°) = S36
let C120 = cos(120°) = -.5
let S120 = sin(120°) = sqrt(3) / 4

icosahedron

ring height h, radius r = sqrt(1-h*h)
point A (0,0,1)
point B (r,0,h)
point C (r*C72,r*S72,h)
point D (r*C144,r*S144,h)
point E (r*C144,-r*S144,h)
point F (r*C72,-r*S72,h)
point G (0,0,-1)
point H (-r,0,-h)
point I (-r*C72,r*S72,-h)
point J (-r*C144,r*S144,-h)
point K (-r*C144-,r*S144,-h)
point L (-r*C72,-r*S72,-h)

solve based on |A-B| = |B-C|

h = .4472135954999579392818347337462552470881236719223051448541794479
 = [0 2 4 4 4 4 4 ...]
 = 1/sqrt(5)
r = .8944271909999158785636694674925104941762473438446102897083588988
 = 2/sqrt(5)
 = 2 * h

dodecahedron

top ring height h1
lower ring height h2

top ring radius r1 = sqrt(1-h1^2)
lower ring radius r2 = sqrt(1-h2^2)

point A1 = (r1,0,h1)
point A2 = (r1*C72,r1*S72,h1)
point A3 = (r1*C144,r1*S144,h1)
point A4 = (r1*C144,-r1*S144,h1)
point A5 = (r1*C72,-r1*S72,h1)
point B1 = (r2,0,h2)
point B2 = (r2*C72,r2*S72,h2)
point B3 = (r2*C144,r2*S144,h2)
point B4 = (r2*C144,-r2*S144,h2)
point B5 = (r2*C72,-r2*S72,h2)
point C1 = (-r2,0,-h2)
point C2 = (-r2*C72,-r2*S72,-h2)
point C3 = (-r2*C144,-r2*S144,-h2)
point C4 = (-r2*C144,r2*S144,-h2)
point C5 = (-r2*C72,r2*S72,-h2)
point D1 = (-r1,0,-h1)
point D2 = (-r1*C72,-r1*S72,-h1)
point D3 = (-r1*C144,-r1*S144,-h1)
point D4 = (-r1*C144,r1*S144,-h1)
point D5 = (-r1*C72,r1*S72,-h1)

constraints: |A1-A2| = |A1-B1| = |B1-C4|

|A1-A2| = 2 r1 S36
|A1-A2|^2 = r1^2 (5 - sqrt(5)) / 2

|A1-B1|^2 = (r1-r2)^2 + (h1-h2)^2

|B1-C4|^2 = (r2 + r2*C144)^2 + (r2 * S144)^2 + (2 h2)^2
 = r2^2 + 2 r2^2 C144 + r2^2 C144^2 + r2^2 S144^2 + 4 h2^2
 = r2^2 + 2 r2^2 C144 + r2^2 (C144^2 + S144^2) + 4 h2^2
 = r2^2 + 2 r2^2 C144 + r2^2 + 4 h2^2
 = 2 r2^2 + 2 r2^2 C144 + 4 h2^2
 = 2 r2^2 + 2 r2^2 C144 + 4 (1 - r2^2)
 = 2 r2^2 + 2 r2^2 C144 + 4 - 4 r2^2
 = 2 r2^2 C144 + 4 - 2 r2^2
 = 4 + 2 r2^2 C144 - 2 r2^2
 = 4 + 2 r2^2 (C144 - 1)
 = 4 + 2 r2^2 (- C36 - 1)
 = 4 - 2 r2^2 (C36 + 1)
 = 4 - 2 r2^2 (((sqrt(5) + 1) / 4) + 1)
 = 4 - 2 r2^2 [((sqrt(5) + 1) / 4) + (4 / 4)]
 = 4 - 2 r2^2 [((sqrt(5) + 1) + 4) / 4]
 = 4 - 2 r2^2 [(sqrt(5) + 5) / 4]
 = 4 - r2^2 [(sqrt(5) + 5) / 2]

so r1^2 (5 - sqrt(5)) / 2 = 4 - r2^2 ((sqrt(5) + 5) / 2)
or r1^2 (5 - sqrt(5)) = 8 - r2^2 (sqrt(5) + 5)
or r1^2 (5 - sqrt(5)) (5 + sqrt(5)) = (8 - r2^2 (sqrt(5) + 5)) (5 + sqrt(5))
or 20 r1^2 = 40 + 8 sqrt(5) - r2^2 (sqrt(5)+5)^2
or r1^2 = 2 + 2 / sqrt(5) - r2^2 (sqrt(5)+5)^2 / 20

(r1-r2)^2 + (h1-h2)^2
 = r1^2 - 2 r1 r2 + r2^2 + h1^2 - 2 h1 h2 + h2^2
 = r1^2 - 2 r1 r2 + r2^2 + (1 - r1^2) - 2 h1 h2 + (1 - r2^2)
 = r1^2 - 2 r1 r2 + r2^2 + 1 - r1^2 - 2 h1 h2 + 1 - r2^2
 = - 2 r1 r2 + 1 - 2 h1 h2 + 1
 = 2 (1 - r1 r2 - h1 h2)
 = 2 (1 - r1 r2 - sqrt(1 - r1^2) sqrt(1 - r2^2))
 = 2 (1 - r1 r2 - sqrt((1 - r1^2) (1 - r2^2)))
 = 2 (1 - r1 r2 - sqrt(1 - r1^2 - r2^2 + r1^2 r2^2))

not clear to me how to solve this analytically

solving numerically with

let(	r5, sqrt(5),
	solve(	let(	r1, sqrt((8-(r2*r2*(r5+5)))/(5-r5)),
			let(	h1, sqrt(1-(r1*r1)),
				h2, sqrt(1-(r2*r2)),
				let(	A1B1, ((r1-r2)*(r1-r2))+((h1-h2)*(h1-h2)),
					((r1*r1*(5-r5))/2) - A1B1
				)
			)
		), r2, .851, .99
	)
)

seems to infinite-loop.  Adding a fuzz factor of 1@-17 yields
.982246946376846024551188207851737388409674167633056640625 as answer;
1@-18 infinite-loops.  Fortunately the 17-place figure is quite
accurate enough for practical purposes.  These lead us to

r2 = .9822469463768460 2 ± 1
h2 = .187592474085079 8907 ± 0524
r1 = .607061998206686 2157 ± 0424
h1 = .794654472291766 1285 ± 0324

pentagons

A1 A5 A4 A3 A2
A1 A2 B2 C4 B1
A2 A3 B3 C5 B2
A3 A4 B4 C1 B3
A4 A5 B5 C2 B4
A5 A1 B1 C3 B5
B1 C4 D4 D3 C3
B2 C5 D5 D4 C4
B3 C1 D1 D5 C5
B4 C2 D2 D1 C1
B5 C3 D3 D2 C2
D1 D2 D3 D4 D5

tetrahedron

top point at
P1 = (0,0,1)
ring of three points at z=h, radius r = sqrt(1-h^2)
points at
P2 = (r,0,h)
P3 = (r*C120,r*S120,h)
P4 = (r*C120,-r*S120,h)

for it to be regular, |P1-P2| = |P2-P3|, or

r^2 + (1-h)^2 = (r (1 - C120))^2 + (r*S120)^2

r^2 + 1 - 2h + h^2 = 2.25 r^2 + .75 r^2
r^2 + h^2 + 1 - 2h = 3 r^2
2 - 2h = 3 (1 - h^2)
2 - 2h = 3 - 3 h^2
3 h^2 - 2 h - 1 = 0
h = (2 ± sqrt(4+12)) / 6
  = (2 ± 4) / 6
  = -1/3 (or 1, which is valid but degenerate)