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add some explanation in aperiodic_pentagons.tes

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Zeno Rogue
2025-12-11 10:12:51 +01:00
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tessellations/sample/aperiodic_pentagons.tes
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@@ -1,8 +1,33 @@
## pentagons that tile the plane aperiodically, by toimine
h2.
# the default value of 'a' yields pentagons with equal edges (i.e., d = d1)
slider(a, 1.383623710490773765269419035368675322851830232551, 0.4, 1.5)
# some advanced formula elements:
# vertex order
# 1
# ^
# / \
#0/ \2
# | |
# |___|
#4 3
# - the shape is symmetric; "big angles" 0 and 2 are 2*a, while the angle at 1 is 2 * pi - 4 * a
# - we need to pick the length of edge, 'd', which makes the angle at vertices 4 and 3 also a
# - for this, we construct a matrix M, using RogueViz matrix formulae:
# lxz(x) is 'Lorentz boost by x', rxy is 'rotate by x radians in xy', etc.
# - angle_from_matrix(M) and dist_from_matrix(M) return am and dm such as M * C0 = rxy(am) * lxz(dm) * C0
# (C0 is the vector (0,0,1) which has 1 on the 'z' coordinate)
# - then we use solve(d=a, b, f(d)) to find the value of d in [a,b] such that f(d) = 0. It is assumed that
# f(a) <= 0 and f(b) >= 0 (bisection method)
let(d = solve(d=0, 3,
a - angle_from_matrix(lxz(d) * ryx(2*a) * lxz(-d) * rxy(4 * a) * lxz(d) * ryx(2*a) * lxz(-d))
))