Vedalken Samurai

Triangular Orthobicupola

z_(n+1) = sin(i^(4t) sin(z_n)) + c if n is even

z_(n+1) = i^(4t) sin(i^(4t) sin(z_n)) + c if n is even

Gyroelongated Square Cupola

z_(n+1) = ½^a * (i^(-8t) * (1 + cos(2πt))^a * z_n⁹ + i^(8sin(4π(t-⅛))t) * (1 + cos(2π(t+¼)))^a * z_n⁴ + i^(8t) * (1 + cos(2π(t+½)))^a * z_n⁵ + i^(8sin(4π(t+⅛))t) * (1 + cos(2π(t+¾)))^a * z_n⁴) + c(-1)^n

z_(n+1) = i^(4t) sin(sin(z_n)) + c if n is even

Dual of Triangular Cupola, or Semibisected Triangular Trapezohedron

z_(n+1) = ½^a * (i^(-8t) * (1 + cos(2πt))^a * z_n^-4 + (1 + cos(2π(t+⅓)))^a * z_n^-7 + i^(8t)(1 + cos(2π(t+⅔)))^a * z_n^-5) + c(-1)^n

a = log_((1+cos(π/3))/2)(½)

Usually I get the vertex coordinates for these from the Polytope Wiki, but it doesn't list them for this polyhedron. So instead, I calculated them using a Polar Set of a triangular cupola with edge length 1 and centroid (average position of the vertices) at (0, 0, 0). This yields the following vertices:

(0, 0, (3√6)/4)

(0, -(6√3)/7, (3√6)/7

(±9/7, (3√3)/7, (3√6)/7)

(±9/8, -(3√3)/8, (3√6)/16)

(0, (3√3)/4, (3√6)/16)

(0, 0, -(3√6)/2)

There are eight vertices, fifteen edges, and nine faces:

Three kites with edge lengths (3√246)/28 ≈ 1.68047 and (9√358)/112 ≈ 1.52043

z_(n+1) = i^(4t) sin(sin(z_n)) + c if n is even

Elongated Triangular Cupola

z_(n+1) = ½^a * (i^(4t) * (1 + cos(2πt))^a * z_n⁷ + (1 + cos(2π(t+⅓)))^a * z_n⁴ + i^(6t) * (1 + cos(2π(t+⅔)))^a * z_n⁵) + c(-1)^n

z_(n+1) = i^(4t)sin(z_n) + c if n ≡ 0 (mod 4)

z_(n+1) = i^(4t)cosh(z_n) + c if n ≡ 1 (mod 4)

z_(n+1) = i^(4t)sinh(z_n) + c if n ≡ 2 (mod 4)

Square Cupola

z_(n+1) = ½^a * (i^(4t) * (1 + cos(2πt))^a * z_n⁴ + i^(8t) * (1 + cos(2π(t+⅓)))^a * z_n⁵ + (i^(12t) * (1 + cos(2π(t+⅔)))^a * z_n⁹) + c(-1)^n

z_(n+1) = i^(4t) sin(z_n) + c if n ≡ 0 (mod 4)

z_(n+1) = i^(-4t) csc(z_n) + c if n ≡ 1 (mod 4)

z_(n+1) = i^(4t) sec(z_n) + c if n ≡ 2 (mod 4)

Sphenocorona

z_(n+1) = ½^a * (i^(-8t) * (1 + cos(2πt))^a * z_n⁴ + (1 + cos(2π(t+¼)))^a * z_n⁵ + i^(8t) * (1 + cos(2π(t+½)))^a * z_n⁴ + (1 + cos(2π(t+¾)))^a * z_n⁵) + c(-1)^n

z_(n+1) = i^(4t) sin(z_n) + c if n ≡ 0 (mod 4)

z_(n+1) = i^(4t) csc(z_n) + c if n ≡ 1 (mod 4)

z_(n+1) = i^(4t) sec(z_n) + c if n ≡ 2 (mod 4)