Jun 26
Elongated Square Pyramid
z_(n+1) = ½^a ((1 + cos(2πt))^a * z_n⁵ + i^(8t) * (1 + cos(2π(t+⅓)))^a * z_n⁴ - (1 + cos(2π(t+⅔)))^a * z_n⁵) + c(-1)^n
a = log_((1+cos(π/3))/2)(½)
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Elongated Square Pyramid
z_(n+1) = ½^a ((1 + cos(2πt))^a * z_n⁵ + i^(8t) * (1 + cos(2π(t+⅓)))^a * z_n⁴ - (1 + cos(2π(t+⅔)))^a * z_n⁵) + c(-1)^n
a = log_((1+cos(π/3))/2)(½)
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so sad that my blorbo, J37 the Pseudo Rhombicuboctahedron is not qualified for this tournament :(
look at him go!!
he almost made it as a 14th Archimedean solid!
I wanted to include him instead of the sphere, because even if he's not always considered an Archimedean solid we must agree that he's more related to the shapes in this bracket than the sphere. But alas, I couldn't separate him from his buddy and dual, the psuedo-deltoidal icositetrahedron.
And if I used both of them then there would be 33 entrants in the bracket, which just wouldn't do. So I used the sphere, which might not be a polyhedron but it is at least a highly symmetrical three-dimensional shape.
I will find a way to put these shapes in a bracket eventually, although I don't yet know how.
(Image credits: Robert Webb's Great Stella Software and Tomruen.)
Gyrobifastigium
z_(n+1) = ½(i^(4t) * (1 + cos(2πt)) * z_n⁴ + i^(6t) * (1 - cos(2πt)) * z_n⁵) + c(-1)^n
Triaugmented Triangular Prism
z_(n+1) = ½^a * (i^(4t) * (1 + cos(2πt))^a * z_n⁴ + (1 + cos(2π(t+⅓)))^a * z_n⁴ + (i^(-4t)(1 + cos(2π(t+⅔)))^a * z_n⁴) + c(-1)^n
a = log_((1+cos(π/3))/2)(½)
Gyroelongated Triangular Cupola
z_(n+1) = ½^a * (i^(4t) * (1 + cos(2πt))^a * z_n⁵ + i^(-8t) * (1 + cos(2π(t+⅓)))^a * z_n⁷ + i^(12t) * (1 + cos(2π(t+⅔)))^a * z_n⁴) + c(-1)^n
a = log_((1+cos(π/3))/2)(½)
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Snub Square Antiprism
z_(n+1) = ½^a * (i^(-8t) * (1 + cos(2πt))^a * z_n⁴ + (1 + cos(2π(t+⅓)))^a * z_n⁵ + i^(8t)(1 + cos(2π(t+⅔)))^a * z_n⁴) + c(-1)^n
a = log_((1+cos(π/3))/2)(½)
Metabidiminished Icosahedron
z_(n+1) = ½^a * (i^(-8t) * (1 + cos(2πt))^a * z_n⁴ + (1 + cos(2π(t+⅓)))^a * z_n⁶ + i^(8t) * (1 + cos(2π(t+⅔)))^a * z_n⁴) + c(-1)^n
a = log_((1+cos(π/3))/2)(½)
Gyroelongated Square Bipyramid
z_(n+1) = 0.5 * (i^(12t) * (1 + cos(2πt)) * z_n⁴ + i^(-12t) * (1 - cos(2πt)) * z_n⁴) + c(-1)^n