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Sunday, June 8, 2014

Day 286 - Torus conformations of 7_1

Today we have two torus knot conformations of 7_1. A torus knot is one that can be drawn on the surface of torus ("inner tube" shape) without intersecting itself. A T(p,q) torus knot wraps around the torus like a clock p times, and around the handle of the torus q times. The standard conformation of 7_1 in the knot table is shown on the left, in the form of a T(2,7) torus knot. The blue conformation on the right is the same knot but in the T(7,2) torus knot conformation.

Settings: Printed on a MakerBot Replicator 2 with our custom knot slicing settings to minimize supports. Printing the knot on its side results in far, far less support material than printing knot horizontally.

Technical notes, OpenSCAD flavor: This knot was printed by JMU student Taylor Meador, who used a modified version of the OpenSCAD code for our trefoil torus knot models from Day 150 (thanks as always to kitwallace).

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/*
// trefoil as the torus knot T(7,2)
// http://mathworld.wolfram.com/Torus.html
// take parameterization of torus (u,v)->R^3
// and let u=2t, v=3t
// scaled to 40mm before tubifying
function f(t) =
[ 3.9*(3+1.6*cos(7*t))*cos(2*t),
3.9*(3+1.6*cos(7*t))*sin(2*t),
3.9*(1.6*sin(7*t))
];
// create the knot with given radius and step
tubify(1.6, 1, 360);
*/

module tubify(r, step, end) {
for (t=[0: step: end+step]) {
hull() {
translate(f(t)) sphere(r);
translate(f(t+step)) sphere(r);
}
};