Today’s knot print is a mosaic projection of 6_1:

Thingiverse link: http://www.thingiverse.com/make:80243

Settings: Printed on a Replicator 2 with .3mm/low and normal support settings.

Technical notes, math flavor: This knot was designed and printed by JMU Student Taylor Meador. The images in her description below are from slides of a Knot Mosaic talk from Lew Ludwig of Denison College.

A mosaic projection of a knot is one that can be constructed as a mosaic using any of the 11 possible mosaic tiles:

We say that a mosaic projection is

*n*-mosaic if it can be enclosed in an*n*x*n*square, and that the mosaic number of a knot is the minimum*n*such that the knot has an*n*-mosaic projection. The mosaic number for today’s knot 6_1 is 5, and our conformation is taken from the figure below right:Interestingly, as you can see in the figure, in order to realize a 5-mosaic projection of 6_1 we had to use an inefficient projection with

*seven*crossings instead of six. In other words, in order to achieve a minimal mosaic number we had to use a projection with a non-minimal crossing number.Technical notes, OpenSCAD flavor: Taylor designed this model in OpenSCAD based on kitwallace’s minimal stick code by constructing (

*x*,*y*,*z*) corner coordinates near the crossings, based on the picture above right. She describes her process as follows:The code is a list of coordinates interpreted directly from the 2D 6_1 mosaic. We added coordinates one at a time to make a path around the knot. I reinterpreted the 5 x 5 mosaic board as an (

*x*,*y*) coordinate plane in OpenSCAD, with each mosaic tile edge representing 4 units on the (*x*,*y*) plane. I used the*z*-coordinate to allow the knot to pass over or under itself at the crossings; overcrossings simply remained at level*z*=1, and undercrossings were adjusted to z=0. For example, traveling across one mosaic tile from left to right while following an undercrossing would result in coordinates that move across, then down, then across, then up, then across:(

*x*,*y*,1) –> (*x*+1,*y*,1) –> (*x*+1,*y*,0) –> (*x*+3,*y*,0) –> (*x*+3,*y*,1) –> (*x*+4,*y*,1).The a and b parameters in the code allow us to scale the (x,y) plane separately from the z-axis, so that we can better adjust the clearance around and inside the crossings.

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