Trefoil This unfortunate looking shape is a soft compact trefoil resting on a hard surface. Can you guess how the 2 dimensional visible surface image of this physically real minimum compact trefoil was created?  The largest image is accomplished from an entirely different frame of view but produces the same result.  The knot transformation is perfectly described by XYXY-1X-1Y-1 = 1.

Watching the resulting images while a laser scanner operates provides a little insight into important symmetry within the shape.  For example, some of the above images are the result of a laser line projected onto portions of the visible surface (4 separate side scans each 90 degrees apart) while the larger image results from a laser dot projected on the surface while the surface rotates.

The minimum compact length of a torus with similar properties would be 4*pi*R (1 loop or 2pi traversed at a 2R radius).  Is the minimum compact length of a trefoil (nobody has a formula for it yet supposedly) 3*squareroot(3)*pi*R ~ 16.3242*R?  That would be pretty convenient.  Regardless, this formula works to the third decimal place.

If this idea has any merit (beware, very likely not!), you may consider it to be the result of traversing through 2 loops (4pi) multiplied by an effective projected radius of one of the loops (in this case, (¾)*sqrt(3)*R).  This would assume that there is no elongation or shrinkage of the neutral axis from the effects of bending and torsion that generate the shape.

It seems somewhat consistent with minimization of the curvature function for the trefoil and the fact that the length of any curve will involve the square root of the sum of squares of 3 coordinate derivatives, the characteristic of each coordinate derivative being identical.

Trefoil curvature function:

c(t) = (4cos2t+2cost, 4sin2t-2sint, sin3t) A laser scan of a large non compact trefoil formed with Loc-Line tubing segments.  The reflectivity of the glossy finish on the tubing causes many of the laser dot projections to be lost.  In some sense, this loss is a little helpful. This surface can be oriented to expose 2 loop projections.

A 2D photographic image of the actual surfaces that were scanned in the above images is shown below: Physics Experiments

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