Knots (with METAFONT) - Gallery

See the page Knots (with METAFONT) for more details.

Click on the images for high resolution versions.

Overview:

• Three common knots
  • Torusknoten (2,3) (Kleeblattschlinge, trefoil knot, knot 3₁)
  • Viererknoten (four-knot, Achterknoten, figure-eight knot, knot 4₁)
  • Torusknoten (2,5) (knot 5₁)
• Torusknoten (2,7) (knot 7₁)
• Some knot and its mirror image (knot 5₂)
• Three knots with 9 crossings
  • A strange knot (knot 9₂₂)
  • Another knot and its mirror image (knot 9₂₉)
  • And yet another knot (knot 9₄₀)
• Torusknoten (3,4) - 3 variants (mirror images, with alternating crossings)
• Torusknoten (4,3) - 3 variants (ditto)
• Torusknoten (6,11) - 3 variants (ditto)
• thin versions…

There is a further section Drawing torus knots.

Drawing torus knots

Cf. Source files (1995).

The torus knots have interpolation points

torusknot(p,q,n): (R+r cos(qt+c))*(cos(pt), sin(pt)) with t=360°k/n, k=0,…,n-1

where r<R are two radii and n is the number of points.

The interpolation is done with METAFONT's "P₀..P₁.. ⋯ ..P_n-1..cycle" (without "tension"). For examples see the images at the end of the page.

The constant c is now set to c=0. In 1995 it was c=30° causing rotations (disturbed for smaller n). (The setting c=30° was perhaps a leftover from experiments for (p,q)=(2,3)).

Torus knot (p,q) has (p-1)q crossings. On the path there are q groups of (p-1) consecutive over-crossings followed by (p-1) consecutive under-crossings.

The shown torus knots are

• trefoil knot - the 1995 version (it is drawn in another way)
• torusknot(2,5,21) (r/R=3/10, c=0, rotated by 90°)
• torusknot(2,7,60) (r/R=3/10, c=0, rotated by 90°)
• torusknot(3,4,20) (r/R=4/10, c=30°, unrotated) - see Variants for torus knot (3,4) below
• torusknot(4,3,20) (r/R=4/10, c=0, rotated by 90°)
• torusknot(6,11,140) (r/R=3/10, c=0, rotated by 90°)

Ideally n is large approximating a smooth curve. However one may experiment with small values of n. Here are torusknot(2,5,n) for n=15,21,30:

Variants for torus knot (3,4)

Here are for comparison

• torusknot(3,4,20) (r/R=4/10, c=0, unrotated)
• torusknot(3,4,20) (r/R=4/10, c=30°, unrotated) - the 1995 drawing which looks better, I think

In the second case the interpolation points do not contain the extremal "outside" points resulting in a different shape (not just a rotation).

[Click on the images for high resolution versions.]

Go to: Markus Rost's Web Page