Knotted Web Tight Origami Tessellation

Backlighting Surprise!

As the name implies, Knotted Web Tight is a close relative of Knotted Web - everything's just one spacing closer together.

Interestingly, the donut shape that's somewhat visible in Knotted Web is much more pronounced in Knotted Web Tight and shows up as overlapping circles in the backlighting.

This was a huge surprise to me when I first saw it while I was folding!

No part of this pattern is actually curved, and yet the backlighting seems to show only curves!

What's more, since this is an isoarea pattern the same backlighting is seen on both sides.

Unusual symmetry pattern

When you look closely at Knotted Web Tight you don't see the circles as clearly as you do from a zoomed-out view.

Instead, I see three triangles and a hexagon forming a larger triangle, which is then arranged in a way that creates alternating large and small triangular gaps.

I see three positions of three-fold rotational symmetry - one in the hexagon, one in the small triangle gap, and one in the large triangle gap (which is actually a hexagon twist on the other side).

I see that every other pleat from the hexagon twists is going to another hexagon twist - not a triangle!

I see that the triangles right next to the hexagons aren't actually on pleats coming directly coming from the hexagon - all pleats from twists on one side connect directly to twists on the other side.

Same on Both Sides

Did you know that you can put hexagons into isoarea arrangements?

If you're using the standard way of connecting them, you'll end up with all hexagons on the same side.

This means they'll all be on either the front or the back of the pattern and you won't have an even split of some on one side and some on the other and your pattern won't be isoarea.

But if the hexagon is connected to three hexagons and three triangles instead of to six triangles, then you can (under limited circumstances) put them on alternating sides!

I was just exploring this tiling further before I teach it to my students in Tessellations by Tiles and these cases of Knotted Web and Knotted Web Tight that end up being the same on both sides are the exception, not the norm.

The more typical case is to take a hexagon and three triangles, then mirror this cluster to form the full tessellation.

The only way to get these clusters on alternating sides is if the pleats match up perfectly when you flip the paper over, which is exceedingly rare.

So even though the ring of six triangle twists is central in the pattern, the specifics of the ring result from the interplay of these clusters of a hexagon and three triangles.

This means that these tessellations can't be designed from the inside out - only from the cluster out!

If you'd like to learn design tips like these in more depth and practice designing your own tessellations with support and guidance, sign up for the Tessellations by Tiles waitlist here to be notified when enrollment opens again.