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Sunday, April 29, 2012

Interlock, Part 2

Interlocking pieces serve us well in many ways. In relationships, parts that fit together to make an integral whole keep us together. In furniture, like in relationships, parts and the way they fit are critical in the integrity of the final result. A simple dovetail joint is not as simple as I thought at first. It will come apart unless we engineer it correctly.

And it won't go together if we over-engineer it. Here is an example of a correctly engineered dovetail joint. One degree of freedom is prevented from slipping by making the pegs into wedges that cannot slip out from left to right.

This dovetail joint is engineered correctly. I can imagine knocking it together with a rubber mallet.

Interlocking components are common in everyday life, but sometimes interlock is too complex to actually engineer. M. C. Escher showed interlocking prison bars in Belvedere. It's one of his more interesting impossible figures.

So I did my best to draw some. When I saw them, I knew that they were impossible to actually engineer. But they could exist in real life, because there's nothing to prevent it physically, once it has been made.

It's just not clear to me how they can go together, since assembly becomes a chicken-an-egg problem. This is an example of something that has been over-engineered so much that it cannot be assembled, I think. No, in order to make this: it must be grown.

It is possible, however, to connect four cotter pins so they interlock. This was no problem for me to construct, but it isn't M. C. Escher's impossible prison bars. It shows also that slippage is possible in interlocking figures: it is not necessary that all degrees of freedom are eliminated.

It is heavily interlocking. And you can make several figures that can never really be assembled out of independent pieces. For example, a relative of the rectangular trefoil knot can be constructed from three pieces that interlock.

Like the impossible prison bars, each piece holds the next piece, which holds the next. This one can actually exist physically. And if it did exist, it can be slipped wider and smaller, a bit like a slip knot.

This is exactly the same as the cotter pins, which can be slipped apart in pairs quite easily.

Actually the image of the three interlocking pieces comes from the mutual overlapping slats that I used to construct when I was a kid. Was it my grandfather that showed me? I can't remember.

We used to make these kinds of interlocking models out of popsicle sticks and toothpicks. Since wood can bend, it made the ideal material.

I remember making models that could also remain rigid by tensegrity, a Buckminster Fuller concept. I considered it to be the work of genius.

Eventually, I constructed geodesic models using thick plastic sheets, using an X-Acto knife and a protractor to get all the angles right and the shapes the right size. Then I used gaffer's tape to put the triangles together and complete the geodesic dome. I was probably 13 when I did this.

Most of my models when I was a child were geometric inventions, a celebration of form. But I also understood function. It's just that I chose not to employ it!

I have created an interlocking monogram for Painter 4 (at the end of the post The Miracle of the Paint Can) and also for Painter 5.

I have created an interlocking monogram for my initials here. As usual, an over-under rule is used, alternating pass-over and pass-under along the line of the letter.

I have designed many chop marks over the years, but I haven't really been satisfied with any of them.

Interlock continues to fascinate me, intuitively. It is a brain teaser that incites us to come up with ways to make these objects in real life. I imagine some of the principles used are useful in industrial design.

It turns out that interlock, useful for locks and keys in the traditional sense, doesn't always imply items that touch and constrain each others movements. We have shown this in the first post on Interlock. I will now continue this theme with some interlocking items which do not touch at all.

Some interlocking items have many degrees of freedom, as these atomic rings show. There used to be a depict for an atom, used in the early days of atomic power, that showed three electrons circling a nucleus. This was naive optimism at its most simplistic and iconic. Especially when you consider the inherent dangers and value of nuclear power. I guess they figured that if you gave it an iconic face, a designer-friendly symbol, you could demystify it in a similarly friendly way.

Anyway, here I have shown three interlocking rings. These do not use the over-under rule, but they are nonetheless interlocked in an inseparable way. No two of them are actually interlocked. Remove the third and they fall apart. This is the most powerful and synergistic form of interlock.


  1. Fascinating, Mr.Zimmer!
    By the way Watercolors and liquid ink in extend, soon? ;) Where did the tints come from?

    Thank you!!

    1. Umm, not sure what you mean? Are you saying you want me to go over watercolors and liquid ink in a new post?

      As for where the tints came from that's easy: (1) new layer, (2) change layer method from default to gel, (3) choose digital airbrush and set opacity and width for coloring, (4) choose the color, (5) color the object in, (6) choose white to clean up when we color outside the lines.

      In other words, it's a coloring book recipe... ;-)

  2. The Gordian knot (click here for an image) may be formed by reducing the degrees-of-freedom ("possible manipulations") by shrinking the rope, e.g. perhaps by baking it in an oven.

  3. the iron bars, I am pretty sure I saw some like that in some windows in Rome.
    Maybe not as complicated as the one Escher drew, but complicated enough to puzzle me for several minutes....

  4. I think I might have seen some there as well. Or perhaps it was Florence or Venice, I can't remember. Anyway, it can be fabricated using a mould, and/or welded together. In Europe, there are more artisans per square kilometer than there are here in the US, partly because of things like cobblestone streets (like those in Florence with the intersecting arch patterns!) and the demand is generated a bit because of all the history around.