Level One

Today we continue our step-by-step walkthrough of building a Level 4 Menger cube out of business cards. If you want to join the fun then sign up now at www.megamenger.com. You can help with an official Level 3 build site if there is one near you, contribute a Level 2 component to a Level 3 site, be your own independent Level 2 mini-site, or even take part as a Level 1 micro-site.

Of course a Level 0 Menger sponge is not very exciting, since it’s just a cube. But things start to get interesting when you attach Level 0’s together, as described nicely in Evelyn Lamb’s recent Scientific American blog post about the MegaMenger project (she’s also hosting a Level 2 build with the AWM chapter at the University of Utah!).

Here’s a Level 1 Menger sponge made of twenty Level 0 Menger sponges attached together by their tabs:

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The MegaMenger site has great set of detailed downloadable worksheets that describe how to make such a construction. The most basic thing you have to know after making a six-card cube is how to attach two cubes together, and how to put a panel card over the tabs of a cube. Here’s the worksheet that tells you how to do just that:


Panels provide stability, lie flat, and just plain look nice. As an added bonus, the MegaMenger build sites are using paneling cards that carry the image of a Level 5 Sierpinski carpet, so that the resulting fractal will look even fractally-er (you can download the panel card design at megamenger.com). The trick is to add panels as soon as possible, but not too soon. It’s easier to link two cubes together if at least one of them has a panel on it somewhere for stability, and it’s important to attach panels before they become inaccessible within the build. However, we don’t want to add panels too early because paneled faces can’t be joined to other cubes.

There are lots of ways to make a Level 1 from twenty Level 0’s; you could build up cube by cube, or level by level, or you could build around four-cube structures called “tripods”, like the one shown in the Example in the worksheet above. The tripod method was pioneered by engineer Jeannine Mosely, who built the first Level 3 Menger sponge out of business cards. (Note: Mosely’s first Level 3 sponge took nine years to construct – which is why it is completely crazy that the MegaMenger project will have twenty sites around the world each trying to make Level 3 sponges in just a week!)

Notice that on the worksheet picture, six of the faces of the tripod are paneled (each of the faces you can’t see in the picture are not paneled). These six paneled faces are precisely the faces that will end up on the inside of our Level 1 sponge after assembly. I think the easiest way to make a interior-paneled tripod is to make three basic cubes that each have two adjacent faces paneled, plus one basic cube with no panels, like this:

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Here’s a key method that is going to help us through all stages of our Menger build this week:

Once you make a cube, panel any two adjacent faces. It doesn’t matter which two, as long as they are adjacent.

Of course you’ll also want to make some blank cubes, but if you’ve got people working with you on a Menger build, the first thing you should have them make is two-paneled cubes.

To make a tripod from these four cubes, attach them so that the non-paneled cube forms a corner that is connected to each of the paneled cubes. In the picture below, the non-paneled cube is hidden behind the three paneled cubes. In addition, only the six faces that face front are paneled, and no others:

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It turns out that tripods have an “orientation”, and that tripods with different orientations will not attach together. For the MegaMenger project we have decided to use the orientation shown above, in which when you look at the tripod you see horizontal tabs on the left leg and vertical tabs on the right leg. This tab orientation will be the same no matter how you turn the object, as long as you are facing the point where the six paneled faces meet. Be careful not to construct “mirror” tripods with the reverse orientation. Starting with three cubes that each have any two adjacent sides paneled, you’ll always be able to make a correctly-oriented tripod. If things look backwards then try turning the offending cube around so that a different face attaches to the center cube.

If you’re managing a build and people are building tripods, then remind them to check for correct orientation:

Make sure that each tripod is oriented correctly, with horizontal tabs on the left and vertical tabs on the right.

To make a Level 1 with the tripod method you’ll need four six-paneled tripods and four basic unpaneled cubes, which looks like this:

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Then put the tripods and cubes together as shown in this handy MegaMenger worksheet:

Tripod construction method for level-1 sponge

If you’re running a MegaMenger build and need to train a lot of people at once, then show them this great video from my hero Katie Steckles (@stecks on Twitter) of ThinkMaths. Katie seems to literally do six impossible things before breakfast every day, and is the person who manages and drives the worldwide MegaMenger project. In this video she’s using the “one cube at a time” construction method:

Next time: Moving on up from Level 1 to Level 2 and beyond…

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