Menger's Cube Cardboard Lampshade Made From Metro Tickets

Here in Paris, the metro network still uses disposable one-use tickets. I always wanted to do something with all those tickets, so I kept them for years, until I found the perfect project for those !

I got the idea from a video of Mickaël Launay, a french youtuber talking about mathematics on his channel Micmaths. In an old video from 2015, he talks about "non-transitive relashionships" in mathematics. Basically, those are non-linear relathionships between 3 or more objects, meaning that it is not possible to deduce a clear order applying to all of them. For example, if we know that A > B and B > C, if we are in a non-transitive relashionship between A, B and C, we cannot automatically deduce that A > C.

An easy way to understand that is the rock-paper-cissors game : we know that rock beats cisors, and cisors beats paper, but it doesn't mean that rock beats paper ! Those three objects have a non-transitive relashionship, each one beats one other and gets beaten by the last one. There is no "absolute" winner, there are in a closed circle.

Here we are going to use non-transitive relationship to build a 3D structure in carboard tickets, with no glue or tape. We are going to build cubes where two sides are going to maintain two other sides and get maintenained by the two remaining ones. If this sounds complicated, don't worry, we are getting to it step by step, and after that you're gonna have a very cool piece of knowledge to drop !

Once we'll have our cardboard cubes, we are going to use them to build an M2 Menger sponge. A Menger sponge is a fractal structure : you start with a cube, you divide each face into 9 squares and remove the center square until you reach the opposing side. You do that for each side and you got an M1 Menger sponge. Then, you repeat the same process on each 8 remaining square on each side. And you repeat. You are left with a cubic structure, where each face has its center square hollowed, and each other square on the face also has its center hollowed, etc. etc.

Obviously, we are not gonna do this process all the way to the microscopical level, we are gonna stop at a level 2 Menger sponge and use the structure as a cool and kinda unique lampshade !

All of this may sound really difficult to understand, but once we get to the practical bulding, it's all gonna make sense !

Let's get to it

Supplies

Used metro tickets (you will need 2400 tickets for an M2 Menger sponge, more if you wanna make some legs to your lamp like I did)

Nothing else, this is a glue/tape free project !

(Note: I used metro tickets from Paris because they are made of thick cardboard and I have tons of them. But you can use any cardboard ticket, or even carboard pieces that you cut to size. The only requierement is that they have to be rectangular, with a quite longer side, to correctly maintain the sides)

Step 1: Collecting the Tickets

First, be a bit picky about the tickets you're keeping. Avoid any ticket that has deep folds or creases. You'll have to create your own folds, so avoid the ones that already have some, like the right one on the picture.

I also asked all my friends and familly to keep and give me their own used tickets to collect all 2400 of them faster.

Step 2: Folding the Tickets

We are going to fold the tickets in their usable form. The steps are simple:

1. place two tickets on top of eachother, centered, forming a 90° angle
2. fold the bottom one over the top one
3. place another ticket on top, centered, forming a 90° angle with the one under it
4. fold again
5. repeat

This will give the tickets their final form, and prevent them from unfolding before you use them. If your tickets have different sides like mine, try to always fold them on the same side, to keep some uniformity in the final project.

Step 3: Make a Cube Using Non-transitive Relashionships

This is the step where we are going to use the non-transitive relashionships.

Take 6 of your tickets and mark the folds :

1. take three of them and place the wings of two of them (n°1 & n°2) behind the third one (n°3)
2. place another ticket (n°4) so that one wing of n°3 is behind it, while its own wings are over n°1 & n°2
3. take another (n°5) and place it on the opposite side of the n°3, with one wing over n°4, and the wings of n°1 & n°2 over it
4. take the last one (n°6) and close the cube : place it on opposite side of n°4, with its wing over n°1 & n°2, and the wings of n°3 & n°5 over it.

And there you have it, a cube made of 6 tickets, with each ticket maintaining two tickets with its wings and being maintained by the wings of two others. Each side has a non-transitive relashionship with the others, creating a self-maintaining cubewith no glue or tape :)

Step 4: Assemble Two Cubes

Here, we are going to connect two cubes, by bulding the second one on top of the first. The process is almost identical, except that you have to start by placing the wings of the two first tickets under the wings of the side of the other cube.

Then, you just continue building the second cube following the same process that you did in Step 3. Always make sure that each side maintains and is maintained by the four sides next to it.

Step 5: Making an M1 Menger Sponge

Now that we know how to make cubes and attach them together, let's combine those skills to create an M1 Menger sponge.

Start by attaching 8 cubes to form a hollow square. Then add a cube in each corner. And finally, attach 8 other cubes on top to compleat you your M1 Menger sponge

Step 6: Note - Attaching the Corner Cubes

One important note :

When building the corner cubes, sometimes the 3 surrounding ones won't align well, making it impossible to attach the cube to all three sides.

In this case, you can cheat a little by folding one wing of some tickets the other way around. This way, you can place the wings under the top wings of the surrounding cube, instead of the side wings. Look at the images to visualize it.

Step 7: Step Up to Make an M2 Menger Sponge

This step is simple but very long : you create a Menger sponge with M1 Menger sponges instead of simple cubes. I sadly don't have any step-by-step photos, because I finished mine a few years ago, but there is nothing complicated.

You have to build and connect each cube one by one, following the same patern : create one M1 Menger sponge after the other, first placing 8 of them to make a hollow square, then 4 sponges on the corners, and then another hollow square made of 8 sponges on top.

This step is not really complicated, but it's really long, especially if you don't have enough tickets and have to collect some more before going back to the building. Don't be impatient, the end result is worth the time !

Step 8: Bulding Legs to Your Lampshade

I want light to be able to come from under the cube in my final lamp, so I took some spare tickets and builded some small cubes on the corners of one side of the sponge. This step is really up to you but I would advise making more than four legs, to add support to the structure.

Step 9: Installing the Lamp

This is finally the last step !

Install a light bulb where you wanna put your lamp. I chose a corner over my desk. Chose carrefully the lightbulb : because it's going to be close to a lot of carboard, be sure to select an LED one, to avoid excessive temperatures.

Place the Menger sponge so that the lightbulb will end up in the hollow cube in the middle, and it's finally done !

You can now enjoy your own glue/tape-free lampshade, made from tickets linked by non-transitive relashionships forming an M2 Menger sponge, that also acts as a great conversation starter ;)