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Here’s a seriously crafty construction project to take on! While it’s impossible to make a Menger sponge in real life, give this approximation a go.

You will need

• At least 120 business cards, all the same size

What to do

Preparing the cards

1. Take two business cards, put one on top of the other. Rotate the top card so they form a cross.

2. 

   Take one end of the bottom card and crease it at the side of the top card. Then do the same for the other end of the bottom card. The bottom card should now be ‘hugging’ the top card.

3. Fold the two ends of the top card over and crease them at the sides of the bottom card. Your two business cards should now make a square.
4. Separate the two cards – they are now ready to be used.
5. You will need six of these cards to make one cube and 120 to make a Menger sponge.

Making a cube

1. 

   Take six prepared business cards. The two folded flaps on the ends are called tabs.

2. Lie one card on the table with its tabs to the sides, facing upwards. This forms the base of the cube.
3. Take another prepared card and tuck one of its tabs under the front of the base. A third should have a tab tucked under the back of the base. These form the front and back of the cube.
4. Take another card to form the left side of the cube. The tabs on this card should go around the front and back of the cube. The left tab on the base needs to go around the left side. Once you’ve finished, all the tabs should be on the outside, not inside the cube.

5. 

   Using a similar technique, use a card to make the right hand side of the cube. The cube should now be open on top and there should be two tabs over the gap, from the front and back.

6. Take a final card to form the top and hold it with its tabs to the sides, facing down. Its tabs need to go on the outside of the two sides and the tabs on the front and back of the cube need to go around the top card. This step is tricky and might take a bit of unfolding.

Joining cubes together

1. Take two cubes and put them side by side. Look at the adjacent sides and make sure the tabs on the left cube run up–down and the tabs on the right cube run front–back.

   

2. The tabs should ‘hug’ each other. Start by tucking the bottom tab on the right cube under both tabs on the left cube. It should hold securely.
3. Now, tuck the top tab under the two left hand tabs. This is a bit tricky and you might need to partially dismantle the right cube to do it.
4. Your two cubes should now be securely joined.

The Menger sponge

1. Start by making 20 cubes.

   

2. The shape is a 3 x 3 x 3 cube, with some cubes missing. We won’t put a cube in the middle of each face and we won’t put one in the centre of the cube.
3. Start by making a 3 x 3 square, with the middle cube missing. This will form the base.
4. Use five more cubes to make the front face, also a 3 x 3 square with the centre missing.
5. Now complete the left and right sides in a similar manner and finally add one cube to join the two to complete the back and top sides. Your Menger sponge is complete!

What’s happening?

What you’ve made is not actually a true Menger sponge – it’s just an approximation. A true Menger sponge is not made of cubes, like in this activity – instead it’s made of Menger sponges. Each of those Menger sponges is also made of Menger sponges, and so on, forever.

It’s impossible to make a Menger sponge in real life. However, you can make closer and closer approximations. A cube is a level-0 approximation. In this activity, you make a level-1 approximation. If you have a lot of business cards, you could make 20 of the sponges in this activity. Then you could make a sponge of sponges – a level-2 approximation.

Menger sponges have some strange properties. Imagine a line of Menger sponge approximations, all the same height. If you look at the front face of each, you’ll notice the area shrinks as the approximations get better. A level-1 approximation has 8/9 the face area of a level-0 approximation and a level-2 approximation has 8/9 the face area of a level-1 approximation. Following this pattern to the end, we find a true Menger sponge has a face with no area!

A similar argument shows a Menger sponge has no volume. Surprisingly, although the face of a Menger sponge has no area, the sponge as a whole has an infinite surface area!

More information

Some good diagrams for folding business card cubes
A third order approximation of a Menger sponge
What happens when you slice a Menger sponge diagonally?

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