Like another project of mine, "Heart of glass" (here), this constitutes an unconventional Valentine's gift, or simply a nice project if you like folding paper.

Technically speaking, it's the first stellation of an icosidodecahedron (info here), but I like to think it's a "paper heart" :)

What

A 25cm tall (almost 1 foot) paper stellated icosidodecahedron.

Bill of materials

You'll need:

1. 30 rectangular sheets of paper. I have used standard A4 paper, 80gr/sqm, red. A rim of 100 sheets was 3€, around 3.5\$.
2. about 3-6 hours to cut and fold, about 1 hour for assembling. The cost of this item depends on your salary :)
3. 1 metric ton of PATIENCE! :) And this is PRICELESS :)

No scissors, no glue, no ruler, no pencil. Just paper, time, and patience.

How

First, you need to prepare 30 sheets of ratio 1:sqrt(3). Then you need to cut and fold 60 modules out of them. Then you need to assemble the heart.

Let's see these three things step by step.

## Step 1: Making a 1:sqrt(3) Sized Sheet of Paper

1. Start from a rectangular sheet.
2. Fold along the long size.
3. Unfold
4. Bring the upper left corner along the fold, such that the upper right corner is divided in two angles
5. Flip the sheet
6. Fold the paper and make it slide until you reach the point where the upper right corner meets the fold as in point 4 above.
7. Fold the paper there and mark the fold with your nail.
8. Gently cut the paper following that fold.
9. You're now left with a 1:sqrt(3) sized sheet of paper, with a fold marking two equal sides.
10. Separate the two sides.

## Step 2: Make One Module

You should repeat the previous step (Step 1) 30 times, so you should end up with 60 sides of a 1:sqrt(3) sized sheet of paper.

For each of them, follow this procedure to obtain a module:

1. Fold along the height
2. Unfold
3. Fold along the width
4. Unfold
5. Bring the lower right corner inside the paper, by folding the lower right quadrant along the line connecting the bottom end of the first fold with the right end of the second fold
6. Now fold the entire bottom half along the just created diagonal line, such that the bottom edge of the paper matches the horizontal fold exactly
7. Unfold once
8. Fold the triangular flap so it matches the diagonal fold
9. Turn the paper upside down and repeat steps 5-8 for the other bottom right (or, upper left) quadrant
10. You should end up with a parallelogram, which is flat on one side, and which has a "pocket" on the other side
11. Bring, one by one, the two acute angled points of the parallelogram inside the paper, creating two equilateral triangles
12. Fold the two equilateral triangles keeping the pocket inside
13. You should obtain a single equilateral triangle (folder many times)
14. Unfold the triangle, you should end up with the parallelogram composed now by 4 evident equilateral triangle, two of which are "flaps" (or "wings"), the other two are pockets

Now keep in mind one thing with these modules: each element plays a role: each fold will be either cut or folded again, each pocket with hold a wing, each wing will go into a pocket. This is the key thing to keep in mind in the next steps.

## Step 3: Start Assembling

This is where the fun begins. So far we have used 30 sheets of paper, and 3-6 hours of your time.

Now you need to use the other ingredient: your PATIENCE, and you'll need a lot of it.

There are many strategies to assemble the heart: you can prepare pentagonal pyramids and connect them, or prepare triangular pyramids and connect them, or you can stick the modules one by one, or you can use a hybrid approach.

I'm going to describe the latter, as it's the most flexible one, and it allows to accommodate for each small leftover flaw made while making the modules.

1. Take two modules
2. Take the wing of a module, insert it into the pocket of the other module.
3. Repeat point 2 as many times as needed for the correct pyramid: you need three modules to create a triangular pyramid, and 5 modules to create a pentagonal pyramid.
4. Make some triangular pyramids, and a pentagonal one.

## Step 4: Grow Your Basis

After you make the pentagonal pyramid, connect 5 different triangular pyramids at its edges, keeping in mind one simple fact: pentagons share edges with triangular pyramids and vice-versa. So, no neighboring pentagons, no neighboring triangular pyramids.

Use the boundaries of the just connected triangular pyramids to create other pentagonal pyramids, and keep building up.

Be flexible, if an area is weak, break it, simplify the basis of that area, make it again, try until the structure is solid.

## Step 5: Dont' Panic

When the basis is to large, the pentagons won't stay closed anymore, that's when you need to flip everything and start using the third dimension. The basis should be robust enough to sustain the weight of the sides.

Keep building the sides.

## Step 6: Okay, Panic!

..and I hope you appreciate the reference to "Airplane!" :)

As you see from the picture, it will be hard to keep the entire structure together while building. Arm yourself with a lot a patience. In the end, you're not using Lego or real bricks, you're connecting paper.

The quality, grain, and humidity of the paper will affect your ability to keep the structure stable and tight while building.

If it collapses from one side, don't worry: remove the unstable modules, build an isolated pyramid out of them, tighten the edges as much as you can, mark twice all the folds with your nails, and connect it back to the structure. Every time you do this you'll end up with a slightly more stable product.

One last thing: the more you get close to the end, the less you'll be able to put your hands inside the structure, and you'll be forced to work on the outside only. Don't worry, this will basically affect only the closing of the last pentagonal pyramid, but at that point all the rest of the structure should be tight enough that you can squeeze and move the whole thing.

## Step 7: Enjoy!

Enjoy the final picture!

More tips for a better result:

1) A4 (or Letter) paper will give you easier times making the modules (less precision needed), but more headache assembling (a larger structure to handle).

2) drier paper will bend less and it will bounce back from the pockets more easily. Try to work in a more humid environment, it will make it easier.

3) you can deviate from the read heart design, in that case I'll suggest to use paper of 5 different colors, trying to build each pentagonal pyramid with all the colors available, while the triangular pyramids will be random. Please share a picture if you manager to do this and have triangular pyramids with all sides of one same color!

4) 80gr/sqm paper is good to have a more robust structure, but it's prone to more mistakes and less ease of assembly w.r.t proper origami paper, or 40gr/sqm paper.

Final thoughts:

With this module, you can build a lot more polyedra. If you start folding a module from the upper left corner, you make a mirrorized module. By folding modules of both kinds, you can build:

1) a simple pyramid (1+1 modules)

2) an octahedron (2+2 modules)

3) a dodecahedron (6+6 modules)

4) a stellated icosahedron (15+15 modules)

To build 1-3 as above, which are not stellated, when you fold the pockets, keep the pockets on the outside, instead of on the inside.

Cheers!