Transformer / Paper Infinity Cube (Tubes) - No Origami Boxes!

I've been obsessing about infinity cubes. I'm trying to wrap my brain around them, so I've made a lot. I've used different materials, different techniques. I even found that the "cube" can be tall or flat, such as the shape of small candy boxes (see https://www.youtube.com/shorts/-3XD0auftzg). 

I had a vague memory of an Instructable that I'd seen some years ago, but I couldn't find it again. I had to reinvent it. The paper is folded into two long, square tubes that are cut and folded strategically. This is MUCH faster to make than the usual approach of 8 origami cubes, fastened together. (I've since relocated the original; see "The Paper Puzzle" (misko13) https://www.instructables.com/The-Paper-Puzzle/ and its video demo.) 

I promised to show this technique at a table in a Maker's Fest. Madly prototyping, I tried again and again to find a good, simple material that was easy to fold, and yet sturdy enough to stand up. I thought I had settled on a tiny, elongated "cube" of construction paper (cardstock did not fold cleanly), and then I discovered something almost magical:

If I stuff each segment with a paper coil, the toy becomes much more sturdy, even at the relatively large and easy-to-handle half-sheet size.

(I also learned that I can mist cardstock or other heavy paper with a tiny bit of water to make it fold better. And I worked out how to make an embossing template out of plastic needlepoint mesh. I'll save the second for a future Instructable.) 

Below are instructions for the most successful construction-paper version, with some notes for other options. You can also watch my video. (I'll come back and insert it here when I figure out how to do that.)

In the photos, the blue/dark one is a "tall cube" and the lighter violet one is a "true cube".

• Paper - 1 sheet construction paper *
• Scissors - not too long, preferably sharp and pointy (for paper, maybe tape)
• Tape - clear/cellophane style, standard 3/4" *
• Hard surface

Optional:

• Pencil (recommended for beginners)
• Paper cutter or sharp letter opener (to supplement scissors) ~
• Folding tool (bone folder, ruler, or a firm, smooth tool handle—possibly the scissors)
• Water mister (if using cardstock or thick paper) *
• Scoring plate and tool (alternative for cardstock) ~
• Quick-Reference Sheet (PaperTubesQuickRefSheet.pdf)

* See the final step for options, variations, and tips. 

~ I don't give specific instructions for these tools. They're just alternative ways to cut or fold paper.

Right now, I just want to get into the how-to, so I'll explain variations in the final step. Start simple; you'll probably want to make more.

The step title says it all. If you're comfortable folding paper, just use your preferred method. If you do origami, this begins as a cupboard fold and then goes one level further. See the first photo of Step 2 for the desired result.

I do this folding in any of several ways, depending on how easily the paper folds, the lighting, my surface, etc. Here's one way, in much more detail than you probably need:

• Bend the sheet over to match the short edges. Align the corners as well as possible and fold gently. Check, adjust if needed, and then run a fingernail or folding tool (possibly a handle of the scissors if you can find a smooth part) along the fold to crease it (make a clean, crisp fold). Open.
• Bend the sheet over to match the long edges. Align the corners and central fold as well as possible, then fold gently, check, and crease firmly as before. Open.
• Match one long edge to the fold. Match the corners, then fold gently, check, and crease firmly as before. Open.
• Repeat with the other long edge.

You should now have three folds and four long sections. Each section needs to be divided in half, so continue:

• Re-fold the central fold and one of the long folds so that they bend to the same side (both mountain folds or both valley folds). Align the folds and crease. (Open or leave folded, as you prefer; I leave them.)
• Repeat with the remaining long fold and the center. (Open or leave folded, as you prefer.)
• Fold a long edge to match the nearest fold; crease. (Open or leave folded, as you prefer.)
• Repeat with the other long edge.
• Open all folds.

Cut the paper in half in the short direction (parallel to a short edge). If you followed the ridiculously detailed instructions in Step One, you'll cut along the very first fold you made.

NOTE: The following instructions make a shape that isn't a perfect cube, but is bigger in one dimension (a "tall cube"). I strongly recommend making a tall cube the first time. However, if you want a true cube (same size all three ways), use one half of the paper to measure out the piece you'll be folding further; see the extra photo for how to start, or watch the later part of the video for more details. Trim it to size, and then continue with these instructions.

Again, use your favorite technique if you have one. (If you do origami, this is a cupboard fold.)

These creases should be at right angles to the previous creases, forming 32 boxes. Fold only one of the two pieces; leave the other with just the original eight long strips.

Here's one way (it's the same as the first part of Step 1):

• Bend the half-sheet so that the (new) short ends align; crease (fold gently, check, adjust if needed, crease firmly). Open.
• Fold one short edge to the center crease. Open.
• Repeat with the second short edge.

You can just follow the diagram; the cuts are the heavy, red lines. They can be cut in whatever sequence you wish. 

Here's one way (again, more detail than you probably need):

• Use a pencil to mark the parts to be cut (on the back of the paper, if the two sides differ). Follow the diagram. Mark the end of each cut so that you don't cut too far. (I usually don't mark, anymore; I just remember the 1–2–1 and 2–1–2 pattern.)
• Fold the piece in half so that the short ends match. Notice how three creases cross this fold. Cut along the two outside creases, just a little distance. (Optional, especially if using short scissors: Also cut the middle crease.) You're making slits to fit a blade of the scissors.
• Open the paper most of the way.
• Tuck the two outside strips down slightly; see the photo.
• Put the bottom blade of the scissors into the slit and cut along the center crease, just between the two side-to-side slits. Do not cut the outside strips! If you do cut them, tape them back together neatly. Re-crease them before continuing.
• From the center slit outward (toward one short end), cut along one box of the first fold, two boxes of the second fold, and one box of the third fold (1–2–1). It's symmetric, so either side can be first. (If you goof and cut 2–1–2 in the center, you can match it, and then cut 1–2–1 into the two outside edges. Make it symmetric, both ways. It'll work; it's just a bit harder to tape.)
• Repeat this step on the other side, cutting from the center slit toward the other short end: 1–2–1. 
• From one of the short ends, cut along the creases 2–1–2: two boxes along the first fold, one box along the second fold, and two boxes along the third fold.
• Repeat with the other short end (2–1–2).

Open the paper and check it. If you follow along one of the long folds, it should be cut, then uncut for one box, then cut, then uncut for one box, then cut. The two sides should be mirror images. The top and bottom should be mirror images. The cuts along the center crease should not be quite the same as the cuts along the other two long creases. If that doesn't make sense, compare your cuts to the diagram.

If the paper has a good side (that you want on the outside), put it face down. If any of the creases fold away from you (mountain folds), refold them so they fold toward you (valley folds); the page should curl upward slightly.

• Position the paper so there are eight parts from side to side and four from top to bottom. You're going to fold the two side edges toward the center and tape them, forming two square tubes (see gray photo in the introduction).
• (Get a piece of tape ready, if you can hold it while folding.)
• Lift the top left corner. Fold two boxes toward the center so that the edge lies along the center crease. Tape the edge to the center crease (see photo). Try to keep the tape neatly along the top strip.
• (Tape ready)
• Lift the top right corner and fold it to the center crease (the same location). Tape. Part of this piece of tape should be on top of the last piece. Top completed.

Rotate the paper so the bottom (toward you) is on top (but the "good" side is still against the surface). Then fold and tape the new top strip, as above.

Form four boxes along the sides:

• (Tape ready)
• Two strips should be sticking out on each side. (Four boxes in a strip, going side to side.) Start on, say, the left side and the upper of the two strips.
• Fold one box in from the outside (left edge, fold to the right).
• On the same strip, fold the centermost box toward the outside (right edge of that strip, fold to meet the previous edge) - see the photo.
• Tape across the two edges to connect them.

Repeat with the lower of the two strips.

Repeat with the upper and lower of the other two strips; rotate the page first, if that's easier.

The general idea: Go over the toy bit by bit, creasing or re-creasing the places that have been taped, and rehearsing the folds that are going to be hinges. None of this is essential, but I find it makes the toy work more smoothly. If it's your first time, you might skip to the Step 9 to see how it works, then come back to this.

I use "rehearse" to mean to fold a crease one way, then open it, then fold it the other way, and then open it again. Maybe do this a few times, until it folds easily. (My thinking: The cube has already learned these folds, but they need to be loosened up through practice. These are the toy's hinges, and they need to work easily and naturally.)

Rehearse the vertical center creases (top and bottom). Each of these has two layers of tape, so it may be stiff, and may not even seem to fold in quite the right place. If needed, match the nearby edges or ends to remake the crease in the right place.

Rehearse the horizontal creases. There are six of them; two along each horizontal line (see the first photo).

The tape tends to flatten out the boxes. Time to make those folds sharp again. (I think: Teach the four center and four end strips how to fold and become 3-D boxes. ...Yeah, I know I'm odd. But these infinity cubes develop personalities!)

First the middle four:

Take each part that sticks out to the side and open it, then collapse/fold it toward the center. The place that was taped needs to become a vertical crease along the center line of the toy; see photo.

Now the top and bottom:

Open the top two boxes (the end will look like a double diamond) and collapse them the other way, so that the crease that was along the left side now matches the center crease, and similar for the other part. See the photo; it's better than my words. If you can think of the toy as two tubes, you're flattening the tubes against each other.

Repeat with the bottom two boxes. Sometimes the second pair tries to fold off to the side rather than aligning along the center. Just open the tubes up a bit and pull them to the center.

While the toy is in this squashed position, I often rehearse the three middle folds again. It's a fan fold one way, then open, then the other way and open. If any of the cuts was a bit off, this extra rehearsal may help fix things.

Open each of the eight boxes until the edges form right angles (roughly). If they're stubborn, I take the scissors and slide the (closed) blades along the inside of the tube, then twist gently to open the boxes.

Roll the two tubes toward each other.

Try to fold the ends forward or backward, gently. Either the toy should fold in half, or the two ends should fold toward the center; continue that motion. Then open it the other way. (If it folded in half, the ends should open in a different direction. If the ends folded in, two halves should open in a different direction.)

If a box collapses, keep going with the motion until you can open the box again. 

Remember that other half of the paper? Find it again. Look on the floor, maybe; mine often ends up there.

If you're doing this with other people and they have paper of different colors than yours, maybe trade.

This half-page should be folded into eight strips from Step 1. Now:

• Separate along the creases to make eight separate strips. (For construction paper, I fan-fold it both ways, then tear. I like the soft edges. Cutting is neater but slower.)
• If needed, trim the strips so that the edges of each strip don't include the crease; it's better for them to be a bit too narrow than too wide. ("Too long" is actually good.)
• Take a strip. If there's a nice side, put it face up. If there's a nice end, start there.
• Starting on one end, roll it into a tube. (The nice part should show on the inside of the tube.)
• Place the tube into one of the boxes of the cube so that the open end is still open (see the photo).
• Repeat with the remaining strips and boxes.

Play again. 

I generally find that the tubes puff out the walls of the boxes just a little bit, so that sometimes the toy seems to open itself. (I like that; it feels like the cube is alive and playing with me!)

Sometimes the tube edges get caught on each other. I gently pull them apart, sometimes twisting a little bit. 

• If it's the stuffing that was caught, I tuck it back. If it's too big, I remove, trim, and replace it.
• If the tube edges catch, I examine them, maybe mark it (if I have to refold the cube to get it into a different position), and trim, being careful to avoid cutting hinges. Trimming the top and bottom edge (original position) is usually safe.
• If I know that a material tends to catch (such as card stock made into true cubes), I cut slots instead of slits. Rather than one cut, I make two. I cut just a tiny bit to each side of the fold; then I cut across the end to remove the little strip.

If the tape on the top or bottom strip fails, it is possible to tape the inside of the tube.

• I cut a piece of tape shorter than the width of two box-sides.
• I get the cube back into the position where the top edge looks like a double diamond (unstuffing it).
• Then I put the almost-squashed cube against a hard surface, slip the tape into the partly-squashed diamond sticky-side-up, and finish the squashing.
• I plan to try using a pair of tweezers, but haven't yet.

As before, I'm using "cube" as short for "infinity cube"; it's not necessarily of equal sides. Is there a friendlier term than "parallelpiped"?

Varying the material:

• I like making tiny cubes from bill envelopes. The insides have pretty patterns, and the paper is stiff, yet easy to fold. 
• I'm currently experimenting with glossy magazine pages, old calendar pages, and similar heavy, decorative papers.
• I've also used wrapping paper (a lot) and the same fold/cut pattern, but glued to solid cubes (wood cubes or foam dice). See https://www.youtube.com/watch?v=gj0CIodd7nI for an example. I especially like the wrapping paper with cut lines on the back (1" spacing) combined with 1" wooden cubes.
• I've even used empty cardboard tubes (you know the ones) instead of cubes. See https://www.youtube.com/watch?v=kgX9GMry_7s for an early attempt.
• Decorative tape can be fun, as long as it's sticky enough and not too thick/stiff. (Washi tape failed me. It came unstuck too easily, especially on the top and bottom. Your results may differ.)
• I tried small colored labels (meant for garage-sale prices, I think), but they came off too easily. Really STICKY stickers ought to work. I have some bacon-patterned bandages I want to try.

Making the cube:

• I don't generally use a folding tool (just a fingernail) unless I'm making a lot at once, or a cube too small for my fingers.
• Sometimes I fold a bit down at the top edge and a bit up at the bottom edge (rather than trimming it off), before making the main folds. The result is slightly harder to tape, but the extra layer makes the toy sturdier.
• If I'm using decorative paper, I sometimes lay the tape down, sticky side up, and fold the flap onto it. For the top and bottom tubes (both sides), I then have to bend the tape back and stick it behind the flap. (That part's hard to describe, and I don't have a photo yet. Sorry!) For the center tubes, I can stick down both flaps of each tube pretty easily. (It's a bit like the second troubleshooting photo.)
• If I use card stock, I spray it with water, very lightly, on one or both sides. I like the very fine mist from a cleaned-out nose-spray bottle. I let it soak in a minute or so before I fold the paper. I let it dry before taping. Sometimes the wet dyes transfer, so I pay attention to my hands, clothes, and work surface.

Stuffing the cube:

• It's best if the coils of stuffing are not quite as tall as the tubes. That happens naturally with the tall cube. With the true cube, I cheat on the fan-folding to make narrower strips, trimming as needed.
• It's more reliable to use the coils as shown, but if they match the width of the boxes (made by folding the other half of the page into 8 strips), they can go in the other way. When I've tried this, they've always bulged out and gotten in the way unless I squash or tape them in some way (coiling extra-tight and taping can work). I don't have a satisfying solution yet, but it's easy to try and then to undo. Just take out a coil, and then insert it back so the side (rather than the open end) shows.
• I've stuffed cubes with torn-up paper, bits of paper towel, cotton balls, and other materials. Color is fun.

Varying the shape and size:

• It's fun to try different sizes. I like tubes that are big enough to get my fingers into, but I also like sturdy cubes that are small enough to fold with one hand.
• I've used everything from very long rectangles to almost-squares. The wider ones are less stable (requiring stuffing or other reinforcement). The taller, less-wide rectangles make sturdier cubes.
• Small cubes with tall tubes may not need stuffing. I tend to start with rectangles about 3x4 or 4x5 finger-widths. This works well with envelopes.
• For small cubes, though, the tape is too big. For the top, I slip a piece of scrap paper underneath and tape right off the top edge. Then I trim off the extra tape and scrap, along with a tiny sliver of the top edge. For the sides, I fold all four side boxes together (two strips), then tape both strips with one piece of tape. Then I fold the cubes to the center (creasing the tape) and recut ONE box from the center toward the side (see photo).
• When planning, I try to match the tape and the size. It's ideal to have tape that's just slightly smaller than 1/8 the width of the page. So 1" tape with 8.5x11" paper is good. The more standard 3/4" tape with a half-sheet is fine on the side boxes, but a little skinny for the top and bottom. If the tape is wider than the tubes, I place it strategically, so that I can trim/cut it.

Here's a YouTube playlist that includes all my cubes and some other people's:

https://www.youtube.com/playlist?list=PLHbuG6dzKF8Jjb318BnuM3oillk-PPi3q

I'm hoping to do more - I'm figuring out how to put drawings (or a finger maze) onto the paper before assembly. Check back or subscribe to the ImprovCrafting YouTube channel!