Introduction: Uheska: a 3D Cardboard Stellation

About: I like mathematical constructs, especially fractals!

I'm going to show you how you can make a fantastic and visually arresting 3D sculpture of my own design. Uheska (Estonian for "9"; I needed a nice pretentious name for this piece) is a stellation of a 60-sided solid called a Rhombidodecacron. The end result is (almost) the product of 5 nested 12 sided triakistetrahedrons. At each point, 9 sides join up to meet. Hence the name. The final piece measures almost 14" from point to point.

Inspired by the work of mathematical artist George W. Hart, I was driven to try to create my own 3D stellation of a geometrical solid. It took me a good 2 weeks of evenings to pull off, from conception to final piece, but the end product came out amazing!

You're gonna need a steady hand and patience if you tackle this puppy. I hope you enjoy!

Step 1: Supplies & Tools

- Olfa knife with lots of blades.
- Hobby hook knife, also with lots of blades.
 - Optionally, you could possibly use a reciprocating saw. I didn't have one. Let me know how that goes!
- Heavy gauge, large-eyed needle (that can poke through the cardboard, and thread copper wire).
- Pliers

- Marker
- 1 Paper sheet
- 1 scrap manila folder (or similar heavy bond paper/cardboard). A cereal box would do in a pinch.
- Cardboard. Lots and lots of cardboard. Pay attention to the thickness! I made mine too thick for the size and ended up needing to notch my pieces. (D'oh!). In the end I used 6 mm (~ 1/4") thick cardboard for pieces that were 27cm (10 11/16") long. A thinner cardboard would work, but you need to ensure it will be strong enough to not bend when really thin.
- 24-26g copper wire.
- Thin twine (makes things a little easier for test fitting, or to temporarily hold some pieces together).
- Spray Paint (optional of course)

Step 2: Design the Shape

Not really a step for this instructable, but I'm including it for completeness, and to include the pictures.

Basically I used Vladimir Bulatov's Polyhedra Stellations Applet to find a shape that I thought would work. Then I used Google Sketchup to first draw the stellated Rhombidodecacron (special thanks to imperar and their tutorial that let me model the base rhomidodecacron) and selectively remove anything extra from the broad triangles. As much art as math, I think.

I've included the Google Sketchup file in this step.

Step 3: Make Your Template.

First, print the piece on standard printer paper, and cut it out leaving as much black line as possible... If you cut it off, your pattern will effectively "shrink" and won't fit together.

Using that pattern piece, trace it onto your manila folder or cereal box. Once traced, cut that piece out as well, cutting on the inside of the line this time.

This is now your template for tracing on to the thick cardboard.

Step 4: Trace! and Trace, and Trace, and Trace.

Now you'll trace your template onto the cardboard. Remember when I said this was a 60 sided piece? You're gonna need 60 cardboard pieces.

They don't trace together too well, so there will be a fair amount of waste. Do yourself a favour, and make sure you trace each piece with the cardboard's ribs running along the long straight edge of the template. The final product will look crisper and be easier to assemble.

Step 5: Make With the Cutting.

Now you'll need to cut. It's a bit tedious. Use the olfa knife for the long straight cuts, and the hook knife for the graceful (and tight) curves. Replace your blades often to keep crisp lines.

If you have a small, fast saw (jigsaw, reciprocating saw, etc), it *might* work. Let me know if that works for you!

Step 6: Notch the Pieces

So, I freely admit that I messed up this shape. I used cardboard that was too thick, and I had collisions between the planes. To alleviate the stress, I notched all of the pieces to makes them fit better. I've included the Google Sketchup model of the notches. When I did it, I had to take apart the Uheska, and notch them after the paint job. (Ouch)

If your cardboard is thin enough, you might get away without this step.

Step 7: Test Assembly

I tested putting it together before I painted and permanently attached the pieces, to make sure I could in fact assemble this shape (I wasn't sure up to this point....). I'm not going to walk you through this, because it was way easier to do it when the pieces were coloured.

First off, I arranged all pieces in pairs and used the thin twine to attach the pieces into pairs. Align two pieces with their straight backs lined up, with their curved legs pointed in opposite directions. These pairs should resemble stylized Xs. You should have 30 of these pairs. Tie a little loosely, as each pair should be able to bend to a ~140 degree angle.

Step 8: Paint!

I used spray paint to give all of the pieces a white back, so that when you peer inside the Uheska, it's much brighter.

I then coloured batches of 12 pieces in 5 different colours: red, orange, yellow, green and blue.

Remember proper ventilation!

Step 9: Assemble Into Pairs

Here's where I start using the copper wire and needle. Arrange two pieces (of the same colour) with their straight edges back to back, so it resembles a stylized "X."

To attach my pieces, I used the copper wire and thread it through a large-eyed needle. I twirled this wire, to give it a bit more strength, then poked the needle through the arms of the cardboard pieces. The pic shows in black generally where I stitched it. I basically ran a circuit around the center hole. Make sure your two loose ends are in between the straight sides of the two pieces. Grab some pliers and twist the ends together. This should tighten it up a little, but don't overdo it. You want the pair of pieces to be able to bend to an angle of 143 degrees.

Repeat for the remaining 29 pairs.

Step 10: Layer 1

This will get tricky.

First, let's establish from terms of reference:
Pair: a pair of like coloured pieces that have already been attached with wire. Looks like a stylized X.
Layer 1: the first batch of 5 pairs.
Layer 2-5: The second through fifth batch of 5 pairs.
Curved Leg: the leg that is curved, jutting away from the two straight legs.
Curved Foot: The end of the Curved Leg.
Straight Legs: The two straight legs of the pair, back to back.
Double Foot: the two joined feet of the Straight Legs.
"h": a single piece, or one side of the pair. Because it looks like an "h"
Top: the end away from the ground.
Bottom: the end of the pair that is closest to the surface you are building on.

Alright, lets start assembling.

Weave 5 pairs together. Refer to the google sketchup model to make sure you can wrap your head around it. Each "h" goes around the previous piece's curved leg; the curved foot should sit against the previous piece's double foot. All 5 pieces should weave together, and they will sit loosely.

Step 11: Layer 2

Tip: When slotting in a new pair, it is generally easier to try slipping it in flat (or unfolded) and then bend the pair along the crease to the right angle.

Now, take 5 more pairs. Each piece you add in this step should touch a piece from Layer 1 at four points (!).
For example:
- Yellow top Curved Foot to the Red Top Double Foot
- Yellow bottom Double foot to Red bottom curved foot
- Yellow top Double foot to Orange Top Curved foot
- Yellow bottom Curved Foot to Green Top Double foot.

Grab the Yellow Pair. Wrap the bottom "h" around Green's top Curved leg (from Layer 1). The Curved foot should sit beside the green double foot and the Yellow Double foot should sit beside the Red. You can then bend the pair so that Yellow's top curved foot touches the Red top Double Foot (and the top Yellow Double foot will also touch the Top Orange Curved Foot

Now we are going to go counter-clokwise and grab the Blue pair. Slide it in from the top such that the Bottom Double foot threads the hole made by Layer 1 Red Top Double Foot and Layer 2 Yellow Curved. Also the Blue Bottom Curved Foot will thread the hole between Layer 1 Top Curved Leg and Layer 2 Yellow Top Double Foot. Refer to the Sketchup diagram.

Similarly, proceed with Green, Orange and Red in order, as above. With the last piece, you may need to unfold the Layer 2 Yellow to get it to fit in easily.

Step 12: Layer 3

Now for a slightly easier step.

Take the Layer 2 Yellow, and "unfold" it along the pair's join (so it's basically flat). This makes a gap between Layer 2 Yellow and Layer 1 Green. From the top, slide Layer 3 Blue straight down. Thread the hole between Layer 1 Green and Layer 2 Red (and Layer 2 Yellow) with the Bottom Curved Leg. Also thread the hole between Layer 1 Orange and Layer 2 Red with the Bottom Double foot.

Bend Layer 2 Yellow back into position. You may need to make a few more gaps to get the piece in smoothly.

Repeat around the circle.

P.S. This is now a crown. Wear it in style!

Step 13: Layer 4

We're half way done the assembly!

Next, let's unfold the Layer 3 Blue. This will make a gap that we can slide the Layer 4 Red through. Then all we need to worry about is slipping the Bottom curved foot behind the Layer 2 Yellow and Blue, through the center of the "star" type shape (look at the second picture), and slot in into the last spot on the Yellow point on the bottom of the entire Uheska.

Repeat around the circle. For the last piece, you will need to pull the Layer 4 red out a little to slot the bottom Double Foot in.

Step 14: Layer 5

We can see the Uheska taking shape now. We're getting close...

Now, for these pairs, we're gonna want to slot them all in more or less at the same time. In the pictures I've isolated one pair in white, to give you a better idea of where it sits in relation to the other pieces. I've also marked 4 pieces to unfold. Unfold them all, all around the Uheska.

Take layer 5's orange pair and slide it down behind layer 3's green piece, but don't slide it all the way! Grab the green pair, and slot it against the layer 5 orange, hooking their "h"s on each other, and push it a little down behind layer 3's blue pair. Continue in this fashion until you have all 5 pairs slotted.

Do NOT push them down yet. Proceed to the next step.

Step 15: Layer 6: Final Boss!

Ok, so you should have Layer 5s pieces still a little above their final position. Unfold all of the upper sides so they are flat. Carefully insert the Layer 6 Blue (the white piece in the first image) into the gap under Layer 5 Red's curved leg, Layer 5 Orange's Straight leg and Layer 4 Yellow's curved leg. You will need to pry the Layer 3 Green and Orange apart to slide the bottom Double foot in place. Proceed around the circle, counter-clockwise, inserting other Layer 6 pieces keeping all pairs flat. Weave all of this layer's upper side's "h"s so they weave together into a star like in Layer 1.

Once you have all 5 pieces in place, then gently slide all of layer 5 and layer 6 down into place, folding the pairs as needed.

Step 16: Tie It All Together

Now we need to secure our pieces into one solid unit.

Using your needle and wire, you're gonna poke your needle through all 9 pieces around the circle at the point. Follow the black line (but pierce through the middle of the edge of the cardboard). Use pliers to pull the needle through. Once all the way through all 9 points, grab the two loose ends with the pliers and give them some tight turns to tighten them up. Snip off any excess and push the end down between two adjacent pieces. (Sorry I don't have a good picture of this).

Repeat for all 19 other points.

Step 17: Finished!

Now admire your work.

For presentation, grab some monofilament. Looking directly at one point, you will see three points that orbit it. Attach three lengths to these points, then tie them together. Then attach one longer length to this knot. Hang and enjoy (and spin it!)!

Or just leave it on a shelf! You may find yourself playing with the Uheska...

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