Saturday, February 21, 2015

Paper Icosahedron

Recently I got a spark to try and create a icosahedron (aka 20-sided polygon). I am familiar with the shape because of 20-sided dice and I've created this 3d shape before using toy magnets; but today I wanted to discover how to create an icosahedron from paper. (Without cheating of course; no internet! This is a fun mental exercise.)

20-sided dice
Moreover as I pursued the problem I wanted to discover the best way to create an icosahedron from paper. What was my criteria for the "best"? I started out without any criteria in mind, but managed to flush it out as I dove into it.

Criteria 1: Complete Shape

First step was to prove to myself what a 20 sided polygon consisted of. To create an icosahedron you need 12 points, 30 edges, and 20 faces. Each face is an equilateral triangle. Thus I need to cut out twenty equilateral triangles; but how would they be assembled in a way that could enforce the proper shape?

I made an observation: each point forms a slightly raised pentagon pyramid; which I could cut out as pentagon 'caps'. Then I could overlap these pentagon caps to form the full 20 sided shape. 

Pentagon "cap"
How many would I needed to cut out? Ideally the minimum number of caps; an intriguing number to prove. 

A quick upper limit was 12 caps, one for each point. But some whiteboarding showed that with a cap at each point, each face would be overlapped 3 times. Too redundant.

Criteria 2: Low Waste

A quick lower limit was 4 caps (4 caps * 5 faces = 20 faces) without any overlapping, but some eyeballing of the 20-sided dice showed that I could only get away with arranging 3 non-overlapping caps before I was forced to overlap in triplicate to cover the remaining faces. The greedy approach of 'avoid overlap whenever possible' produced an awkward/uneven arrangement with at least 6 caps.

New approach; conceding to the fact that any arrangement of pentagon caps must include overlap, let's see if there is a more even/symmetrical overlapping arrangement; let's see if we can come up with an arrangement where each cap only overlaps one other cap. I observed that two overlapping pentagon caps will always overlap on 2 faces, thus after the first cap, all subsequent caps could at most cover 3 more faces. Therefore the best we could hope for in this approach is:
5 + 3 + 3 + 3 + 3 + 3 => (6 caps; 10 faces are overlapped once, 10 faces are non-overlapped. )
Turns out there is an arrangement that matches this approach! And after more futzing I settled that 6 is the minimum number of caps needed. (Again empirically determined, I held off on a graph theory proof for the time being :P)

So, how to craft the pentagon caps? How to map the 3d shape to a 2d piece of paper?

Hexagon pac-man.
The shaded area is waste; cut it out and tape the edges together between 1 to 5.
Simple. Low number of cuts and taping. Fits in a hexagon which tessellates easily.
(With this came an interesting realization that you cannot create a pyramid with more than 5 equilateral triangles)

Criteria 3: Contiguous Shape

All said an done, I ditched the pentagon cap approach. The assembly of the 6 caps was clumsy and frustrating and because I had cut out all the caps equal size the overlapping didn't fit quite right. And quite frankly I should be able to assemble the shape without any overlap.

After going through the exercise of flattening out the pentagon cap onto paper, unwrapping the whole icosahedron onto paper wasn't as scary a task. (I later found out that is known as a polygon's net). Rather than guessing and checking a complex sprawl of triangles, I simplified the problem by noticing that an icosahedron can be reduced to two pentagon caps and a ring/ribbon of 10 alternating triangles between them. 

Two pentagon caps and a ribbon of triangles.
Thus so long as each pentagon is connect to each side of the ribbon I have a working cutout. Which leaves me with about 12 different cutouts to choose from; which one should I choose?

Criteria 4: Tessellating Cutout

As I began to layout cutout diagrams on a piece of paper I began noticing the waste that would be generated. I then couldn't help but thinking 'What if I were a manufacturer in charge of mass producing millions?' In that case the best cutout would be one that tessellates without any gaps. More whiteboarding!

Attempt 1: I discovered many of the 2-cap-1-ribbon layouts could self tesselate in columns, but adjacent columns would need to be a different layout variation to reduce waste. After the 4th or 5th column a gave up on converging this tessellation. If I could get this approach to work, I estimated gaps in tessellation 5%.
Failed attempt. Tessellation didn't converge.
Shaded triangles are waste.
Attempt 2: I discovered a layout that tessellated in both columns and rows. Estimated gaps in tessellation 10%.
Successful attempt. Cactus arms layout.
Shaded triangles are waste.
Attempt 3: The troublesome part of the tessellation were the pac-man shape of the two pentagon caps. I realized instead that I could spread out the cap along the whole ribbon. Thus the final solution; perfectly tessellating and doesn't require large area to actualize the tessellation.

Final solution. Simpler to actualize too.

Red and blue icosahedron nets.
(utilized the scrape for green pentagon caps; later futzing)

Cut out and creased.
Final product.
Taped at the top point of each pentagon cap.

Final Thoughts

A fun exercise in crafting 3d shapes from 2d materials. Some headway was made by geometry and whiteboarding alone, though many aspects came down to trial and error. Parts of this process reiterated the contrast between academic/theoretical and practical, and the benefits of early implementations. As I began to assemble my designs I found additional requirements for the design; part feature creep and part discovering unknown requirements. For instance if I were to go through this exercise again I might consider adding extra overlapping flaps to provide spots for gluing or I might drop the contiguous layout criteria in favor of a design that is quicker to physically cut out.

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