Instructables

Rhombic Triacontahedron

Make an origami polyhedron. The rhombic triacontahedron is easy to make because it is both face uniform and edge uniform. I came up with this folding scheme by modifying some instructions for a rhombic dodecahedron (http://www.ii.uib.no/~arntzen/kalender/), (http://www.paperfolder.info/diagrams/a4unit.htm).
 
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Step 1: Paper size

Pick a piece of paper with aspect ratio as close to the golden ratio as possible. You can do this by picking two consecutive numbers in the fibonacci sequence. (Shown here is a 8.5"x5.5" sheet, the aspect ratio of 17:11 isn't very close to the golden ratio.) You will need 30 sheets. Experiment with different colors and patterns.

Step 2: Folding

Fold in half, and then into quarters.

Step 3: Folding

Fold in half the other direction

Step 4: Folding

Fold the corners to the center so that the fold goes from the midpoint on the side to the opposite corner. Repeat for all four corners to make a grid pattern.

Step 5: Folding

Open up the sheet to show the grid of golden rhombi. Fold the corners up in toward the center to make pockets on the sides and tabs at the end. A piece of tape across the center helps to keep the pockets tight.

Step 6: Assembly

Put tabs into pockets, alternating three and five rhombi.

Step 7: Examples

The pink triacontahedron was made from 5.5"x8.5" sheets (aspect ratio of 17:11), and is too loose to stay together. The white triacontahedron was made from readily available 5"x8" index cards and is very sturdy. The multicolored triacontahedron was made from sheets cut to 4.25"x2.625" (aspect ratio of 34:21) and is very tight.
When Leonardo painted the Last Supper, he didn't paint it on a square. No, he used a Golden Rectangle where the width was 1.618 times the height. And, when he painted the Mona Lisa, he didn't use a square. He used a Golden Rectangle where the height was 1.618 times the width. Grab an ordinary calculator and divide 1 by 1.618 and get 0.61805. Now try 1/1.618025 ("half way point") and get 1.618037. Now try 1/1/1.618031 and get 0.618035. Now try 1/1/6.18033 and get 0.618034. Finally take 1/0.618034 and get 1.6180339. So we can consider 1.618034 accurate for many situations, arrived at here by "successive approximations"...
Equation wise, this would be X = 1 + 1/X, where X can be found to be 1+sqrt(5) all divided by 2 or 1.6180339... The Golden Ratio is also key to the Pentagon and Pentagram but that's another story.
All of this is meant only for added information but not meant to improve on the fine work of the author here...
It took me a while to figure this one out but I finally did and now I'm going try to explain it using some overlays on the images from the instructions. Image 1: Start by putting together five rhombi that meet at a center point. Each rhombus's tab is inserted into the pocket of the one next to it. This will leave you with a pinwheel form with five tabs exposed. Image 2: This shows that tab structure for constructing the five rhombi form. Image 3: From each exposed tab add another rhombus and tuck one of it's tabs into the rhombus next to it to create the three rhombi forms that surround the five. From this point add two rhombi to each "arm" to create more five rhombi forms. This makes sense if you are doing it physically rather than trying to wrap your mind around it before doing it. Just try it. I hope this helps clarify the original instruction, which were lacking on the most important step. Patrick
1.jpg2.jpg3.jpg
AcidDaze7 years ago
first, eassin, i'd like to thank you, i love origami and this has been my fav piece so far. i enjoyed it so much i made two of them then i thought 'can i join them?' this is what i got Front veiwSide View, then from the same pocket-tab pieces i made this, frontside does anyone have a name for this shape?
eassin (author)  AcidDaze7 years ago
Your second shape is a rhombic dodecahedron; it's easier with a different size sheet of paper. I started out with the bottom half of this page and then figured out this shape; just the opposite of you.

Your first shape doesn't have a specific name it's a kind of penrose tiling
sandeepy7 years ago
com'on man ... cant understand wht 2 do after step 4 ... i got the golden rhombi ... wht then ... can anyone explain ? or does anyone have better pictures
endolith7 years ago
Here's an alternative way to get the right paper size without measuring:

http://www.josephwu.com/Files/PDF/silver-gold.pdf
Danny7 years ago
could you just say for example... make ot aproximately *cm long and *cm wide or something please
LasVegas7 years ago
Cool! You photographed this instructable 2 years in advance! Great to be prepared! :) You mentioned "The Golden Ratio" a couple times. What exactly is it and why?
eassin (author)  LasVegas7 years ago
I got a two year old camera for Christmas - that's preparation in my family ;) - and I thought I'd try it out with an instructable. Like Trans_Am says, the golden ratio is about 1.618..., or (sqrt(5) plus 1)/2. It shows up a lot in geometry so it gets its own name. The rhombi should be 1.618... times as long as wide so that it actually folds together tightly. The easiest way to do that is with a "continued fraction approximation." Check wikipedia or the math section in a newer machinist's handbook, because it doesn't make much sense to me.
LasVegas eassin7 years ago
Thanks. Some of us had HS geometry in the early 70's and don't have quite that good of recall. It's interesting to be reminded of it periodically.
Trans_Am7 years ago
The "Golden Ratio" is a ratio of length to width/height found often in nature, and some consider it almost "magical" *from my Grade 9 math textbook* "the golden ratio is about 1.618" "some people claim that the golden ratio can be found in many measurements of the human body" theres about two pages about this, an how to construct the golden rectangle.
PetervG7 years ago
Huh?