The Epcot Ball

8/11/2019 The Epcot Ball

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The Epcot Ball!

Preliminary instructions:

As I said, these models are all made up of a number of absolutelyidentical paper pieces. (The word "origami" therefore applies verytenously, you see; it's more of a mechanical process.) Therefore,making a model involves three steps: choosing which model you

want to make, constructing and folding the appropriate number ofpieces, and then assembling them according to a pattern. No step ishard; they just take some time. I've gotten it down to the pointwhere I can fold one piece in about one minute; assembly into amodel takes a couple of hours at most for the most complex modeland mere minutes for the simpler models. First I will detail how tomake a single piece (step-by-step) and then I will outline how toassemble them into a model of a polyhedron.

Paper type and paper size:

Unlike for the fighter jet paper airplane, I strongly recommend thatyou use lined filler paper for these models. (I prefer college rule,but of course that doesn't matter). Typing paper or printer paper,


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while appropriate for airplanes, is too thick and stiff for thesemodels. (However, read below for cases when typing paper isappropriate.) Each piece starts its life as a small square ofpaper.The size of the square determines the eventual size of the

polyhedron model (and the model's physical strength, and how easythe model is to assemble, and how easy the pieces are to assemble,and so forth). For average models, I absolutely recommendusing squares that have edges of 1.5 inch. Larger sizes, such as2-inch squares, may be appropriate for models that you wish tomake larger. Too large squares (3-inch and above) may lead toflimsy models unless you use stiff paper, such as typing paper. (Ihave indeed dreamed of making an Epcot ball out of posterboards,but it would require a LOT of posterboards.) And of course, thelarger the square, the more sheets of paper you'll have to consume

to make the squares. Smaller squares, such as 1-inch squares, areABSOLUTELY TERRIBLE to work with and would only be appropriateif you want to show off how cool you are. (And assembling an Epcotball out of 1-inchers would be absolute torture). The 1.5 inch sizeallows for sturdy, quick construction and does not penalizeinaccuracies too much.

Accuracy issues and making the squares:

Basically, all the rules that apply to my paper airplane instructionsapply here. All folds should be absolutely accurate and creasedsharp; all measurements must be absolutely accurate and all cutsmust be absolutely accurate. If you make a small error, the pieceswill not be too adversely affected, but remember that you aremaking a large number of (supposedly) absolutely identicalpieces. If you are inaccurate in your folding, cutting, or measuring,the pieces will be irregular and will not fit together nicely. (1-inch

pieces demand absolute perfection, in contrast.) You may find thatyou have to discard irregular pieces. When you make a model (youaren't, yet, because you don't know how which models are possible;let's concentrate on making a single piece first), you'll gather alarge number of sheets of paper and draw very precise grids (using,say, a mechanical pencil or something) which divide the paper into


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1.5" square regions. Then you'll take sharp scissors and cut exactlyon the lines, producing X number of squares (where X is the numberof pieces you'll need). Then you'll go through and fold X pieces(repetitively), and finally assemble the model. And

remember: accuracy is everything!Now, on to how to fold apiece.

Folding a single piece:

(The convention here is different from my paper airplane page: solidlines mark valley folds, and no mountain folds are involvedanywhere. It will be obvious; don't worry. Also, do not label your

square in any manner while folding it. The only time a pencil isneeded is when dividing up a sheet of paper into squares to be cut.All folds can be made without any markings; my markings are tomake my instructions clear.) Also, when I refer to spinning thepaper 180 degrees around, I mean rotating it around on the table.You actually will need to turn the paper over near the end of theprocess, but I'll make it clear then.

A completed piece.

This is your final goal for this section: to make a completed piece.This process is actually very easy and quick; once mastered, itought to take you one minute per piece. (For the purposes ofphotographing this process, I used a 3-inch square; you should usea 1.5-inch square. This is why my photos will have more blue lineson them than your squares will have.) So, start with a 1.5-inchsquare of paper:


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A freshly cut square of paper.

Make a precise and creased fold lengthwise. Here is what I mean:

Dividing the square in half.

Here is the process halfway through completion.


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The actual purpose of this fold is just to give you a reference tomake the next two folds. Unfold the paper and lay it flat. Take thebottom edge of the paper and fold it to the center crease; then spinthe paper 180 degrees and do the same. Here is what I mean:

The folds that you'll be making.

Here is the process halfway through completion (both folds areshown simultaneously; you should make them one at a time, of

course).

Okay. Now, unfold the paper and lay it flat. (You will be folding thepaper here again; you just need to do some things in the meantime.

I'll refer to these folds, rather uncreatively, as "the second and third

folds" later on.) Take the bottom-right corner of the paper and foldit into a triangle so that what was the left side of the paper now lieson top of the second fold you made. Leave that folded, spin thepaper 180 degrees and make the same fold. Here is what I mean:


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Folding two triangles.

This is the traditional fold you make when producing a (lousy)needlenose paper airplane. Now, take the bottom-right corner of thepaper and make another needlenose-type fold. That means bringingthe fold that you just made to lie exactly on top of the second foldyou made. Then rotate the paper 180 degrees and make the samefold. The following image's bottom-right corner shows the end result

of this process; the upper-left corner shows it halfway throughcompletion.


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Another "needlenose" type fold.

That fold was hard to describe but easy to perform; it's used in theproduction of lousy needlenose paper airplanes everywhere. Now isthe time to remake the second and third folds you made:

Okay. Now, take the bottom-left corner of the paper and fold it sothat what was the left edge of the paper now lies on top of the top

edge of the paper, producing a triangle, like this:

Rotate the paper 180 degrees and repeat. A parallelogram! Now,you must tuck in that large triangle fold into the paper. I have noway to easily describe this in words. Here is what I mean:


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The left fold is tucked in, while the right fold is not.

Then rotate the paper 180 degrees and tuck in the other fold,resulting in:

Good. Now flip the paper over and rotate it so that it looks like this:

The backside of the paper.

Fold the bottom point of the paper straight up to meet anothervertex of the parallelogram, like this:

Then rotate the paper 180 degrees and repeat, producing this:


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And here they are locked together, corner in pocket:

Here is a third piece, placed over the first two:

And here the third piece is locked in:

There is a free corner and free pocket that can be locked together.Doing so necessitates forming the three pieces into a three-dimension configuration that I call a peak:


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It is vitally important to understand what I mean when I say "peak",because peaks are the founding blocks of your models. (Although

you should assemble your models piece-by-piece and not make abunch of 3-piece peaks and then assemble the peaks. The formerworks; the latter won't. Example: the cube contains three peaks,but only requires six pieces. Solution: pieces can form more thanone peak. Trust me: go piece-by-piece.) Now you should be able toassemble a cube. Here is a cube, pictured with a peak at the centerof the image:

A cube.

If you were unable to assemble the cube, then read on, because Iwill make it even clearer. (I need to demonstrate the next fact withan octahedron; cubes are too small). Now, here is a what I call(somewhat confusingly), a "point". This is my own terminology; callit what you will. Around every "point" in a model there are three ormore peaks. The lens flare in the following image shows where apoint is located relative to a peak:


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This is what I mean by "point".

Here is a picture of an octahedron, with a point more or less in thecenter of the image. See how four peaks are arranged around thepoint?

An octahedron.

And here is an icosahedron. Icosahedra have five peaks aroundevery point:

An icosahedron.

Forming the first three models, therefore, is quite easy. Simply startout with the required number of pieces (6, 12, or 30) and place 3,4, or 5 peaks around every point until you've run out of pieces andclose the model. You will end up with a cube, an octahedron, or anicosahedron, respectively. (It's hard to show with a single picturehow three peaks surround every "point" of a cube.) Here is a picture


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to make this even more clear; you can see two points very clearly,around both of which are five peaks (there are also two more pointson this icosahedron for which you can clearly see the fivesurrounding peaks, but they're at more of an angle):

Hopefully that should make what I'm trying to say absolutely clear.

Now, here's that icosahedron missing three pieces. This is what youshould see when your model is almost completed. (Actually, the lastpiece is the hardest to put in.)

To form a model, simply start with a peak, and add pieces to formpeaks in a circular fashion, until you have a point surrounded by 3,4, or 5 peaks. Then form more points by adding more pieces tomake more peaks. Gradually this will cause the model to curve in onitself, until finally it's almost completed like in the image above. It'sactually a self-assembling process once you understand what you'redoing. If you get really good and precise, you can form models out

of 1-inch squares, anything from cubes to icosahedra. Here are thethree 1-inch icosahedra I've made:


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If you use construction paper, you can make colorful models. Here'sa blue-and-red icosahedron I whipped up:

You can also make it so that three colors of pieces make up everypeak, for a similarly cool effect.

Now, if you try to form a model with 6 peaks around a point, youwill find that it cannot be done. (Don't try it; it's a waste of time.)You end up with sheet of peaks. Mathematically, this corresponds tohexagons tiling a plane. There are no other regular polyhedra that

you can form. (Tetrahedra are too small, and I do not believe thatdodecahedra can be formed with these pieces.) However, have youever seen a soccer ball? It's made of hexagons, but with twelvepentagons that give it enough curvature to be a ball. (To makematters easy, the hexagons are colored white while the pentagonsare black.) You can replicate this here. This mathematical shape,the truncated icosahedron, is also the structure of C60, the


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buckminsterfullerene (which also goes by the names "fullerene" and"buckyball"). I derived how to make this shape out of pieces on myown, and now I'll teach it to you. I call it the "Epcot ball", for itsobvious similarity to the Epcot Center in Disneyworld. Here is a top-

on view of an Epcot ball. Notice how a 5-peak point is at the exactcenter, while 6-peak points surround it in every direction:

To start making an Epcot ball, surround a 5-point peak with six-point peaks. However, you must also know where to put the other5-point peaks. The following image shows how. Two 5-peak points

are marked by green toothpicks. (By the way, if anyone is stillhaving trouble understanding what I mean by "point", there it is.Marked by a green toothpick.) Conceptually draw a straight linebetween those 5-peak points. In between, marked by lens flares,are precisely TWO6-peak points. No more, no less. Here is what Imean:


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The key to making an Epcot ball, therefore, is to construct a 5-peak

point, surround it with 6-peak points, and then surround it withanother ring of 6-peak points, and then add exactly five 5-peakpoints so that they are two 6-peak points away from the original 5-peak point. Repeating this process EXACTLY over the entire surfaceof the model will produce an Epcot Ball. If you fail to do it exactly,you'll end up with a mutant model that will refuse to close in onitself. Not fun. Epcot Balls require 270 pieces and a substantialamount of time dedicated to making them. (Note: That figure of270 is hazily remembered by me. Actually, I've never counted the

pieces before making an Epcot ball. I just follow the pattern andmake pieces as I go. I once derived a simple formula for figuringout the number of pieces one of these things has, but I was neversure that the formula was correct, and in any case I've forgotten theformula and its results. If you indeed discover that 270 pieces is notenough to continue the pattern I've detailed here, then by all meansmake more pieces. The pattern, not the number of pieces, is key.Although I would be incredibly surprised if 270 is not the correctnumber. I do remember the number being in the 200's, and being amultiple of 30.) Due to the fact that they're mostly made of 6-peakpoints, they are not nearly as rigid as the smaller models and canbe damagedEASILYby a jolt or by a drop. Be careful with the EpcotBalls that you make. I have made exactly four Epcot Balls in mylife; I didn't take a photograph of all of them together (it would belarge!), so you'll have to take my word for it. I still have all four.Here is a gratuitous picture of one of them:


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The Epcot Ball!

Now I hope you understand what all those lengthy explanationsabout peaks and points were for. They allow me to communicatehow to make the Epcot Ball in a compressed sentence like "...thensurround it with another ring of 6-peak points, and then add exactlyfive 5-peak points...". Please E-mail me [email protected] you hadsignificant trouble following these instructions, so that I may revisethis page and make it easier to understand. Also feel free to E-mailme if you've made models according to these instructions andenjoyed it. I would also really like to know if you've made an EpcotBall. Agnes Harnisch used these instructions to create avery

beautiful Epcot ball- note the exquisite precision of the folds. Evenmy own models don't fit together that well. ^_^ Have fun!
mailto:[email protected]:[email protected]:[email protected]://klimsmurf.de/gefaltet/modulares/?view=004_epcot.jpghttp://klimsmurf.de/gefaltet/modulares/?view=004_epcot.jpghttp://klimsmurf.de/gefaltet/modulares/?view=004_epcot.jpghttp://klimsmurf.de/gefaltet/modulares/?view=004_epcot.jpghttp://klimsmurf.de/gefaltet/modulares/?view=004_epcot.jpghttp://klimsmurf.de/gefaltet/modulares/?view=004_epcot.jpgmailto:[email protected]

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The Epcot Ball